{"url":"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean","commit":"","full_name":"Unitization.starLift_range","start":[366,0],"end":[370,7],"file_path":"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nC : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : StarRing R\ninst✝⁸ : StarRing A\ninst✝⁷ : Module R A\ninst✝⁶ : SMulCommClass R A A\ninst✝⁵ : IsScalarTower R A A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring C\ninst✝² : StarRing C\ninst✝¹ : Algebra R C\ninst✝ : StarModule R C\nf : A →⋆ₙₐ[R] C\nc : StarSubalgebra R C\n⊢ (starLift f).range ≤ c ↔ StarAlgebra.adjoin R ↑(NonUnitalStarAlgHom.range f) ≤ c","state_after":"R : Type u_1\nA : Type u_2\nC : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : StarRing R\ninst✝⁸ : StarRing A\ninst✝⁷ : Module R A\ninst✝⁶ : SMulCommClass R A A\ninst✝⁵ : IsScalarTower R A A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring C\ninst✝² : StarRing C\ninst✝¹ : Algebra R C\ninst✝ : StarModule R C\nf : A →⋆ₙₐ[R] C\nc : StarSubalgebra R C\n⊢ NonUnitalStarAlgHom.range f ≤ c.toNonUnitalStarSubalgebra ↔ ↑(NonUnitalStarAlgHom.range f) ⊆ ↑c","tactic":"rw [starLift_range_le, StarAlgebra.adjoin_le_iff]","premises":[{"full_name":"StarAlgebra.adjoin_le_iff","def_path":"Mathlib/Algebra/Star/Subalgebra.lean","def_pos":[424,8],"def_end_pos":[424,21]},{"full_name":"Unitization.starLift_range_le","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean","def_pos":[356,8],"def_end_pos":[356,25]}]},{"state_before":"R : Type u_1\nA : Type u_2\nC : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : StarRing R\ninst✝⁸ : StarRing A\ninst✝⁷ : Module R A\ninst✝⁶ : SMulCommClass R A A\ninst✝⁵ : IsScalarTower R A A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring C\ninst✝² : StarRing C\ninst✝¹ : Algebra R C\ninst✝ : StarModule R C\nf : A →⋆ₙₐ[R] C\nc : StarSubalgebra R C\n⊢ NonUnitalStarAlgHom.range f ≤ c.toNonUnitalStarSubalgebra ↔ ↑(NonUnitalStarAlgHom.range f) ⊆ ↑c","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Algebra/GeomSum.lean","commit":"","full_name":"geom_sum_pos_and_lt_one","start":[416,0],"end":[426,37],"file_path":"Mathlib/Algebra/GeomSum.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nhn : 1 < n\n⊢ 0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1","state_after":"case refine_1\nα : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nhn : 1 < n\n⊢ 0 < ∑ i ∈ range 2, x ^ i ∧ ∑ i ∈ range 2, x ^ i < 1\n\ncase refine_2\nα : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nhn : 1 < n\n⊢ ∀ (n : ℕ),\n    2 ≤ n →\n      0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1 →\n        0 < ∑ i ∈ range (n + 1), x ^ i ∧ ∑ i ∈ range (n + 1), x ^ i < 1","tactic":"refine Nat.le_induction ?_ ?_ n (show 2 ≤ n from hn)","premises":[{"full_name":"Nat.le_induction","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[829,6],"def_end_pos":[829,18]}]},{"state_before":"case refine_2\nα : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nhn : 1 < n\n⊢ ∀ (n : ℕ),\n    2 ≤ n →\n      0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1 →\n        0 < ∑ i ∈ range (n + 1), x ^ i ∧ ∑ i ∈ range (n + 1), x ^ i < 1","state_after":"case refine_2\nα : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\n⊢ ∀ (n : ℕ),\n    2 ≤ n →\n      0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1 →\n        0 < ∑ i ∈ range (n + 1), x ^ i ∧ ∑ i ∈ range (n + 1), x ^ i < 1","tactic":"clear hn","premises":[]},{"state_before":"case refine_2\nα : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\n⊢ ∀ (n : ℕ),\n    2 ≤ n →\n      0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1 →\n        0 < ∑ i ∈ range (n + 1), x ^ i ∧ ∑ i ∈ range (n + 1), x ^ i < 1","state_after":"case refine_2\nα : Type u\nβ : Type u_1\nn✝ : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nn : ℕ\nhmn✝ : 2 ≤ n\nihn : 0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1\n⊢ 0 < ∑ i ∈ range (n + 1), x ^ i ∧ ∑ i ∈ range (n + 1), x ^ i < 1","tactic":"intro n _ ihn","premises":[]},{"state_before":"case refine_2\nα : Type u\nβ : Type u_1\nn✝ : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nn : ℕ\nhmn✝ : 2 ≤ n\nihn : 0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1\n⊢ 0 < ∑ i ∈ range (n + 1), x ^ i ∧ ∑ i ∈ range (n + 1), x ^ i < 1","state_after":"case refine_2\nα : Type u\nβ : Type u_1\nn✝ : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nn : ℕ\nhmn✝ : 2 ≤ n\nihn : 0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1\n⊢ -x * ∑ i ∈ range n, x ^ i < 1 ∧ x * ∑ i ∈ range n, x ^ i < 0","tactic":"rw [geom_sum_succ, add_lt_iff_neg_right, ← neg_lt_iff_pos_add', neg_mul_eq_neg_mul]","premises":[{"full_name":"add_lt_iff_neg_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[480,29],"def_end_pos":[480,49]},{"full_name":"geom_sum_succ","def_path":"Mathlib/Algebra/GeomSum.lean","def_pos":[45,8],"def_end_pos":[45,21]},{"full_name":"neg_lt_iff_pos_add'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[101,2],"def_end_pos":[101,13]},{"full_name":"neg_mul_eq_neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[277,8],"def_end_pos":[277,26]}]},{"state_before":"case refine_2\nα : Type u\nβ : Type u_1\nn✝ : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nn : ℕ\nhmn✝ : 2 ≤ n\nihn : 0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1\n⊢ -x * ∑ i ∈ range n, x ^ i < 1 ∧ x * ∑ i ∈ range n, x ^ i < 0","state_after":"no goals","tactic":"exact\n    ⟨mul_lt_one_of_nonneg_of_lt_one_left (neg_nonneg.2 hx.le) (neg_lt_iff_pos_add'.2 hx') ihn.2.le,\n      mul_neg_of_neg_of_pos hx ihn.1⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"mul_lt_one_of_nonneg_of_lt_one_left","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[226,8],"def_end_pos":[226,43]},{"full_name":"mul_neg_of_neg_of_pos","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[418,8],"def_end_pos":[418,29]},{"full_name":"neg_lt_iff_pos_add'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[101,2],"def_end_pos":[101,13]}]}]}
{"url":"Mathlib/Data/Sign.lean","commit":"","full_name":"SignType.map_cast","start":[232,0],"end":[236,26],"file_path":"Mathlib/Data/Sign.lean","tactics":[{"state_before":"α✝ : Type u_1\ninst✝⁸ : Zero α✝\ninst✝⁷ : One α✝\ninst✝⁶ : Neg α✝\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝⁵ : AddGroupWithOne α\ninst✝⁴ : One β\ninst✝³ : SubtractionMonoid β\ninst✝² : FunLike F α β\ninst✝¹ : AddMonoidHomClass F α β\ninst✝ : OneHomClass F α β\nf : F\ns : SignType\n⊢ f ↑s = ↑s","state_after":"no goals","tactic":"apply map_cast' <;> simp","premises":[{"full_name":"SignType.map_cast'","def_path":"Mathlib/Data/Sign.lean","def_pos":[227,6],"def_end_pos":[227,15]}]}]}
{"url":"Mathlib/Deprecated/Subgroup.lean","commit":"","full_name":"AddGroup.closure_subset","start":[436,0],"end":[438,83],"file_path":"Mathlib/Deprecated/Subgroup.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b c : G\ninst✝ : Group G\ns✝ s t : Set G\nht : IsSubgroup t\nh : s ⊆ t\na : G\nha : a ∈ closure s\n⊢ a ∈ t","state_after":"no goals","tactic":"induction ha <;> simp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]","premises":[{"full_name":"IsSubgroup.inv_mem_iff","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[122,8],"def_end_pos":[122,19]},{"full_name":"IsSubmonoid.mul_mem","def_path":"Mathlib/Deprecated/Submonoid.lean","def_pos":[52,2],"def_end_pos":[52,9]},{"full_name":"IsSubmonoid.one_mem","def_path":"Mathlib/Deprecated/Submonoid.lean","def_pos":[50,2],"def_end_pos":[50,9]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.Y_ne_negY_of_Y_ne'","start":[531,0],"end":[537,87],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W'.negY Q * P z ^ 3\n⊢ P y ≠ W'.negY P","state_after":"R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W'.negY Q * P z ^ 3\nhy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0\n⊢ P y ≠ W'.negY P","tactic":"have hy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0 :=\n    (mul_eq_zero.mp <| Y_sub_Y_mul_Y_sub_negY hP hQ hx).resolve_right <| sub_ne_zero_of_ne hy","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]},{"full_name":"WeierstrassCurve.Jacobian.Y_sub_Y_mul_Y_sub_negY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[494,6],"def_end_pos":[494,28]},{"full_name":"mul_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[248,8],"def_end_pos":[248,19]},{"full_name":"sub_ne_zero_of_ne","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[387,2],"def_end_pos":[387,13]}]},{"state_before":"R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W'.negY Q * P z ^ 3\nhy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0\n⊢ P y ≠ W'.negY P","state_after":"R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0\nhy : P y = W'.negY P\n⊢ P y * Q z ^ 3 = W'.negY Q * P z ^ 3","tactic":"contrapose! hy","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0\nhy : P y = W'.negY P\n⊢ P y * Q z ^ 3 = W'.negY Q * P z ^ 3","state_after":"no goals","tactic":"linear_combination (norm := ring1) Y_sub_Y_add_Y_sub_negY P Q hx + Q z ^ 3 * hy - hy'","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Mathlib.Tactic.LinearCombination.add_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[41,8],"def_end_pos":[41,14]},{"full_name":"Mathlib.Tactic.LinearCombination.c_mul_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[47,8],"def_end_pos":[47,16]},{"full_name":"Mathlib.Tactic.LinearCombination.eq_of_add","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[111,8],"def_end_pos":[111,17]},{"full_name":"Mathlib.Tactic.LinearCombination.sub_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[44,8],"def_end_pos":[44,14]},{"full_name":"WeierstrassCurve.Jacobian.Y_sub_Y_add_Y_sub_negY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[518,6],"def_end_pos":[518,28]}]}]}
{"url":"Mathlib/Analysis/BoxIntegral/Partition/Split.lean","commit":"","full_name":"BoxIntegral.Box.splitUpper_eq_bot","start":[108,0],"end":[111,36],"file_path":"Mathlib/Analysis/BoxIntegral/Partition/Split.lean","tactics":[{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ I.splitUpper i x = ⊥ ↔ I.upper i ≤ x","state_after":"ι : Type u_1\nM : Type u_2\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ (I.upper i ≤ max x (I.lower i) ∨ ∃ x, x ≠ i ∧ I.upper x ≤ I.lower x) ↔ I.upper i ≤ x","tactic":"rw [splitUpper, mk'_eq_bot, exists_update_iff I.lower fun j y => I.upper j ≤ y]","premises":[{"full_name":"BoxIntegral.Box.lower","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[75,3],"def_end_pos":[75,8]},{"full_name":"BoxIntegral.Box.mk'_eq_bot","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[285,8],"def_end_pos":[285,18]},{"full_name":"BoxIntegral.Box.splitUpper","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Split.lean","def_pos":[93,4],"def_end_pos":[93,14]},{"full_name":"BoxIntegral.Box.upper","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[75,9],"def_end_pos":[75,14]},{"full_name":"Function.exists_update_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[497,8],"def_end_pos":[497,25]}]},{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ (I.upper i ≤ max x (I.lower i) ∨ ∃ x, x ≠ i ∧ I.upper x ≤ I.lower x) ↔ I.upper i ≤ x","state_after":"no goals","tactic":"simp [(I.lower_lt_upper _).not_le]","premises":[{"full_name":"BoxIntegral.Box.lower_lt_upper","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[77,2],"def_end_pos":[77,16]}]}]}
{"url":"Mathlib/CategoryTheory/Equivalence.lean","commit":"","full_name":"CategoryTheory.Equivalence.changeFunctor_refl","start":[469,0],"end":[470,89],"file_path":"Mathlib/CategoryTheory/Equivalence.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\ne : C ≌ D\n⊢ e.changeFunctor (Iso.refl e.functor) = e","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Order/Disjointed.lean","commit":"","full_name":"preimage_find_eq_disjointed","start":[160,0],"end":[163,51],"file_path":"Mathlib/Order/Disjointed.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\n⊢ (fun x => Nat.find ⋯) ⁻¹' {n} = disjointed s n","state_after":"case h\nα : Type u_1\nβ : Type u_2\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\nx : α\n⊢ x ∈ (fun x => Nat.find ⋯) ⁻¹' {n} ↔ x ∈ disjointed s n","tactic":"ext x","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\nx : α\n⊢ x ∈ (fun x => Nat.find ⋯) ⁻¹' {n} ↔ x ∈ disjointed s n","state_after":"no goals","tactic":"simp [Nat.find_eq_iff, disjointed_eq_inter_compl]","premises":[{"full_name":"Nat.find_eq_iff","def_path":"Mathlib/Data/Nat/Find.lean","def_pos":[76,6],"def_end_pos":[76,17]},{"full_name":"disjointed_eq_inter_compl","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[156,8],"def_end_pos":[156,33]}]}]}
{"url":"Mathlib/CategoryTheory/Types.lean","commit":"","full_name":"CategoryTheory.mono_iff_injective","start":[226,0],"end":[235,48],"file_path":"Mathlib/CategoryTheory/Types.lean","tactics":[{"state_before":"X Y : Type u\nf : X ⟶ Y\n⊢ Mono f ↔ Function.Injective f","state_after":"case mp\nX Y : Type u\nf : X ⟶ Y\n⊢ Mono f → Function.Injective f\n\ncase mpr\nX Y : Type u\nf : X ⟶ Y\n⊢ Function.Injective f → Mono f","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/PiTensorProduct.lean","commit":"","full_name":"PiTensorProduct.lifts_add","start":[336,0],"end":[342,13],"file_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","tactics":[{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx y : ⨂[R] (i : ι), s i\np q : FreeAddMonoid (R × ((i : ι) → s i))\nhp : p ∈ x.lifts\nhq : q ∈ y.lifts\n⊢ p + q ∈ (x + y).lifts","state_after":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx y : ⨂[R] (i : ι), s i\np q : FreeAddMonoid (R × ((i : ι) → s i))\nhp : p ∈ x.lifts\nhq : q ∈ y.lifts\n⊢ ↑p + ↑q = x + y","tactic":"simp only [lifts, Set.mem_setOf_eq, AddCon.coe_add]","premises":[{"full_name":"AddCon.coe_add","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[320,2],"def_end_pos":[320,13]},{"full_name":"PiTensorProduct.lifts","def_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","def_pos":[312,4],"def_end_pos":[312,9]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx y : ⨂[R] (i : ι), s i\np q : FreeAddMonoid (R × ((i : ι) → s i))\nhp : p ∈ x.lifts\nhq : q ∈ y.lifts\n⊢ ↑p + ↑q = x + y","state_after":"no goals","tactic":"rw [hp, hq]","premises":[]}]}
{"url":"Mathlib/Order/Filter/Ker.lean","commit":"","full_name":"Filter.ker_eq_univ","start":[42,0],"end":[42,97],"file_path":"Mathlib/Order/Filter/Ker.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nf g : Filter α\ns : Set α\na : α\n⊢ 𝓟 ⊤ ≤ f ↔ f = ⊤","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.RightHomologyData.ofZeros_ι","start":[190,0],"end":[203,36],"file_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.30360, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ S.f ≫ 𝟙 S.X₂ = 0","state_after":"no goals","tactic":"rw [comp_id, hf]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.30360, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ 𝟙 S.X₂ ≫ (CokernelCofork.IsColimit.ofId S.f hf).desc (CokernelCofork.ofπ S.g ⋯) = 0","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.30360, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ 𝟙 S.X₂ ≫ S.g = 0","tactic":"change 𝟙 _ ≫ S.g = 0","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.30360, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ 𝟙 S.X₂ ≫ S.g = 0","state_after":"no goals","tactic":"simp only [hg, comp_zero]","premises":[{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]}]}]}
{"url":"Mathlib/Data/NNReal/Basic.lean","commit":"","full_name":"Real.toNNReal_pos","start":[566,0],"end":[568,39],"file_path":"Mathlib/Data/NNReal/Basic.lean","tactics":[{"state_before":"r : ℝ\n⊢ 0 < r.toNNReal ↔ 0 < r","state_after":"no goals","tactic":"simp [← NNReal.coe_lt_coe, lt_irrefl]","premises":[{"full_name":"NNReal.coe_lt_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[327,25],"def_end_pos":[327,35]},{"full_name":"lt_irrefl","def_path":"Mathlib/Order/Defs.lean","def_pos":[65,8],"def_end_pos":[65,17]}]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","commit":"","full_name":"ContDiffOn.derivWithin","start":[1855,0],"end":[1863,67],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : m + 1 ≤ n\n⊢ ContDiffOn 𝕜 m (derivWithin f₂ s₂) s₂","state_after":"case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\n⊢ ContDiffOn 𝕜 ⊤ (derivWithin f₂ s₂) s₂\n\ncase coe\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\na✝ : ℕ\nhmn : ↑a✝ + 1 ≤ n\n⊢ ContDiffOn 𝕜 (↑a✝) (derivWithin f₂ s₂) s₂","tactic":"cases m","premises":[]}]}
{"url":"Mathlib/Algebra/Periodic.lean","commit":"","full_name":"Function.Antiperiodic.sub_zsmul_eq","start":[384,0],"end":[386,85],"file_path":"Mathlib/Algebra/Periodic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : AddGroup β\nh : Antiperiodic f c\nn : ℤ\n⊢ f (x - n • c) = ↑n.negOnePow • f x","state_after":"no goals","tactic":"simpa only [sub_eq_add_neg, neg_zsmul, Int.negOnePow_neg] using h.add_zsmul_eq (-n)","premises":[{"full_name":"Function.Antiperiodic.add_zsmul_eq","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[377,8],"def_end_pos":[377,33]},{"full_name":"Int.negOnePow_neg","def_path":"Mathlib/Algebra/Ring/NegOnePow.lean","def_pos":[78,6],"def_end_pos":[78,19]},{"full_name":"neg_zsmul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[428,29],"def_end_pos":[428,38]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/Data/Finset/Powerset.lean","commit":"","full_name":"Finset.mem_ssubsets","start":[132,0],"end":[134,73],"file_path":"Mathlib/Data/Finset/Powerset.lean","tactics":[{"state_before":"α : Type u_1\ns✝ t✝ : Finset α\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ t ∈ s.ssubsets ↔ t ⊂ s","state_after":"no goals","tactic":"rw [ssubsets, mem_erase, mem_powerset, ssubset_iff_subset_ne, and_comm]","premises":[{"full_name":"Finset.mem_erase","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1582,8],"def_end_pos":[1582,17]},{"full_name":"Finset.mem_powerset","def_path":"Mathlib/Data/Finset/Powerset.lean","def_pos":[31,8],"def_end_pos":[31,20]},{"full_name":"Finset.ssubset_iff_subset_ne","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[372,8],"def_end_pos":[372,29]},{"full_name":"Finset.ssubsets","def_path":"Mathlib/Data/Finset/Powerset.lean","def_pos":[129,4],"def_end_pos":[129,12]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean","commit":"","full_name":"EuclideanGeometry.preimage_inversion_perpBisector","start":[56,0],"end":[59,37],"file_path":"Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc x y : P\nR : ℝ\nhR : R ≠ 0\nhy : y ≠ c\n⊢ inversion c R ⁻¹' ↑(perpBisector c y) = sphere (inversion c R y) (R ^ 2 / dist y c) \\ {c}","state_after":"no goals","tactic":"rw [← dist_inversion_center, ← preimage_inversion_perpBisector_inversion hR,\n    inversion_inversion] <;> simp [*]","premises":[{"full_name":"EuclideanGeometry.dist_inversion_center","def_path":"Mathlib/Geometry/Euclidean/Inversion/Basic.lean","def_pos":[91,8],"def_end_pos":[91,29]},{"full_name":"EuclideanGeometry.inversion_inversion","def_path":"Mathlib/Geometry/Euclidean/Inversion/Basic.lean","def_pos":[103,8],"def_end_pos":[103,27]},{"full_name":"EuclideanGeometry.preimage_inversion_perpBisector_inversion","def_path":"Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean","def_pos":[52,8],"def_end_pos":[52,49]}]}]}
{"url":"Mathlib/Algebra/Group/Defs.lean","commit":"","full_name":"neg_add_cancel_comm","start":[1113,0],"end":[1114,36],"file_path":"Mathlib/Algebra/Group/Defs.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : CommGroup G\na b : G\n⊢ a⁻¹ * b * a = b","state_after":"no goals","tactic":"rw [mul_comm, mul_inv_cancel_left]","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_inv_cancel_left","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1063,8],"def_end_pos":[1063,27]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","commit":"","full_name":"CategoryTheory.CostructuredArrow.mkPrecomp_comp","start":[498,0],"end":[500,11],"file_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nf : S.obj Y ⟶ T\ng : Y' ⟶ Y\ng' : Y'' ⟶ Y'\n⊢ mk (S.map (g' ≫ g) ≫ f) = mk (S.map g' ≫ S.map g ≫ f)","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nf : S.obj Y ⟶ T\ng : Y' ⟶ Y\ng' : Y'' ⟶ Y'\n⊢ mkPrecomp f (g' ≫ g) = eqToHom ⋯ ≫ mkPrecomp (S.map g ≫ f) g' ≫ mkPrecomp f g","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/RingTheory/Jacobson.lean","commit":"","full_name":"Ideal.Polynomial.jacobson_bot_of_integral_localization","start":[336,0],"end":[378,99],"file_path":"Mathlib/RingTheory/Jacobson.lean","tactics":[{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\n⊢ ⊥.jacobson = ⊥","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\n⊢ ⊥.jacobson = ⊥","tactic":"have hM : ((Submonoid.powers x).map φ : Submonoid S) ≤ nonZeroDivisors S :=\n    map_le_nonZeroDivisors_of_injective φ hφ (powers_le_nonZeroDivisors_of_noZeroDivisors hx)","premises":[{"full_name":"Submonoid","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[88,10],"def_end_pos":[88,19]},{"full_name":"Submonoid.map","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[194,4],"def_end_pos":[194,7]},{"full_name":"Submonoid.powers","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[393,4],"def_end_pos":[393,10]},{"full_name":"map_le_nonZeroDivisors_of_injective","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[221,8],"def_end_pos":[221,43]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]},{"full_name":"powers_le_nonZeroDivisors_of_noZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[217,8],"def_end_pos":[217,51]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\n⊢ ⊥.jacobson = ⊥","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n⊢ ⊥.jacobson = ⊥","tactic":"letI : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors _ hM","premises":[{"full_name":"IsDomain","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[413,6],"def_end_pos":[413,14]},{"full_name":"IsLocalization.isDomain_of_le_nonZeroDivisors","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[1023,8],"def_end_pos":[1023,38]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n⊢ ⊥.jacobson = ⊥","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\n⊢ ⊥.jacobson = ⊥","tactic":"let φ' : Rₘ →+* Sₘ := IsLocalization.map _ φ (Submonoid.powers x).le_comap_map","premises":[{"full_name":"IsLocalization.map","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[559,18],"def_end_pos":[559,21]},{"full_name":"RingHom","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[297,10],"def_end_pos":[297,17]},{"full_name":"Submonoid.le_comap_map","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[246,8],"def_end_pos":[246,20]},{"full_name":"Submonoid.powers","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[393,4],"def_end_pos":[393,10]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\n⊢ ⊥.jacobson = ⊥","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\n⊢ ∀ (I : Ideal Sₘ), I.IsMaximal → (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"suffices ∀ I : Ideal Sₘ, I.IsMaximal → (I.comap (algebraMap S Sₘ)).IsMaximal by\n    have hϕ' : comap (algebraMap S Sₘ) (⊥ : Ideal Sₘ) = (⊥ : Ideal S) := by\n      rw [← RingHom.ker_eq_comap_bot, ← RingHom.injective_iff_ker_eq_bot]\n      exact IsLocalization.injective Sₘ hM\n    have hSₘ : IsJacobson Sₘ := isJacobson_of_isIntegral' φ' hφ' (isJacobson_localization x)\n    refine eq_bot_iff.mpr (le_trans ?_ (le_of_eq hϕ'))\n    rw [← hSₘ.out isRadical_bot_of_noZeroDivisors, comap_jacobson]\n    exact sInf_le_sInf fun j hj => ⟨bot_le,\n      let ⟨J, hJ⟩ := hj\n      hJ.2 ▸ this J hJ.1.2⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Ideal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[39,7],"def_end_pos":[39,12]},{"full_name":"Ideal.IsJacobson","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[50,6],"def_end_pos":[50,16]},{"full_name":"Ideal.IsJacobson.out","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[57,8],"def_end_pos":[57,22]},{"full_name":"Ideal.IsMaximal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[246,6],"def_end_pos":[246,15]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.comap_jacobson","def_path":"Mathlib/RingTheory/JacobsonIdeal.lean","def_pos":[188,8],"def_end_pos":[188,22]},{"full_name":"Ideal.isJacobson_localization","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[212,8],"def_end_pos":[212,31]},{"full_name":"Ideal.isJacobson_of_isIntegral'","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[135,8],"def_end_pos":[135,33]},{"full_name":"Ideal.isRadical_bot_of_noZeroDivisors","def_path":"Mathlib/RingTheory/Ideal/Operations.lean","def_pos":[934,8],"def_end_pos":[934,39]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsLocalization.injective","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[978,18],"def_end_pos":[978,27]},{"full_name":"RingHom.injective_iff_ker_eq_bot","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[558,8],"def_end_pos":[558,32]},{"full_name":"RingHom.ker_eq_comap_bot","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[529,8],"def_end_pos":[529,24]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]},{"full_name":"eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[282,8],"def_end_pos":[282,18]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"sInf_le_sInf","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[153,8],"def_end_pos":[153,20]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\n⊢ ∀ (I : Ideal Sₘ), I.IsMaximal → (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"intro I hI","premises":[]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis : (comap (algebraMap S Sₘ) I).IsPrime\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"haveI : (I.comap (algebraMap S Sₘ)).IsPrime := comap_isPrime _ I","premises":[{"full_name":"Ideal.IsPrime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[203,6],"def_end_pos":[203,13]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.comap_isPrime","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[193,8],"def_end_pos":[193,21]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis : (comap (algebraMap S Sₘ) I).IsPrime\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝¹ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝ : (comap (algebraMap S Sₘ) I).IsPrime\nthis : (comap φ' I).IsPrime\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"haveI : (I.comap φ').IsPrime := comap_isPrime φ' I","premises":[{"full_name":"Ideal.IsPrime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[203,6],"def_end_pos":[203,13]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.comap_isPrime","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[193,8],"def_end_pos":[193,21]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝¹ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝ : (comap (algebraMap S Sₘ) I).IsPrime\nthis : (comap φ' I).IsPrime\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"haveI : (⊥ : Ideal (S ⧸ I.comap (algebraMap S Sₘ))).IsPrime := bot_prime","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Ideal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[39,7],"def_end_pos":[39,12]},{"full_name":"Ideal.IsPrime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[203,6],"def_end_pos":[203,13]},{"full_name":"Ideal.bot_prime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[241,8],"def_end_pos":[241,17]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"have hcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ := IsLocalization.map_comp _","premises":[{"full_name":"IsLocalization.map_comp","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[570,8],"def_end_pos":[570,16]},{"full_name":"RingHom.comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[563,4],"def_end_pos":[563,8]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"let f := quotientMap (I.comap (algebraMap S Sₘ)) φ le_rfl","premises":[{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.quotientMap","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[464,4],"def_end_pos":[464,15]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"let g := quotientMap I (algebraMap S Sₘ) le_rfl","premises":[{"full_name":"Ideal.quotientMap","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[464,4],"def_end_pos":[464,15]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝² : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝¹ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝ : (comap φ' I).IsPrime\nthis : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝³ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝² : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝¹ : (comap φ' I).IsPrime\nthis✝ : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis : (comap φ' I).IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"have := isMaximal_comap_of_isIntegral_of_isMaximal' φ' hφ' I","premises":[{"full_name":"Ideal.isMaximal_comap_of_isIntegral_of_isMaximal'","def_path":"Mathlib/RingTheory/Ideal/Over.lean","def_pos":[267,8],"def_end_pos":[267,51]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝³ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝² : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝¹ : (comap φ' I).IsPrime\nthis✝ : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis : (comap φ' I).IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁴ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝³ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝² : (comap φ' I).IsPrime\nthis✝¹ : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝ : (comap φ' I).IsMaximal\nthis : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"have := ((isMaximal_iff_isMaximal_disjoint Rₘ x _).1 this).left","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Ideal.isMaximal_iff_isMaximal_disjoint","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[156,8],"def_end_pos":[156,40]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁴ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝³ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝² : (comap φ' I).IsPrime\nthis✝¹ : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝ : (comap φ' I).IsMaximal\nthis : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝³ : (comap φ' I).IsPrime\nthis✝² : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝¹ : (comap φ' I).IsMaximal\nthis✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\nthis : (comap φ (comap (algebraMap S Sₘ) I)).IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","tactic":"have : ((I.comap (algebraMap S Sₘ)).comap φ).IsMaximal := by\n    rwa [comap_comap, hcomm, ← comap_comap] at this","premises":[{"full_name":"Ideal.IsMaximal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[246,6],"def_end_pos":[246,15]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.comap_comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[118,8],"def_end_pos":[118,19]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝³ : (comap φ' I).IsPrime\nthis✝² : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝¹ : (comap φ' I).IsMaximal\nthis✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\nthis : (comap φ (comap (algebraMap S Sₘ) I)).IsMaximal\n⊢ (comap (algebraMap S Sₘ) I).IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝³ : (comap φ' I).IsPrime\nthis✝² : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝¹ : (comap φ' I).IsMaximal\nthis✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\nthis : ⊥.IsMaximal\n⊢ ⊥.IsMaximal","tactic":"rw [← bot_quotient_isMaximal_iff] at this ⊢","premises":[{"full_name":"Ideal.bot_quotient_isMaximal_iff","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[157,8],"def_end_pos":[157,34]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝³ : (comap φ' I).IsPrime\nthis✝² : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝¹ : (comap φ' I).IsMaximal\nthis✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\nthis : ⊥.IsMaximal\n⊢ ⊥.IsMaximal","state_after":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝³ : (comap φ' I).IsPrime\nthis✝² : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝¹ : (comap φ' I).IsMaximal\nthis✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\nthis : ⊥.IsMaximal\n⊢ f.IsIntegral","tactic":"refine isMaximal_of_isIntegral_of_isMaximal_comap' f ?_ ⊥\n    ((eq_bot_iff.2 (comap_bot_le_of_injective f quotientMap_injective)).symm ▸ this)","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Ideal.comap_bot_le_of_injective","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[299,8],"def_end_pos":[299,33]},{"full_name":"Ideal.isMaximal_of_isIntegral_of_isMaximal_comap'","def_path":"Mathlib/RingTheory/Ideal/Over.lean","def_pos":[236,8],"def_end_pos":[236,51]},{"full_name":"Ideal.quotientMap_injective","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[526,8],"def_end_pos":[526,29]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[282,8],"def_end_pos":[282,18]}]},{"state_before":"R✝ : Type u_1\nS : Type u_2\ninst✝¹³ : CommRing R✝\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\nRₘ✝ : Type u_3\nSₘ✝ : Type u_4\ninst✝¹⁰ : CommRing Rₘ✝\ninst✝⁹ : CommRing Sₘ✝\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobson R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization.Away x Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ\nhφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral\nhM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S\nthis✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\nφ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯\nI : Ideal Sₘ\nhI : I.IsMaximal\nthis✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime\nthis✝³ : (comap φ' I).IsPrime\nthis✝² : ⊥.IsPrime\nhcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ\nf : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=\n  quotientMap (comap (algebraMap S Sₘ) I) φ ⋯\ng : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯\nthis✝¹ : (comap φ' I).IsMaximal\nthis✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal\nthis : ⊥.IsMaximal\n⊢ f.IsIntegral","state_after":"no goals","tactic":"exact RingHom.IsIntegral.tower_bot f g quotientMap_injective\n    ((comp_quotientMap_eq_of_comp_eq hcomm I).symm ▸\n      (RingHom.isIntegral_of_surjective _\n        (IsLocalization.surjective_quotientMap_of_maximal_of_localization (Submonoid.powers x) Rₘ\n          (by rwa [comap_comap, hcomm, ← bot_quotient_isMaximal_iff]))).trans _ _ (hφ'.quotient _))","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Ideal.bot_quotient_isMaximal_iff","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[157,8],"def_end_pos":[157,34]},{"full_name":"Ideal.comap_comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[118,8],"def_end_pos":[118,19]},{"full_name":"Ideal.comp_quotientMap_eq_of_comp_eq","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[537,8],"def_end_pos":[537,38]},{"full_name":"Ideal.quotientMap_injective","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[526,8],"def_end_pos":[526,29]},{"full_name":"IsLocalization.surjective_quotientMap_of_maximal_of_localization","def_path":"Mathlib/RingTheory/Localization/Ideal.lean","def_pos":[170,8],"def_end_pos":[170,57]},{"full_name":"RingHom.IsIntegral.quotient","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[635,8],"def_end_pos":[635,35]},{"full_name":"RingHom.IsIntegral.tower_bot","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[616,15],"def_end_pos":[616,43]},{"full_name":"RingHom.IsIntegral.trans","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[584,18],"def_end_pos":[584,42]},{"full_name":"RingHom.isIntegral_of_surjective","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[603,8],"def_end_pos":[603,40]},{"full_name":"Submonoid.powers","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[393,4],"def_end_pos":[393,10]}]}]}
{"url":"Mathlib/Data/Finset/Lattice.lean","commit":"","full_name":"Finset.sup'_lt_iff","start":[1131,0],"end":[1134,53],"file_path":"Mathlib/Data/Finset/Lattice.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : LinearOrder α\ns : Finset ι\nH : s.Nonempty\nf : ι → α\na : α\n⊢ s.sup' H f < a ↔ ∀ i ∈ s, f i < a","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : LinearOrder α\ns : Finset ι\nH : s.Nonempty\nf : ι → α\na : α\n⊢ (∀ b ∈ s, (WithBot.some ∘ f) b < ↑a) ↔ ∀ i ∈ s, f i < a","tactic":"rw [← WithBot.coe_lt_coe, coe_sup', Finset.sup_lt_iff (WithBot.bot_lt_coe a)]","premises":[{"full_name":"Finset.coe_sup'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[703,8],"def_end_pos":[703,16]},{"full_name":"Finset.sup_lt_iff","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[630,18],"def_end_pos":[630,28]},{"full_name":"WithBot.bot_lt_coe","def_path":"Mathlib/Order/WithBot.lean","def_pos":[266,8],"def_end_pos":[266,18]},{"full_name":"WithBot.coe_lt_coe","def_path":"Mathlib/Order/WithBot.lean","def_pos":[262,8],"def_end_pos":[262,18]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : LinearOrder α\ns : Finset ι\nH : s.Nonempty\nf : ι → α\na : α\n⊢ (∀ b ∈ s, (WithBot.some ∘ f) b < ↑a) ↔ ∀ i ∈ s, f i < a","state_after":"no goals","tactic":"exact forall₂_congr (fun _ _ => WithBot.coe_lt_coe)","premises":[{"full_name":"WithBot.coe_lt_coe","def_path":"Mathlib/Order/WithBot.lean","def_pos":[262,8],"def_end_pos":[262,18]},{"full_name":"forall₂_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[215,8],"def_end_pos":[215,21]}]}]}
{"url":"Mathlib/SetTheory/ZFC/Basic.lean","commit":"","full_name":"PSet.toSet_empty","start":[314,0],"end":[315,52],"file_path":"Mathlib/SetTheory/ZFC/Basic.lean","tactics":[{"state_before":"⊢ ∅.toSet = ∅","state_after":"no goals","tactic":"simp [toSet]","premises":[{"full_name":"PSet.toSet","def_path":"Mathlib/SetTheory/ZFC/Basic.lean","def_pos":[263,4],"def_end_pos":[263,9]}]}]}
{"url":"Mathlib/Data/Set/Image.lean","commit":"","full_name":"Disjoint.of_preimage","start":[1359,0],"end":[1362,16],"file_path":"Mathlib/Data/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ns✝ t✝ : Set α\nhf : Surjective f\ns t : Set β\nh : Disjoint (f ⁻¹' s) (f ⁻¹' t)\n⊢ Disjoint s t","state_after":"no goals","tactic":"rw [disjoint_iff_inter_eq_empty, ← image_preimage_eq (_ ∩ _) hf, preimage_inter, h.inter_eq,\n    image_empty]","premises":[{"full_name":"Disjoint.inter_eq","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1213,8],"def_end_pos":[1213,32]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.disjoint_iff_inter_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1210,8],"def_end_pos":[1210,35]},{"full_name":"Set.image_empty","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[265,8],"def_end_pos":[265,19]},{"full_name":"Set.image_preimage_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[421,8],"def_end_pos":[421,25]},{"full_name":"Set.preimage_inter","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[66,8],"def_end_pos":[66,22]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/CompatiblePlus.lean","commit":"","full_name":"CategoryTheory.GrothendieckTopology.toPlus_comp_plusCompIso_inv","start":[185,0],"end":[187,99],"file_path":"Mathlib/CategoryTheory/Sites/CompatiblePlus.lean","tactics":[{"state_before":"C : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F\nP : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F\n⊢ J.toPlus (P ⋙ F) ≫ (J.plusCompIso F P).inv = whiskerRight (J.toPlus P) F","state_after":"no goals","tactic":"simp [Iso.comp_inv_eq]","premises":[{"full_name":"CategoryTheory.Iso.comp_inv_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[184,8],"def_end_pos":[184,19]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"intervalIntegral.intervalIntegral_eq_integral_uIoc","start":[443,0],"end":[447,88],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : Measure ℝ\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ","state_after":"case pos\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : Measure ℝ\nh : a ≤ b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 1 • ∫ (x : ℝ) in Ι a b, f x ∂μ\n\ncase neg\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : Measure ℝ\nh : ¬a ≤ b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = -1 • ∫ (x : ℝ) in Ι a b, f x ∂μ","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean","commit":"","full_name":"CategoryTheory.IsKernelPair.cancel_right_of_mono","start":[120,0],"end":[126,70],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\nf₁ : X ⟶ Y\nf₂ : Y ⟶ Z\ninst✝ : Mono f₂\nbig_k : IsKernelPair (f₁ ≫ f₂) a b\n⊢ a ≫ f₁ = b ≫ f₁","state_after":"no goals","tactic":"rw [← cancel_mono f₂, assoc, assoc, big_k.w]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CommSq.w","def_path":"Mathlib/CategoryTheory/CommSq.lean","def_pos":[44,2],"def_end_pos":[44,3]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]}]}
{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.MeasurePreserving.integrable_comp","start":[552,0],"end":[556,67],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nν : Measure δ\ng : δ → β\nf : α → δ\nhf : MeasurePreserving f μ ν\nhg : AEStronglyMeasurable g ν\n⊢ Integrable (g ∘ f) μ ↔ Integrable g ν","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nν : Measure δ\ng : δ → β\nf : α → δ\nhf : MeasurePreserving f μ ν\nhg : AEStronglyMeasurable g (Measure.map f μ)\n⊢ Integrable (g ∘ f) μ ↔ Integrable g (Measure.map f μ)","tactic":"rw [← hf.map_eq] at hg ⊢","premises":[{"full_name":"MeasureTheory.MeasurePreserving.map_eq","def_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","def_pos":[43,12],"def_end_pos":[43,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nν : Measure δ\ng : δ → β\nf : α → δ\nhf : MeasurePreserving f μ ν\nhg : AEStronglyMeasurable g (Measure.map f μ)\n⊢ Integrable (g ∘ f) μ ↔ Integrable g (Measure.map f μ)","state_after":"no goals","tactic":"exact (integrable_map_measure hg hf.measurable.aemeasurable).symm","premises":[{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"Measurable.aemeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[380,8],"def_end_pos":[380,31]},{"full_name":"MeasureTheory.MeasurePreserving.measurable","def_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","def_pos":[42,12],"def_end_pos":[42,22]},{"full_name":"MeasureTheory.integrable_map_measure","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[528,8],"def_end_pos":[528,30]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Intervals.lean","commit":"","full_name":"Finset.sum_Ioi_add_eq_sum_Ici","start":[53,0],"end":[55,41],"file_path":"Mathlib/Algebra/BigOperators/Intervals.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : CommMonoid M\nf : α → M\na✝ b : α\ninst✝ : LocallyFiniteOrderTop α\na : α\n⊢ (∏ x ∈ Ioi a, f x) * f a = ∏ x ∈ Ici a, f x","state_after":"no goals","tactic":"rw [mul_comm, mul_prod_Ioi_eq_prod_Ici]","premises":[{"full_name":"Finset.mul_prod_Ioi_eq_prod_Ici","def_path":"Mathlib/Algebra/BigOperators/Intervals.lean","def_pos":[50,6],"def_end_pos":[50,30]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.exists_degree_le_of_mem_span_of_finite","start":[193,0],"end":[203,45],"file_path":"Mathlib/RingTheory/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\n⊢ ∃ p' ∈ s, ∀ p ∈ Submodule.span R s, p.degree ≤ p'.degree","state_after":"case intro.intro\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\n⊢ ∃ p' ∈ s, ∀ p ∈ Submodule.span R s, p.degree ≤ p'.degree","tactic":"rcases Set.Finite.exists_maximal_wrt degree s s_fin hs with ⟨a, has, hmax⟩","premises":[{"full_name":"Polynomial.degree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[48,4],"def_end_pos":[48,10]},{"full_name":"Set.Finite.exists_maximal_wrt","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[1417,8],"def_end_pos":[1417,33]}]},{"state_before":"case intro.intro\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\n⊢ ∃ p' ∈ s, ∀ p ∈ Submodule.span R s, p.degree ≤ p'.degree","state_after":"case intro.intro\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\np : R[X]\nhp : p ∈ Submodule.span R s\n⊢ p.degree ≤ a.degree","tactic":"refine ⟨a, has, fun p hp => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\np : R[X]\nhp : p ∈ Submodule.span R s\n⊢ p.degree ≤ a.degree","state_after":"case intro.intro.intro\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\np : R[X]\nhp : p ∈ Submodule.span R s\np' : R[X]\nhp' : p' ∈ s ∧ p.degree ≤ p'.degree\n⊢ p.degree ≤ a.degree","tactic":"rcases exists_degree_le_of_mem_span hs hp with ⟨p', hp'⟩","premises":[{"full_name":"Polynomial.exists_degree_le_of_mem_span","def_path":"Mathlib/RingTheory/Polynomial/Basic.lean","def_pos":[178,8],"def_end_pos":[178,36]}]},{"state_before":"case intro.intro.intro\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\np : R[X]\nhp : p ∈ Submodule.span R s\np' : R[X]\nhp' : p' ∈ s ∧ p.degree ≤ p'.degree\n⊢ p.degree ≤ a.degree","state_after":"case pos\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\np : R[X]\nhp : p ∈ Submodule.span R s\np' : R[X]\nhp' : p' ∈ s ∧ p.degree ≤ p'.degree\nh : a.degree ≤ p'.degree\n⊢ p.degree ≤ a.degree\n\ncase neg\nR : Type u\nS : Type u_1\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\nhs : s.Nonempty\na : R[X]\nhas : a ∈ s\nhmax : ∀ a' ∈ s, a.degree ≤ a'.degree → a.degree = a'.degree\np : R[X]\nhp : p ∈ Submodule.span R s\np' : R[X]\nhp' : p' ∈ s ∧ p.degree ≤ p'.degree\nh : ¬a.degree ≤ p'.degree\n⊢ p.degree ≤ a.degree","tactic":"by_cases h : degree a ≤ degree p'","premises":[{"full_name":"Polynomial.degree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[48,4],"def_end_pos":[48,10]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Multiset/Sum.lean","commit":"","full_name":"Multiset.inl_mem_disjSum","start":[47,0],"end":[53,21],"file_path":"Mathlib/Data/Multiset/Sum.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx : α ⊕ β\n⊢ inl a ∈ s.disjSum t ↔ a ∈ s","state_after":"α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx : α ⊕ β\n⊢ (∃ a_1 ∈ s, inl a_1 = inl a) ↔ a ∈ s\n\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx : α ⊕ β\n⊢ ¬∃ b ∈ t, inr b = inl a","tactic":"rw [mem_disjSum, or_iff_left]","premises":[{"full_name":"Multiset.mem_disjSum","def_path":"Mathlib/Data/Multiset/Sum.lean","def_pos":[44,8],"def_end_pos":[44,19]},{"full_name":"or_iff_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[91,8],"def_end_pos":[91,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx : α ⊕ β\n⊢ ¬∃ b ∈ t, inr b = inl a","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb✝ : β\nx : α ⊕ β\nb : β\nleft✝ : b ∈ t\nhb : inr b = inl a\n⊢ False","tactic":"rintro ⟨b, _, hb⟩","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb✝ : β\nx : α ⊕ β\nb : β\nleft✝ : b ∈ t\nhb : inr b = inl a\n⊢ False","state_after":"no goals","tactic":"exact inr_ne_inl hb","premises":[{"full_name":"Sum.inr_ne_inl","def_path":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","def_pos":[95,8],"def_end_pos":[95,18]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","commit":"","full_name":"SimpleGraph.Subgraph.deleteEdges_le_of_le","start":[992,0],"end":[996,37],"file_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","tactics":[{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : G.Subgraph\ns✝ s s' : Set (Sym2 V)\nh : s ⊆ s'\n⊢ G'.deleteEdges s' ≤ G'.deleteEdges s","state_after":"case right\nι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : G.Subgraph\ns✝ s s' : Set (Sym2 V)\nh : s ⊆ s'\n⊢ ∀ ⦃v w : V⦄, G'.Adj v w → s(v, w) ∉ s' → s(v, w) ∉ s","tactic":"constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj,\n    true_and_iff, and_imp, subset_rfl]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"SimpleGraph.Subgraph.deleteEdges_adj","def_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","def_pos":[950,8],"def_end_pos":[950,23]},{"full_name":"SimpleGraph.Subgraph.deleteEdges_verts","def_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","def_pos":[946,8],"def_end_pos":[946,25]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"subset_rfl","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[535,6],"def_end_pos":[535,16]},{"full_name":"true_and_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[94,8],"def_end_pos":[94,20]}]},{"state_before":"case right\nι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : G.Subgraph\ns✝ s s' : Set (Sym2 V)\nh : s ⊆ s'\n⊢ ∀ ⦃v w : V⦄, G'.Adj v w → s(v, w) ∉ s' → s(v, w) ∉ s","state_after":"no goals","tactic":"exact fun _ _ _ hs' hs ↦ hs' (h hs)","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Measure/Restrict.lean","commit":"","full_name":"MeasureTheory.Measure.restrict_apply'","start":[90,0],"end":[97,60],"file_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","tactics":[{"state_before":"R : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nhs : MeasurableSet s\n⊢ (μ.restrict s) t = μ (t ∩ s)","state_after":"no goals","tactic":"rw [← toOuterMeasure_apply,\n    Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict hs,\n    OuterMeasure.restrict_apply s t _, toOuterMeasure_apply]","premises":[{"full_name":"MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[53,8],"def_end_pos":[53,58]},{"full_name":"MeasureTheory.Measure.toOuterMeasure_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[145,8],"def_end_pos":[145,36]},{"full_name":"MeasureTheory.OuterMeasure.restrict_apply","def_path":"Mathlib/MeasureTheory/OuterMeasure/Operations.lean","def_pos":[294,8],"def_end_pos":[294,22]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Girth.lean","commit":"","full_name":"SimpleGraph.three_le_egirth","start":[50,0],"end":[57,81],"file_path":"Mathlib/Combinatorics/SimpleGraph/Girth.lean","tactics":[{"state_before":"α : Type u_1\nG : SimpleGraph α\n⊢ 3 ≤ G.egirth","state_after":"case pos\nα : Type u_1\nG : SimpleGraph α\nh : G.IsAcyclic\n⊢ 3 ≤ G.egirth\n\ncase neg\nα : Type u_1\nG : SimpleGraph α\nh : ¬G.IsAcyclic\n⊢ 3 ≤ G.egirth","tactic":"by_cases h : G.IsAcyclic","premises":[{"full_name":"SimpleGraph.IsAcyclic","def_path":"Mathlib/Combinatorics/SimpleGraph/Acyclic.lean","def_pos":[50,4],"def_end_pos":[50,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"PartialHomeomorph.tendsto_extend_comp_iff","start":[897,0],"end":[904,70],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\n⊢ Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y)) ↔ Tendsto g l (𝓝 y)","state_after":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\n⊢ g ⁻¹' u ∈ l","tactic":"refine ⟨fun h u hu ↦ mem_map.2 ?_, (continuousAt_extend _ _ hy).tendsto.comp⟩","premises":[{"full_name":"ContinuousAt.tendsto","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1338,8],"def_end_pos":[1338,28]},{"full_name":"Filter.Tendsto.comp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2652,8],"def_end_pos":[2652,20]},{"full_name":"Filter.mem_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1658,8],"def_end_pos":[1658,15]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"PartialHomeomorph.continuousAt_extend","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[799,8],"def_end_pos":[799,27]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\n⊢ g ⁻¹' u ∈ l","state_after":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\nthis : Tendsto (↑(f.extend I).symm ∘ ↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I).symm (↑(f.extend I) y)))\n⊢ g ⁻¹' u ∈ l","tactic":"have := (f.continuousAt_extend_symm I hy).tendsto.comp h","premises":[{"full_name":"ContinuousAt.tendsto","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1338,8],"def_end_pos":[1338,28]},{"full_name":"Filter.Tendsto.comp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2652,8],"def_end_pos":[2652,20]},{"full_name":"PartialHomeomorph.continuousAt_extend_symm","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[844,8],"def_end_pos":[844,32]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\nthis : Tendsto (↑(f.extend I).symm ∘ ↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I).symm (↑(f.extend I) y)))\n⊢ g ⁻¹' u ∈ l","state_after":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\nthis : Tendsto (↑(f.extend I).symm ∘ ↑(f.extend I) ∘ g) l (𝓝 y)\n⊢ g ⁻¹' u ∈ l","tactic":"rw [extend_left_inv _ _ hy] at this","premises":[{"full_name":"PartialHomeomorph.extend_left_inv","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[781,8],"def_end_pos":[781,23]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\nthis : Tendsto (↑(f.extend I).symm ∘ ↑(f.extend I) ∘ g) l (𝓝 y)\n⊢ g ⁻¹' u ∈ l","state_after":"case h\n𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\nthis : Tendsto (↑(f.extend I).symm ∘ ↑(f.extend I) ∘ g) l (𝓝 y)\nz : α\nhz : g z ∈ f.source\nhzu : z ∈ ↑(f.extend I).symm ∘ ↑(f.extend I) ∘ g ⁻¹' u\n⊢ z ∈ g ⁻¹' u","tactic":"filter_upwards [hg, mem_map.1 (this hu)] with z hz hzu","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mem_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1658,8],"def_end_pos":[1658,15]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","commit":"","full_name":"Real.continuous_negMulLog","start":[103,0],"end":[104,60],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","tactics":[{"state_before":"⊢ Continuous negMulLog","state_after":"no goals","tactic":"simpa only [negMulLog_eq_neg] using continuous_mul_log.neg","premises":[{"full_name":"Continuous.neg","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[223,2],"def_end_pos":[223,13]},{"full_name":"Real.continuous_mul_log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","def_pos":[26,6],"def_end_pos":[26,24]},{"full_name":"Real.negMulLog_eq_neg","def_path":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","def_pos":[85,6],"def_end_pos":[85,22]}]}]}
{"url":"Mathlib/SetTheory/ZFC/Basic.lean","commit":"","full_name":"PSet.toSet_sUnion","start":[391,0],"end":[394,6],"file_path":"Mathlib/SetTheory/ZFC/Basic.lean","tactics":[{"state_before":"x : PSet\n⊢ (⋃₀ x).toSet = ⋃₀ (toSet '' x.toSet)","state_after":"case h\nx x✝ : PSet\n⊢ x✝ ∈ (⋃₀ x).toSet ↔ x✝ ∈ ⋃₀ (toSet '' x.toSet)","tactic":"ext","premises":[]},{"state_before":"case h\nx x✝ : PSet\n⊢ x✝ ∈ (⋃₀ x).toSet ↔ x✝ ∈ ⋃₀ (toSet '' x.toSet)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/SetTheory/Cardinal/Ordinal.lean","commit":"","full_name":"Cardinal.mk_bounded_set_le_of_infinite","start":[1119,0],"end":[1149,36],"file_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","tactics":[{"state_before":"α : Type u\ninst✝ : Infinite α\nc : Cardinal.{u}\n⊢ #{ t // #↑t ≤ c } ≤ #α ^ c","state_after":"α : Type u\ninst✝ : Infinite α\nc : Cardinal.{u}\n⊢ #{ t // #↑t ≤ c } ≤ (#α + 1) ^ c","tactic":"refine le_trans ?_ (by rw [← add_one_eq (aleph0_le_mk α)])","premises":[{"full_name":"Cardinal.add_one_eq","def_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","def_pos":[721,8],"def_end_pos":[721,18]},{"full_name":"Cardinal.aleph0_le_mk","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1491,8],"def_end_pos":[1491,20]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]},{"state_before":"α : Type u\ninst✝ : Infinite α\nc : Cardinal.{u}\n⊢ #{ t // #↑t ≤ c } ≤ (#α + 1) ^ c","state_after":"case h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\n⊢ #{ t // #↑t ≤ #β } ≤ (#α + 1) ^ #β","tactic":"induction' c using Cardinal.inductionOn with β","premises":[{"full_name":"Cardinal.inductionOn","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[123,8],"def_end_pos":[123,19]}]},{"state_before":"case h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\n⊢ #{ t // #↑t ≤ #β } ≤ (#α + 1) ^ #β","state_after":"case h.f\nα : Type u\ninst✝ : Infinite α\nβ : Type u\n⊢ (fun α β => β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β → { t // #↑t ≤ #β }\n\ncase h.hf\nα : Type u\ninst✝ : Infinite α\nβ : Type u\n⊢ Surjective ?h.f","tactic":"fapply mk_le_of_surjective","premises":[{"full_name":"Cardinal.mk_le_of_surjective","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[251,8],"def_end_pos":[251,27]}]},{"state_before":"case h.hf\nα : Type u\ninst✝ : Infinite α\nβ : Type u\n⊢ Surjective fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩","state_after":"case h.hf.mk.intro\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ∃ a, (fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) a = ⟨s, ⋯⟩","tactic":"rintro ⟨s, ⟨g⟩⟩","premises":[]},{"state_before":"case h.hf.mk.intro\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ∃ a, (fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) a = ⟨s, ⋯⟩","state_after":"case h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n      if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) =\n    ⟨s, ⋯⟩","tactic":"use fun y => if h : ∃ x : s, g x = y then Sum.inl (Classical.choose h).val\n               else Sum.inr (ULift.up 0)","premises":[{"full_name":"Classical.choose","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[25,18],"def_end_pos":[25,24]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]},{"full_name":"Sum.inl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[132,4],"def_end_pos":[132,7]},{"full_name":"Sum.inr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[134,4],"def_end_pos":[134,7]},{"full_name":"ULift.up","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[810,40],"def_end_pos":[810,42]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n      if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) =\n    ⟨s, ⋯⟩","state_after":"case h.a\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ↑((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n        if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) =\n    ↑⟨s, ⋯⟩","tactic":"apply Subtype.eq","premises":[{"full_name":"Subtype.eq","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1110,18],"def_end_pos":[1110,20]}]},{"state_before":"case h.a\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ↑((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n        if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) =\n    ↑⟨s, ⋯⟩","state_after":"case h.a.h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\nx : α\n⊢ x ∈\n      ↑((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) ↔\n    x ∈ ↑⟨s, ⋯⟩","tactic":"ext x","premises":[]},{"state_before":"case h.a.h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\nx : α\n⊢ x ∈\n      ↑((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) ↔\n    x ∈ ↑⟨s, ⋯⟩","state_after":"case h.a.h.mp\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\nx : α\n⊢ x ∈\n      ↑((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) →\n    x ∈ ↑⟨s, ⋯⟩\n\ncase h.a.h.mpr\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\nx : α\n⊢ x ∈ ↑⟨s, ⋯⟩ →\n    x ∈\n      ↑((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 })","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Sites/Sieves.lean","commit":"","full_name":"CategoryTheory.Presieve.ofArrows_bind","start":[148,0],"end":[160,53],"file_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z✝ : C\nf : Y ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\n⊢ ((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) =\n    ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf : Y✝ ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\nY : C\n⊢ ((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) =\n    ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst","tactic":"funext Y","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf : Y✝ ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\nY : C\n⊢ ((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) =\n    ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst","state_after":"case h.h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf✝ : Y✝ ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\nY : C\nf : Y ⟶ X\n⊢ (f ∈ (ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) ↔\n    f ∈ ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst","tactic":"ext f","premises":[]},{"state_before":"case h.h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf✝ : Y✝ ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\nY : C\nf : Y ⟶ X\n⊢ (f ∈ (ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) ↔\n    f ∈ ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst","state_after":"case h.h.mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf✝ : Y✝ ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\nY : C\nf : Y ⟶ X\n⊢ (f ∈ (ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) →\n    f ∈ ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst\n\ncase h.h.mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf✝ : Y✝ ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\nY : C\nf : Y ⟶ X\n⊢ (f ∈ ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst) →\n    f ∈ (ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/Star/Subalgebra.lean","commit":"","full_name":"StarSubalgebra.sInf_toSubalgebra","start":[618,0],"end":[621,34],"file_path":"Mathlib/Algebra/Star/Subalgebra.lean","tactics":[{"state_before":"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Set (StarSubalgebra R A)\n⊢ ↑(sInf S).toSubalgebra = ↑(sInf (toSubalgebra '' S))","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/ModelTheory/DirectLimit.lean","commit":"","full_name":"FirstOrder.Language.DirectLimit.funMap_quotient_mk'_sigma_mk'","start":[237,0],"end":[245,5],"file_path":"Mathlib/ModelTheory/DirectLimit.lean","tactics":[{"state_before":"L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\n⊢ (funMap F fun a => ⟦Structure.Sigma.mk f i (x a)⟧) = ⟦Structure.Sigma.mk f i (funMap F x)⟧","state_after":"L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\n⊢ (funMap F fun i_1 => Structure.Sigma.mk f i (x i_1)) ≈ Structure.Sigma.mk f i (funMap F x)","tactic":"simp only [funMap_quotient_mk', Quotient.eq]","premises":[{"full_name":"FirstOrder.Language.funMap_quotient_mk'","def_path":"Mathlib/ModelTheory/Quotients.lean","def_pos":[50,8],"def_end_pos":[50,27]},{"full_name":"Quotient.eq","def_path":"Mathlib/Data/Quot.lean","def_pos":[270,8],"def_end_pos":[270,19]}]},{"state_before":"case intro.intro\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\nk : ι\nik : i ≤ k\njk : Classical.choose ⋯ ≤ k\n⊢ (funMap F fun i_1 => Structure.Sigma.mk f i (x i_1)) ≈ Structure.Sigma.mk f i (funMap F x)","state_after":"case intro.intro\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\nk : ι\nik : i ≤ k\njk : Classical.choose ⋯ ≤ k\n⊢ (f (Classical.choose ⋯) k jk)\n      (funMap F (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) (Classical.choose ⋯) ⋯)) =\n    (f i k ik) (funMap F x)","tactic":"refine ⟨k, jk, ik, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\nk : ι\nik : i ≤ k\njk : Classical.choose ⋯ ≤ k\n⊢ (f (Classical.choose ⋯) k jk)\n      (funMap F (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) (Classical.choose ⋯) ⋯)) =\n    (f i k ik) (funMap F x)","state_after":"case intro.intro\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\nk : ι\nik : i ≤ k\njk : Classical.choose ⋯ ≤ k\n⊢ funMap F (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) k ⋯) = funMap F (⇑(f i k ik) ∘ x)","tactic":"simp only [Embedding.map_fun, comp_unify]","premises":[{"full_name":"FirstOrder.Language.DirectLimit.comp_unify","def_path":"Mathlib/ModelTheory/DirectLimit.lean","def_pos":[111,8],"def_end_pos":[111,18]},{"full_name":"FirstOrder.Language.Embedding.map_fun","def_path":"Mathlib/ModelTheory/Basic.lean","def_pos":[535,8],"def_end_pos":[535,15]}]},{"state_before":"case intro.intro\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\nk : ι\nik : i ≤ k\njk : Classical.choose ⋯ ≤ k\n⊢ funMap F (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) k ⋯) = funMap F (⇑(f i k ik) ∘ x)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Field/Basic.lean","commit":"","full_name":"inv_lt_one_iff","start":[200,0],"end":[203,65],"file_path":"Mathlib/Algebra/Order/Field/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\n⊢ a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a","state_after":"case inl\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : a ≤ 0\n⊢ a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a\n\ncase inr\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\n⊢ a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a","tactic":"rcases le_or_lt a 0 with ha | ha","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]}]}
{"url":"Mathlib/Probability/Kernel/RadonNikodym.lean","commit":"","full_name":"ProbabilityTheory.Kernel.singularPart_eq_zero_iff_measure_eq_zero","start":[450,0],"end":[460,52],"file_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","tactics":[{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","tactic":"have h_eq_add := rnDeriv_add_singularPart κ η","premises":[{"full_name":"ProbabilityTheory.Kernel.rnDeriv_add_singularPart","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[396,6],"def_end_pos":[396,30]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ∀ (a : α) (s : Set γ), MeasurableSet s → ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","tactic":"simp_rw [ext_iff, Measure.ext_iff] at h_eq_add","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.Measure.ext_iff","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[132,8],"def_end_pos":[132,15]},{"full_name":"ProbabilityTheory.Kernel.ext_iff","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[190,8],"def_end_pos":[190,15]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ∀ (a : α) (s : Set γ), MeasurableSet s → ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add :\n  ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) =\n    (κ a) (κ.mutuallySingularSetSlice η a)\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","tactic":"specialize h_eq_add a (mutuallySingularSetSlice κ η a)\n    (measurableSet_mutuallySingularSetSlice κ η a)","premises":[{"full_name":"ProbabilityTheory.Kernel.measurableSet_mutuallySingularSetSlice","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[213,6],"def_end_pos":[213,44]},{"full_name":"ProbabilityTheory.Kernel.mutuallySingularSetSlice","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[198,4],"def_end_pos":[198,28]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add :\n  ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) =\n    (κ a) (κ.mutuallySingularSetSlice η a)\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = (κ a) (κ.mutuallySingularSetSlice η a)\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","tactic":"simp only [coe_add, Pi.add_apply, Measure.coe_add,\n    withDensity_rnDeriv_mutuallySingularSetSlice κ η, zero_add] at h_eq_add","premises":[{"full_name":"MeasureTheory.Measure.coe_add","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[781,8],"def_end_pos":[781,15]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"ProbabilityTheory.Kernel.coe_add","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[86,25],"def_end_pos":[86,32]},{"full_name":"ProbabilityTheory.Kernel.withDensity_rnDeriv_mutuallySingularSetSlice","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[337,6],"def_end_pos":[337,50]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = (κ a) (κ.mutuallySingularSetSlice η a)\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = (κ a) (κ.mutuallySingularSetSlice η a)\n⊢ (κ.singularPart η) a = 0 ↔ ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = 0","tactic":"rw [← h_eq_add]","premises":[]},{"state_before":"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = (κ a) (κ.mutuallySingularSetSlice η a)\n⊢ (κ.singularPart η) a = 0 ↔ ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = 0","state_after":"no goals","tactic":"exact singularPart_eq_zero_iff_apply_eq_zero κ η a","premises":[{"full_name":"ProbabilityTheory.Kernel.singularPart_eq_zero_iff_apply_eq_zero","def_path":"Mathlib/Probability/Kernel/RadonNikodym.lean","def_pos":[410,6],"def_end_pos":[410,44]}]}]}
{"url":"Mathlib/Data/Set/Prod.lean","commit":"","full_name":"Set.disjoint_univ_pi","start":[660,0],"end":[662,83],"file_path":"Mathlib/Data/Set/Prod.lean","tactics":[{"state_before":"ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\n⊢ Disjoint (univ.pi t₁) (univ.pi t₂) ↔ ∃ i, Disjoint (t₁ i) (t₂ i)","state_after":"no goals","tactic":"simp only [disjoint_iff_inter_eq_empty, ← pi_inter_distrib, univ_pi_eq_empty_iff]","premises":[{"full_name":"Set.disjoint_iff_inter_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1210,8],"def_end_pos":[1210,35]},{"full_name":"Set.pi_inter_distrib","def_path":"Mathlib/Data/Set/Prod.lean","def_pos":[626,8],"def_end_pos":[626,24]},{"full_name":"Set.univ_pi_eq_empty_iff","def_path":"Mathlib/Data/Set/Prod.lean","def_pos":[653,8],"def_end_pos":[653,28]}]}]}
{"url":"Mathlib/Data/Nat/Choose/Basic.lean","commit":"","full_name":"Nat.descFactorial_eq_factorial_mul_choose","start":[236,0],"end":[241,90],"file_path":"Mathlib/Data/Nat/Choose/Basic.lean","tactics":[{"state_before":"n k : ℕ\n⊢ n.descFactorial k = k ! * n.choose k","state_after":"case inl\nn k : ℕ\nh : n < k\n⊢ n.descFactorial k = k ! * n.choose k\n\ncase inr\nn k : ℕ\nh : n ≥ k\n⊢ n.descFactorial k = k ! * n.choose k","tactic":"obtain h | h := Nat.lt_or_ge n k","premises":[{"full_name":"Nat.lt_or_ge","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1729,18],"def_end_pos":[1729,30]}]},{"state_before":"case inr\nn k : ℕ\nh : n ≥ k\n⊢ n.descFactorial k = k ! * n.choose k","state_after":"case inr\nn k : ℕ\nh : n ≥ k\n⊢ n.descFactorial k = n.choose k * k !","tactic":"rw [Nat.mul_comm]","premises":[{"full_name":"Nat.mul_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[217,18],"def_end_pos":[217,26]}]},{"state_before":"case inr\nn k : ℕ\nh : n ≥ k\n⊢ n.descFactorial k = n.choose k * k !","state_after":"case inr\nn k : ℕ\nh : n ≥ k\n⊢ n.descFactorial k * (n - k)! = n.choose k * k ! * (n - k)!","tactic":"apply Nat.mul_right_cancel (n - k).factorial_pos","premises":[{"full_name":"Nat.factorial_pos","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[56,8],"def_end_pos":[56,21]},{"full_name":"Nat.mul_right_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[452,18],"def_end_pos":[452,34]}]},{"state_before":"case inr\nn k : ℕ\nh : n ≥ k\n⊢ n.descFactorial k * (n - k)! = n.choose k * k ! * (n - k)!","state_after":"no goals","tactic":"rw [choose_mul_factorial_mul_factorial h, ← factorial_mul_descFactorial h, Nat.mul_comm]","premises":[{"full_name":"Nat.choose_mul_factorial_mul_factorial","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[110,8],"def_end_pos":[110,42]},{"full_name":"Nat.factorial_mul_descFactorial","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[348,8],"def_end_pos":[348,35]},{"full_name":"Nat.mul_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[217,18],"def_end_pos":[217,26]}]}]}
{"url":"Mathlib/Algebra/Module/Submodule/Ker.lean","commit":"","full_name":"LinearMap.le_ker_iff_map","start":[108,0],"end":[109,75],"file_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nK : Type u_5\nM : Type u_6\nM₁ : Type u_7\nM₂ : Type u_8\nM₃ : Type u_9\nV : Type u_10\nV₂ : Type u_11\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring R₂\ninst✝¹¹ : Semiring R₃\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : Module R M\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝³ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_12\ninst✝² : FunLike F M M₂\ninst✝¹ : SemilinearMapClass F τ₁₂ M M₂\ninst✝ : RingHomSurjective τ₁₂\nf : F\np : Submodule R M\n⊢ p ≤ ker f ↔ map f p = ⊥","state_after":"no goals","tactic":"rw [ker, eq_bot_iff, map_le_iff_le_comap]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LinearMap.ker","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Submodule.map_le_iff_le_comap","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[202,8],"def_end_pos":[202,27]},{"full_name":"eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[282,8],"def_end_pos":[282,18]}]}]}
{"url":"Mathlib/FieldTheory/Separable.lean","commit":"","full_name":"Polynomial.separable_X_sub_C","start":[193,0],"end":[194,65],"file_path":"Mathlib/FieldTheory/Separable.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nx : R\n⊢ (X - C x).Separable","state_after":"no goals","tactic":"simpa only [sub_eq_add_neg, C_neg] using separable_X_add_C (-x)","premises":[{"full_name":"Polynomial.C_neg","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[1038,8],"def_end_pos":[1038,13]},{"full_name":"Polynomial.separable_X_add_C","def_path":"Mathlib/FieldTheory/Separable.lean","def_pos":[66,8],"def_end_pos":[66,25]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/Arrow.lean","commit":"","full_name":"CategoryTheory.Arrow.inv_hom_id_left","start":[226,0],"end":[228,41],"file_path":"Mathlib/CategoryTheory/Comma/Arrow.lean","tactics":[{"state_before":"T : Type u\ninst✝ : Category.{v, u} T\nf g : Arrow T\nsq : f ⟶ g\ne : f ≅ g\n⊢ e.inv.left ≫ e.hom.left = 𝟙 g.left","state_after":"no goals","tactic":"rw [← comp_left, e.inv_hom_id, id_left]","premises":[{"full_name":"CategoryTheory.Arrow.comp_left","def_path":"Mathlib/CategoryTheory/Comma/Arrow.lean","def_pos":[63,8],"def_end_pos":[63,17]},{"full_name":"CategoryTheory.Arrow.id_left","def_path":"Mathlib/CategoryTheory/Comma/Arrow.lean","def_pos":[54,8],"def_end_pos":[54,15]},{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]}]}]}
{"url":"Mathlib/Topology/Instances/ENNReal.lean","commit":"","full_name":"isClosed_setOf_lipschitzWith","start":[1327,0],"end":[1329,68],"file_path":"Mathlib/Topology/Instances/ENNReal.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\ninst✝² : PseudoEMetricSpace α✝\nα : Type u_4\nβ : Type u_5\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\n⊢ IsClosed {f | LipschitzWith K f}","state_after":"no goals","tactic":"simp only [← lipschitzOnWith_univ, isClosed_setOf_lipschitzOnWith]","premises":[{"full_name":"isClosed_setOf_lipschitzOnWith","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[1321,8],"def_end_pos":[1321,38]},{"full_name":"lipschitzOnWith_univ","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[84,14],"def_end_pos":[84,34]}]}]}
{"url":"Mathlib/RingTheory/Ideal/Operations.lean","commit":"","full_name":"Ideal.sup_mul_eq_of_coprime_left","start":[584,0],"end":[589,45],"file_path":"Mathlib/RingTheory/Ideal/Operations.lean","tactics":[{"state_before":"R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : I ⊔ J = ⊤\ni : R\nhi : i ∈ I ⊔ K\n⊢ i ∈ I ⊔ J * K","state_after":"R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : 1 ∈ I ⊔ J\ni : R\nhi : i ∈ I ⊔ K\n⊢ i ∈ I ⊔ J * K","tactic":"rw [eq_top_iff_one] at h","premises":[{"full_name":"Ideal.eq_top_iff_one","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[87,8],"def_end_pos":[87,22]}]},{"state_before":"R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : 1 ∈ I ⊔ J\ni : R\nhi : i ∈ I ⊔ K\n⊢ i ∈ I ⊔ J * K","state_after":"R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : ∃ y ∈ I, ∃ z ∈ J, y + z = 1\ni : R\nhi : ∃ y ∈ I, ∃ z ∈ K, y + z = i\n⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i","tactic":"rw [Submodule.mem_sup] at h hi ⊢","premises":[{"full_name":"Submodule.mem_sup","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[374,8],"def_end_pos":[374,15]}]},{"state_before":"R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : ∃ y ∈ I, ∃ z ∈ J, y + z = 1\ni : R\nhi : ∃ y ∈ I, ∃ z ∈ K, y + z = i\n⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i","state_after":"case intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni : R\nhi : ∃ y ∈ I, ∃ z ∈ K, y + z = i\ni1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\n⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i","tactic":"obtain ⟨i1, hi1, j, hj, h⟩ := h","premises":[]},{"state_before":"case intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni : R\nhi : ∃ y ∈ I, ∃ z ∈ K, y + z = i\ni1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\n⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\ni' : R\nhi' : i' ∈ I\nk : R\nhk : k ∈ K\nhi : i' + k = i\n⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i","tactic":"obtain ⟨i', hi', k, hk, hi⟩ := hi","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\ni' : R\nhi' : i' ∈ I\nk : R\nhk : k ∈ K\nhi : i' + k = i\n⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\ni' : R\nhi' : i' ∈ I\nk : R\nhk : k ∈ K\nhi : i' + k = i\n⊢ i' + i1 * k + j * k = i","tactic":"refine ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, ?_⟩","premises":[{"full_name":"AddMemClass.add_mem","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[69,2],"def_end_pos":[69,9]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.mul_mem_mul","def_path":"Mathlib/RingTheory/Ideal/Operations.lean","def_pos":[385,8],"def_end_pos":[385,19]},{"full_name":"Ideal.mul_mem_right","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[486,8],"def_end_pos":[486,21]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\ni' : R\nhi' : i' ∈ I\nk : R\nhk : k ∈ K\nhi : i' + k = i\n⊢ i' + i1 * k + j * k = i","state_after":"no goals","tactic":"rw [add_assoc, ← add_mul, h, one_mul, hi]","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
{"url":"Mathlib/Data/Finset/Card.lean","commit":"","full_name":"Finset.card_union_add_card_inter","start":[466,0],"end":[469,45],"file_path":"Mathlib/Data/Finset/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nR : Type u_3\ns✝ t✝ u : Finset α\nf : α → β\nn : ℕ\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ (s ∪ ∅).card + (s ∩ ∅).card = s.card + ∅.card","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nR : Type u_3\ns✝ t✝ u : Finset α\nf : α → β\nn : ℕ\ninst✝ : DecidableEq α\ns t : Finset α\na : α\nr : Finset α\nhar : a ∉ r\nh : (s ∪ r).card + (s ∩ r).card = s.card + r.card\n⊢ (s ∪ insert a r).card + (s ∩ insert a r).card = s.card + (insert a r).card","state_after":"no goals","tactic":"by_cases a ∈ s <;>\n    simp [*, ← add_assoc, add_right_comm _ 1]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[155,2],"def_end_pos":[155,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"AddSubgroup.coe_eq_singleton","start":[697,0],"end":[704,40],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\nH✝ K H : Subgroup G\nx✝ : ∃ g, ↑H = {g}\ng : G\nhg : ↑H = {g}\n⊢ Subsingleton ↑↑H","state_after":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\nH✝ K H : Subgroup G\nx✝ : ∃ g, ↑H = {g}\ng : G\nhg : ↑H = {g}\n⊢ Subsingleton ↑{g}","tactic":"rw [hg]","premises":[]},{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\nH✝ K H : Subgroup G\nx✝ : ∃ g, ↑H = {g}\ng : G\nhg : ↑H = {g}\n⊢ Subsingleton ↑{g}","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.zip_inits_tails","start":[619,0],"end":[622,16],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ zip appendStream' s.inits s.tails = const s","state_after":"case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ ∀ (n : ℕ), (zip appendStream' s.inits s.tails).get n = (const s).get n","tactic":"apply Stream'.ext","premises":[{"full_name":"Stream'.ext","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[32,18],"def_end_pos":[32,21]}]},{"state_before":"case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ ∀ (n : ℕ), (zip appendStream' s.inits s.tails).get n = (const s).get n","state_after":"case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn : ℕ\n⊢ (zip appendStream' s.inits s.tails).get n = (const s).get n","tactic":"intro n","premises":[]},{"state_before":"case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn : ℕ\n⊢ (zip appendStream' s.inits s.tails).get n = (const s).get n","state_after":"no goals","tactic":"rw [get_zip, get_inits, get_tails, get_const, take_succ, cons_append_stream, append_take_drop,\n    Stream'.eta]","premises":[{"full_name":"Stream'.append_take_drop","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[517,8],"def_end_pos":[517,24]},{"full_name":"Stream'.cons_append_stream","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[436,8],"def_end_pos":[436,26]},{"full_name":"Stream'.eta","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[28,18],"def_end_pos":[28,21]},{"full_name":"Stream'.get_const","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[210,8],"def_end_pos":[210,17]},{"full_name":"Stream'.get_inits","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[604,8],"def_end_pos":[604,17]},{"full_name":"Stream'.get_tails","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[569,8],"def_end_pos":[569,17]},{"full_name":"Stream'.get_zip","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[168,8],"def_end_pos":[168,15]},{"full_name":"Stream'.take_succ","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[479,8],"def_end_pos":[479,17]}]}]}
{"url":"Mathlib/Analysis/Matrix.lean","commit":"","full_name":"Matrix.frobenius_nnnorm_map_eq","start":[523,0],"end":[525,80],"file_path":"Mathlib/Analysis/Matrix.lean","tactics":[{"state_before":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁵ : Fintype l\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Unique ι\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nA : Matrix m n α\nf : α → β\nhf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊\n⊢ ‖A.map f‖₊ = ‖A‖₊","state_after":"no goals","tactic":"simp_rw [frobenius_nnnorm_def, Matrix.map_apply, hf]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Matrix.frobenius_nnnorm_def","def_path":"Mathlib/Analysis/Matrix.lean","def_pos":[512,8],"def_end_pos":[512,28]},{"full_name":"Matrix.map_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","commit":"","full_name":"CategoryTheory.CostructuredArrow.IsUniversal.existsUnique","start":[719,0],"end":[721,57],"file_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nf✝ g✝ : CostructuredArrow S T\nh : f✝.IsUniversal\ng : CostructuredArrow S T\nf : g.left ⟶ f✝.left\nw : (fun η => S.map η ≫ f✝.hom = g.hom) f\n⊢ S.map f ≫ f✝.hom = S.map (h.lift g) ≫ f✝.hom","state_after":"no goals","tactic":"simp [w]","premises":[]}]}
{"url":"Mathlib/Logic/Equiv/Basic.lean","commit":"","full_name":"Equiv.symm_trans_swap_trans","start":[1437,0],"end":[1445,22],"file_path":"Mathlib/Logic/Equiv/Basic.lean","tactics":[{"state_before":"α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\n⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x","state_after":"α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\nthis : ∀ (a : α), e.symm x = a ↔ x = e a\n⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x","tactic":"have : ∀ a, e.symm x = a ↔ x = e a := fun a => by\n      rw [@eq_comm _ (e.symm x)]\n      constructor <;> intros <;> simp_all","premises":[{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]},{"state_before":"α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\nthis : ∀ (a : α), e.symm x = a ↔ x = e a\n⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x","state_after":"α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\nthis : ∀ (a : α), e.symm x = a ↔ x = e a\n⊢ e (if x = e a then b else if x = e b then a else e.symm x) = if x = e a then e b else if x = e b then e a else x","tactic":"simp only [trans_apply, swap_apply_def, this]","premises":[{"full_name":"Equiv.swap_apply_def","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1394,8],"def_end_pos":[1394,22]},{"full_name":"Equiv.trans_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[237,16],"def_end_pos":[237,27]}]},{"state_before":"α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\nthis : ∀ (a : α), e.symm x = a ↔ x = e a\n⊢ e (if x = e a then b else if x = e b then a else e.symm x) = if x = e a then e b else if x = e b then e a else x","state_after":"no goals","tactic":"split_ifs <;> simp","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"inv_zpow","start":[442,0],"end":[445,61],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na✝ b c d a : α\nn : ℕ\n⊢ a⁻¹ ^ ↑n = (a ^ ↑n)⁻¹","state_after":"no goals","tactic":"rw [zpow_natCast, zpow_natCast, inv_pow]","premises":[{"full_name":"inv_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[418,6],"def_end_pos":[418,13]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na✝ b c d a : α\nn : ℕ\n⊢ a⁻¹ ^ Int.negSucc n = (a ^ Int.negSucc n)⁻¹","state_after":"no goals","tactic":"rw [zpow_negSucc, zpow_negSucc, inv_pow]","premises":[{"full_name":"inv_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[418,6],"def_end_pos":[418,13]},{"full_name":"zpow_negSucc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[890,8],"def_end_pos":[890,20]}]}]}
{"url":"Mathlib/Topology/Order/T5.lean","commit":"","full_name":"Set.ordT5Nhd_mem_nhdsSet","start":[76,0],"end":[81,51],"file_path":"Mathlib/Topology/Order/T5.lean","tactics":[{"state_before":"X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nx : X\nhx : x ∈ s\n⊢ tᶜ ∈ 𝓝 x","state_after":"X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nx : X\nhx : x ∈ s\n⊢ x ∈ (closure t)ᶜ","tactic":"rw [← mem_interior_iff_mem_nhds, interior_compl]","premises":[{"full_name":"interior_compl","def_path":"Mathlib/Topology/Basic.lean","def_pos":[462,8],"def_end_pos":[462,22]},{"full_name":"mem_interior_iff_mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[730,8],"def_end_pos":[730,33]}]},{"state_before":"X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nx : X\nhx : x ∈ s\n⊢ x ∈ (closure t)ᶜ","state_after":"no goals","tactic":"exact disjoint_left.1 hd hx","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.disjoint_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1216,8],"def_end_pos":[1216,21]}]}]}
{"url":"Mathlib/CategoryTheory/Monad/Adjunction.lean","commit":"","full_name":"CategoryTheory.Coreflective.comparison_full","start":[364,0],"end":[366,52],"file_path":"Mathlib/CategoryTheory/Monad/Adjunction.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL✝ : C ⥤ D\nR : D ⥤ C\ninst✝ : R.Full\nL : C ⥤ D\nadj : R ⊣ L\nX✝ Y✝ : D\nf : (Comonad.comparison adj).obj X✝ ⟶ (Comonad.comparison adj).obj Y✝\n⊢ (Comonad.comparison adj).map (R.preimage f.f) = f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean","commit":"","full_name":"mfderivWithin_snd","start":[298,0],"end":[302,93],"file_path":"Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns✝ : Set M\nx✝ : M\ns : Set (M × M')\nx : M × M'\nhxs : UniqueMDiffWithinAt (I.prod I') s x\n⊢ mfderivWithin (I.prod I') I' Prod.snd s x = ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns✝ : Set M\nx✝ : M\ns : Set (M × M')\nx : M × M'\nhxs : UniqueMDiffWithinAt (I.prod I') s x\n⊢ mfderiv (I.prod I') I' Prod.snd x = ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)","tactic":"rw [MDifferentiable.mfderivWithin (mdifferentiableAt_snd I I') hxs]","premises":[{"full_name":"MDifferentiable.mfderivWithin","def_path":"Mathlib/Geometry/Manifold/MFDeriv/Basic.lean","def_pos":[239,8],"def_end_pos":[239,37]},{"full_name":"mdifferentiableAt_snd","def_path":"Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean","def_pos":[279,8],"def_end_pos":[279,29]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\ninst✝¹² : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝⁹ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H' M'\ninst✝⁶ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝⁵ : NormedAddCommGroup E''\ninst✝⁴ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝³ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝² : TopologicalSpace M''\ninst✝¹ : ChartedSpace H'' M''\ninst✝ : SmoothManifoldWithCorners I'' M''\ns✝ : Set M\nx✝ : M\ns : Set (M × M')\nx : M × M'\nhxs : UniqueMDiffWithinAt (I.prod I') s x\n⊢ mfderiv (I.prod I') I' Prod.snd x = ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)","state_after":"no goals","tactic":"exact mfderiv_snd I I'","premises":[{"full_name":"mfderiv_snd","def_path":"Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean","def_pos":[293,8],"def_end_pos":[293,19]}]}]}
{"url":"Mathlib/Data/Fintype/Fin.lean","commit":"","full_name":"Fin.card_filter_univ_succ","start":[61,0],"end":[64,71],"file_path":"Mathlib/Data/Fintype/Fin.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nn : ℕ\np : Fin (n + 1) → Prop\ninst✝ : DecidablePred p\n⊢ (if p 0 then 1 else 0) + (filter (p ∘ succ) univ).card =\n    if p 0 then (filter (p ∘ succ) univ).card + 1 else (filter (p ∘ succ) univ).card","state_after":"no goals","tactic":"split_ifs <;> simp [add_comm 1]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Positive.lean","commit":"","full_name":"ContinuousLinearMap.IsPositive.conj_adjoint","start":[79,0],"end":[83,30],"file_path":"Mathlib/Analysis/InnerProductSpace/Positive.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\n⊢ (S.comp (T.comp (adjoint S))).IsPositive","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\nx : F\n⊢ 0 ≤ (S.comp (T.comp (adjoint S))).reApplyInnerSelf x","tactic":"refine ⟨hT.isSelfAdjoint.conj_adjoint S, fun x => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"ContinuousLinearMap.IsPositive.isSelfAdjoint","def_path":"Mathlib/Analysis/InnerProductSpace/Positive.lean","def_pos":[55,8],"def_end_pos":[55,32]},{"full_name":"IsSelfAdjoint.conj_adjoint","def_path":"Mathlib/Analysis/InnerProductSpace/Adjoint.lean","def_pos":[247,8],"def_end_pos":[247,20]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\nx : F\n⊢ 0 ≤ (S.comp (T.comp (adjoint S))).reApplyInnerSelf x","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\nx : F\n⊢ 0 ≤ re ⟪(T.comp (adjoint S)) x, (adjoint S) x⟫_𝕜","tactic":"rw [reApplyInnerSelf, comp_apply, ← adjoint_inner_right]","premises":[{"full_name":"ContinuousLinearMap.adjoint_inner_right","def_path":"Mathlib/Analysis/InnerProductSpace/Adjoint.lean","def_pos":[115,8],"def_end_pos":[115,27]},{"full_name":"ContinuousLinearMap.comp_apply","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[702,8],"def_end_pos":[702,18]},{"full_name":"ContinuousLinearMap.reApplyInnerSelf","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[2022,4],"def_end_pos":[2022,40]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\nx : F\n⊢ 0 ≤ re ⟪(T.comp (adjoint S)) x, (adjoint S) x⟫_𝕜","state_after":"no goals","tactic":"exact hT.inner_nonneg_left _","premises":[{"full_name":"ContinuousLinearMap.IsPositive.inner_nonneg_left","def_path":"Mathlib/Analysis/InnerProductSpace/Positive.lean","def_pos":[58,8],"def_end_pos":[58,36]}]}]}
{"url":"Mathlib/Topology/Bases.lean","commit":"","full_name":"QuotientMap.secondCountableTopology","start":[940,0],"end":[946,53],"file_path":"Mathlib/Topology/Bases.lean","tactics":[{"state_before":"α : Type u\nt : TopologicalSpace α\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nπ : X → Y\ninst✝ : SecondCountableTopology X\nh' : QuotientMap π\nh : IsOpenMap π\n⊢ ∃ b, b.Countable ∧ inst✝¹ = generateFrom b","state_after":"case intro.intro.intro\nα : Type u\nt : TopologicalSpace α\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nπ : X → Y\ninst✝ : SecondCountableTopology X\nh' : QuotientMap π\nh : IsOpenMap π\nV : Set (Set X)\nV_countable : V.Countable\nV_generates : IsTopologicalBasis V\n⊢ ∃ b, b.Countable ∧ inst✝¹ = generateFrom b","tactic":"obtain ⟨V, V_countable, -, V_generates⟩ := exists_countable_basis X","premises":[{"full_name":"TopologicalSpace.exists_countable_basis","def_path":"Mathlib/Topology/Bases.lean","def_pos":[724,8],"def_end_pos":[724,30]}]},{"state_before":"case intro.intro.intro\nα : Type u\nt : TopologicalSpace α\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nπ : X → Y\ninst✝ : SecondCountableTopology X\nh' : QuotientMap π\nh : IsOpenMap π\nV : Set (Set X)\nV_countable : V.Countable\nV_generates : IsTopologicalBasis V\n⊢ ∃ b, b.Countable ∧ inst✝¹ = generateFrom b","state_after":"no goals","tactic":"exact ⟨Set.image π '' V, V_countable.image (Set.image π),\n      (V_generates.quotientMap h' h).eq_generateFrom⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.Countable.image","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[149,8],"def_end_pos":[149,23]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"TopologicalSpace.IsTopologicalBasis.eq_generateFrom","def_path":"Mathlib/Topology/Bases.lean","def_pos":[73,2],"def_end_pos":[73,17]},{"full_name":"TopologicalSpace.IsTopologicalBasis.quotientMap","def_path":"Mathlib/Topology/Bases.lean","def_pos":[926,8],"def_end_pos":[926,38]}]}]}
{"url":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","commit":"","full_name":"measurable_iSup","start":[724,0],"end":[740,12],"file_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\n⊢ Measurable fun b => ⨆ i, f i b","state_after":"case inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : IsEmpty ι\n⊢ Measurable fun b => ⨆ i, f i b\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\n⊢ Measurable fun b => ⨆ i, f i b","tactic":"rcases isEmpty_or_nonempty ι with hι|hι","premises":[{"full_name":"isEmpty_or_nonempty","def_path":"Mathlib/Logic/IsEmpty.lean","def_pos":[195,8],"def_end_pos":[195,27]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\n⊢ Measurable fun b => ⨆ i, f i b","state_after":"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\n⊢ Measurable fun b => ⨆ i, f i b","tactic":"have A : MeasurableSet {b | BddAbove (range (fun i ↦ f i b))} :=\n    measurableSet_bddAbove_range hf","premises":[{"full_name":"BddAbove","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[52,4],"def_end_pos":[52,12]},{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"measurableSet_bddAbove_range","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[669,6],"def_end_pos":[669,34]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\n⊢ Measurable fun b => ⨆ i, f i b","state_after":"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup ∅\n⊢ Measurable fun b => ⨆ i, f i b","tactic":"have : Measurable (fun (_b : δ) ↦ sSup (∅ : Set α)) := measurable_const","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"SupSet.sSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[42,2],"def_end_pos":[42,6]},{"full_name":"measurable_const","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[510,8],"def_end_pos":[510,24]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup ∅\n⊢ Measurable fun b => ⨆ i, f i b","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup ∅\n⊢ ∀ b ∈ {b | BddAbove (range fun i => f i b)}, IsLUB {a | ∃ i, f i b = a} (⨆ i, f i b)\n\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhι : Nonempty ι\nA : MeasurableSet {b | BddAbove (range fun i => f i b)}\nthis : Measurable fun _b => sSup ∅\n⊢ EqOn (fun b => ⨆ i, f i b) (fun _b => sSup ∅) {b | BddAbove (range fun i => f i b)}ᶜ","tactic":"apply Measurable.isLUB_of_mem hf A _ _ this","premises":[{"full_name":"Measurable.isLUB_of_mem","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[527,8],"def_end_pos":[527,31]}]}]}
{"url":"Mathlib/FieldTheory/Adjoin.lean","commit":"","full_name":"IntermediateField.sInf_toSubalgebra","start":[148,0],"end":[151,53],"file_path":"Mathlib/FieldTheory/Adjoin.lean","tactics":[{"state_before":"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\n⊢ ↑(sInf S).toSubalgebra = ↑(sInf (toSubalgebra '' S))","state_after":"no goals","tactic":"simp [Set.sUnion_image]","premises":[{"full_name":"Set.sUnion_image","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[971,8],"def_end_pos":[971,20]}]}]}
{"url":"Mathlib/Data/Set/Function.lean","commit":"","full_name":"Set.MapsTo.piecewise_ite","start":[1413,0],"end":[1419,64],"file_path":"Mathlib/Data/Set/Function.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nδ : α → Sort u_6\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ MapsTo (s.piecewise f₁ f₂) (s.ite s₁ s₂) (t.ite t₁ t₂)","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nδ : α → Sort u_6\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₁ (s.piecewise f₁ f₂) (s₁ ∩ s)\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nδ : α → Sort u_6\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₂ (s.piecewise f₁ f₂) (s₂ ∩ sᶜ)","tactic":"refine (h₁.congr ?_).union_union (h₂.congr ?_)","premises":[{"full_name":"Set.MapsTo.congr","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[360,8],"def_end_pos":[360,20]},{"full_name":"Set.MapsTo.union_union","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[402,8],"def_end_pos":[402,26]}]},{"state_before":"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nδ : α → Sort u_6\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₁ (s.piecewise f₁ f₂) (s₁ ∩ s)\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nδ : α → Sort u_6\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₂ (s.piecewise f₁ f₂) (s₂ ∩ sᶜ)","state_after":"no goals","tactic":"exacts [(piecewise_eqOn s f₁ f₂).symm.mono inter_subset_right,\n    (piecewise_eqOn_compl s f₁ f₂).symm.mono inter_subset_right]","premises":[{"full_name":"Set.EqOn.mono","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[199,8],"def_end_pos":[199,17]},{"full_name":"Set.EqOn.symm","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[172,8],"def_end_pos":[172,17]},{"full_name":"Set.inter_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[764,8],"def_end_pos":[764,26]},{"full_name":"Set.piecewise_eqOn","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[1380,8],"def_end_pos":[1380,22]},{"full_name":"Set.piecewise_eqOn_compl","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[1383,8],"def_end_pos":[1383,28]}]}]}
{"url":"Mathlib/Analysis/BoxIntegral/Partition/Split.lean","commit":"","full_name":"BoxIntegral.Prepartition.split_of_not_mem_Ioo","start":[181,0],"end":[189,41],"file_path":"Mathlib/Analysis/BoxIntegral/Partition/Split.lean","tactics":[{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\nh : x ∉ Ioo (I.lower i) (I.upper i)\n⊢ split I i x = ⊤","state_after":"ι : Type u_1\nM : Type u_2\nn : ℕ\nI J✝ : Box ι\ni : ι\nx : ℝ\nh : x ∉ Ioo (I.lower i) (I.upper i)\nJ : Box ι\nhJ : J ∈ ⊤.boxes\n⊢ J ∈ (split I i x).boxes","tactic":"refine ((isPartitionTop I).eq_of_boxes_subset fun J hJ => ?_).symm","premises":[{"full_name":"BoxIntegral.Prepartition.IsPartition.eq_of_boxes_subset","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[657,8],"def_end_pos":[657,26]},{"full_name":"BoxIntegral.Prepartition.isPartitionTop","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[635,8],"def_end_pos":[635,22]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]},{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nI J✝ : Box ι\ni : ι\nx : ℝ\nh : x ∉ Ioo (I.lower i) (I.upper i)\nJ : Box ι\nhJ : J ∈ ⊤.boxes\n⊢ J ∈ (split I i x).boxes","state_after":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ∉ Ioo (J.lower i) (J.upper i)\nhJ : J ∈ ⊤.boxes\n⊢ J ∈ (split J i x).boxes","tactic":"rcases mem_top.1 hJ with rfl","premises":[{"full_name":"BoxIntegral.Prepartition.mem_top","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[148,8],"def_end_pos":[148,15]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ∉ Ioo (J.lower i) (J.upper i)\nhJ : J ∈ ⊤.boxes\n⊢ J ∈ (split J i x).boxes","state_after":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ∉ Ioo (J.lower i) (J.upper i)\n⊢ J ∈ (split J i x).boxes","tactic":"clear hJ","premises":[]},{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ∉ Ioo (J.lower i) (J.upper i)\n⊢ J ∈ (split J i x).boxes","state_after":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ∉ Ioo (J.lower i) (J.upper i)\n⊢ ↑J = J.splitLower i x ∨ ↑J = J.splitUpper i x","tactic":"rw [mem_boxes, mem_split_iff]","premises":[{"full_name":"BoxIntegral.Prepartition.mem_boxes","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[69,8],"def_end_pos":[69,17]},{"full_name":"BoxIntegral.Prepartition.mem_split_iff","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Split.lean","def_pos":[161,8],"def_end_pos":[161,21]}]},{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ∉ Ioo (J.lower i) (J.upper i)\n⊢ ↑J = J.splitLower i x ∨ ↑J = J.splitUpper i x","state_after":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ≤ J.lower i ∨ J.upper i ≤ x\n⊢ ↑J = J.splitLower i x ∨ ↑J = J.splitUpper i x","tactic":"rw [mem_Ioo, not_and_or, not_lt, not_lt] at h","premises":[{"full_name":"Set.mem_Ioo","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[98,8],"def_end_pos":[98,15]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"ι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh : x ≤ J.lower i ∨ J.upper i ≤ x\n⊢ ↑J = J.splitLower i x ∨ ↑J = J.splitUpper i x","state_after":"case inl.h\nι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh✝ : x ≤ J.lower i\n⊢ ↑J = J.splitUpper i x\n\ncase inr.h\nι : Type u_1\nM : Type u_2\nn : ℕ\nJ✝ : Box ι\ni : ι\nx : ℝ\nJ : Box ι\nh✝ : J.upper i ≤ x\n⊢ ↑J = J.splitLower i x","tactic":"cases h <;> [right; left]","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"intervalIntegrable_iff'","start":[92,0],"end":[94,89],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\na b : ℝ\nμ : Measure ℝ\ninst✝ : NoAtoms μ\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f [[a, b]] μ","state_after":"no goals","tactic":"rw [intervalIntegrable_iff, ← Icc_min_max, uIoc, integrableOn_Icc_iff_integrableOn_Ioc]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.Icc_min_max","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[192,8],"def_end_pos":[192,19]},{"full_name":"Set.uIoc","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[238,4],"def_end_pos":[238,8]},{"full_name":"integrableOn_Icc_iff_integrableOn_Ioc","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[668,8],"def_end_pos":[668,45]},{"full_name":"intervalIntegrable_iff","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[80,8],"def_end_pos":[80,30]}]}]}
{"url":"Mathlib/Algebra/Order/ToIntervalMod.lean","commit":"","full_name":"toIocMod_add_toIcoMod_zero","start":[658,0],"end":[660,50],"file_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocMod hp 0 (a - b) + toIcoMod hp 0 (b - a) = p","state_after":"no goals","tactic":"rw [_root_.add_comm, toIcoMod_add_toIocMod_zero]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"toIcoMod_add_toIocMod_zero","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[654,8],"def_end_pos":[654,34]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean","commit":"","full_name":"CategoryTheory.Functor.mem_homologicalKernel_iff","start":[115,0],"end":[118,5],"file_path":"Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝¹⁶ : Category.{u_4, u_1} C\ninst✝¹⁵ : HasZeroObject C\ninst✝¹⁴ : HasShift C ℤ\ninst✝¹³ : Preadditive C\ninst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹¹ : Pretriangulated C\ninst✝¹⁰ : Category.{?u.18320, u_2} D\ninst✝⁹ : HasZeroObject D\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : Preadditive D\ninst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive\ninst✝⁵ : Pretriangulated D\ninst✝⁴ : Category.{u_5, u_3} A\ninst✝³ : Abelian A\nF : C ⥤ A\ninst✝² inst✝¹ : F.IsHomological\ninst✝ : F.ShiftSequence ℤ\nX : C\n⊢ F.homologicalKernel.P X ↔ ∀ (n : ℤ), IsZero ((F.shift n).obj X)","state_after":"C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝¹⁶ : Category.{u_4, u_1} C\ninst✝¹⁵ : HasZeroObject C\ninst✝¹⁴ : HasShift C ℤ\ninst✝¹³ : Preadditive C\ninst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹¹ : Pretriangulated C\ninst✝¹⁰ : Category.{?u.18320, u_2} D\ninst✝⁹ : HasZeroObject D\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : Preadditive D\ninst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive\ninst✝⁵ : Pretriangulated D\ninst✝⁴ : Category.{u_5, u_3} A\ninst✝³ : Abelian A\nF : C ⥤ A\ninst✝² inst✝¹ : F.IsHomological\ninst✝ : F.ShiftSequence ℤ\nX : C\n⊢ F.homologicalKernel.P X ↔ ∀ (n : ℤ), IsZero ((shiftFunctor C n ⋙ F).obj X)","tactic":"simp only [← fun (n : ℤ) => Iso.isZero_iff ((F.isoShift n).app X)]","premises":[{"full_name":"CategoryTheory.Functor.isoShift","def_path":"Mathlib/CategoryTheory/Shift/ShiftSequence.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"CategoryTheory.Iso.app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[51,4],"def_end_pos":[51,7]},{"full_name":"CategoryTheory.Iso.isZero_iff","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean","def_pos":[123,8],"def_end_pos":[123,22]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]}]},{"state_before":"C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝¹⁶ : Category.{u_4, u_1} C\ninst✝¹⁵ : HasZeroObject C\ninst✝¹⁴ : HasShift C ℤ\ninst✝¹³ : Preadditive C\ninst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹¹ : Pretriangulated C\ninst✝¹⁰ : Category.{?u.18320, u_2} D\ninst✝⁹ : HasZeroObject D\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : Preadditive D\ninst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive\ninst✝⁵ : Pretriangulated D\ninst✝⁴ : Category.{u_5, u_3} A\ninst✝³ : Abelian A\nF : C ⥤ A\ninst✝² inst✝¹ : F.IsHomological\ninst✝ : F.ShiftSequence ℤ\nX : C\n⊢ F.homologicalKernel.P X ↔ ∀ (n : ℤ), IsZero ((shiftFunctor C n ⋙ F).obj X)","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Analysis/Normed/Module/Ray.lean","commit":"","full_name":"SameRay.norm_smul_eq","start":[45,0],"end":[48,46],"file_path":"Mathlib/Analysis/Normed/Module/Ray.lean","tactics":[{"state_before":"E : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : E\nh : SameRay ℝ x y\n⊢ ‖x‖ • y = ‖y‖ • x","state_after":"case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nu : E\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nh : SameRay ℝ (a • u) (b • u)\n⊢ ‖a • u‖ • b • u = ‖b • u‖ • a • u","tactic":"rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩","premises":[{"full_name":"SameRay.exists_eq_smul","def_path":"Mathlib/LinearAlgebra/Ray.lean","def_pos":[609,8],"def_end_pos":[609,22]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nu : E\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nh : SameRay ℝ (a • u) (b • u)\n⊢ ‖a • u‖ • b • u = ‖b • u‖ • a • u","state_after":"case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nu : E\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nh : SameRay ℝ (a • u) (b • u)\n⊢ a • ‖u‖ • b • u = b • ‖u‖ • a • u","tactic":"simp only [norm_smul_of_nonneg, *, mul_smul]","premises":[{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"norm_smul_of_nonneg","def_path":"Mathlib/Analysis/NormedSpace/Real.lean","def_pos":[42,8],"def_end_pos":[42,27]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nu : E\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nh : SameRay ℝ (a • u) (b • u)\n⊢ a • ‖u‖ • b • u = b • ‖u‖ • a • u","state_after":"no goals","tactic":"rw [smul_comm, smul_comm b, smul_comm a b u]","premises":[{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]}]}]}
{"url":"Mathlib/RingTheory/PrincipalIdealDomain.lean","commit":"","full_name":"gcd_isUnit_iff","start":[402,0],"end":[403,92],"file_path":"Mathlib/RingTheory/PrincipalIdealDomain.lean","tactics":[{"state_before":"R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsBezout R\ninst✝ : GCDMonoid R\nx y : R\n⊢ IsUnit (gcd x y) ↔ IsCoprime x y","state_after":"no goals","tactic":"rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]","premises":[{"full_name":"Ideal.eq_top_iff_one","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[87,8],"def_end_pos":[87,22]},{"full_name":"Ideal.mem_span_pair","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[312,8],"def_end_pos":[312,21]},{"full_name":"Ideal.span_singleton_eq_top","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[448,8],"def_end_pos":[448,29]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"IsCoprime","def_path":"Mathlib/RingTheory/Coprime/Basic.lean","def_pos":[38,4],"def_end_pos":[38,13]},{"full_name":"span_gcd","def_path":"Mathlib/RingTheory/PrincipalIdealDomain.lean","def_pos":[391,8],"def_end_pos":[391,16]}]}]}
{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"accPt_iff_frequently","start":[981,0],"end":[984,80],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx✝ : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\nx : X\nC : Set X\n⊢ AccPt x (𝓟 C) ↔ ∃ᶠ (y : X) in 𝓝 x, y ≠ x ∧ y ∈ C","state_after":"no goals","tactic":"simp [acc_principal_iff_cluster, clusterPt_principal_iff_frequently, and_comm]","premises":[{"full_name":"acc_principal_iff_cluster","def_path":"Mathlib/Topology/Basic.lean","def_pos":[971,8],"def_end_pos":[971,33]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"clusterPt_principal_iff_frequently","def_path":"Mathlib/Topology/Basic.lean","def_pos":[906,8],"def_end_pos":[906,42]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.ofFinsupp_add","start":[148,0],"end":[150,34],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na✝ b✝ : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\na b : R[ℕ]\n⊢ { toFinsupp := a + b } = Polynomial.add { toFinsupp := a } { toFinsupp := b }","state_after":"no goals","tactic":"rw [add_def]","premises":[{"full_name":"_private.Mathlib.Algebra.Polynomial.Basic.0.Polynomial.add_def","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[102,24],"def_end_pos":[102,27]}]}]}
{"url":"Mathlib/Data/Part.lean","commit":"","full_name":"Part.append_mem_append","start":[668,0],"end":[669,52],"file_path":"Mathlib/Data/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ma ++ mb ∈ a ++ b","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1 ∈ a, ∃ a ∈ b, a_1 ++ a = ma ++ mb","tactic":"simp [append_def]","premises":[{"full_name":"Part.append_def","def_path":"Mathlib/Data/Part.lean","def_pos":[600,8],"def_end_pos":[600,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1 ∈ a, ∃ a ∈ b, a_1 ++ a = ma ++ mb","state_after":"no goals","tactic":"aesop","premises":[]}]}
{"url":"Mathlib/Computability/Primrec.lean","commit":"","full_name":"Primrec.of_equiv_symm","start":[244,0],"end":[248,85],"file_path":"Mathlib/Computability/Primrec.lean","tactics":[{"state_before":"α : Type u_1\nβ✝ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β✝\ninst✝ : Primcodable σ\nβ : Type u_4\ne : β ≃ α\nthis : Primcodable β := Primcodable.ofEquiv α e\n⊢ Primrec fun a => encode (e (e.symm a))","state_after":"no goals","tactic":"simp [Primrec.encode]","premises":[{"full_name":"Primrec.encode","def_path":"Mathlib/Computability/Primrec.lean","def_pos":[184,18],"def_end_pos":[184,24]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/IntegralNormalization.lean","commit":"","full_name":"Polynomial.integralNormalization_coeff","start":[44,0],"end":[49,20],"file_path":"Mathlib/RingTheory/Polynomial/IntegralNormalization.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\ni : ℕ\n⊢ f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)","state_after":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\ni : ℕ\nthis : f.coeff i = 0 → f.degree ≠ ↑i\n⊢ f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)","tactic":"have : f.coeff i = 0 → f.degree ≠ i := fun hc hd => coeff_ne_zero_of_eq_degree hd hc","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.coeff_ne_zero_of_eq_degree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[370,8],"def_end_pos":[370,34]},{"full_name":"Polynomial.degree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[48,4],"def_end_pos":[48,10]}]},{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\ni : ℕ\nthis : f.coeff i = 0 → f.degree ≠ ↑i\n⊢ f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)","state_after":"no goals","tactic":"simp (config := { contextual := true }) [integralNormalization, coeff_monomial, this,\n    mem_support_iff]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Polynomial.coeff_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[574,8],"def_end_pos":[574,22]},{"full_name":"Polynomial.integralNormalization","def_path":"Mathlib/RingTheory/Polynomial/IntegralNormalization.lean","def_pos":[36,18],"def_end_pos":[36,39]},{"full_name":"Polynomial.mem_support_iff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[607,8],"def_end_pos":[607,23]}]}]}
{"url":"Mathlib/CategoryTheory/Square.lean","commit":"","full_name":"CategoryTheory.Square.toArrowArrowFunctor_map_left_right","start":[170,0],"end":[177,51],"file_path":"Mathlib/CategoryTheory/Square.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : Square C\nφ : X✝ ⟶ Y✝\n⊢ Arrow.homMk ⋯ ≫ ((fun sq => Arrow.mk (Arrow.homMk ⋯)) Y✝).hom =\n    ((fun sq => Arrow.mk (Arrow.homMk ⋯)) X✝).hom ≫ Arrow.homMk ⋯","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","commit":"","full_name":"Sum.getRight?_swap","start":[169,0],"end":[169,96],"file_path":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nx : α ⊕ β\n⊢ x.swap.getRight? = x.getLeft?","state_after":"no goals","tactic":"cases x <;> rfl","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Comon_.lean","commit":"","full_name":"Comon_.Mon_OpOpToComon_obj'_X","start":[206,0],"end":[218,7],"file_path":"Mathlib/CategoryTheory/Monoidal/Comon_.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ A.mul.unop ≫ A.one.unop ▷ unop A.X = (λ_ (unop A.X)).inv","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ (λ_ A.X).hom.unop = (λ_ (unop A.X)).inv","tactic":"rw [← unop_whiskerRight, ← unop_comp, Mon_.one_mul]","premises":[{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"CategoryTheory.unop_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Opposite.lean","def_pos":[186,14],"def_end_pos":[186,31]},{"full_name":"Mon_.one_mul","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[36,2],"def_end_pos":[36,9]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ (λ_ A.X).hom.unop = (λ_ (unop A.X)).inv","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ A.mul.unop ≫ unop A.X ◁ A.one.unop = (ρ_ (unop A.X)).inv","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ (ρ_ A.X).hom.unop = (ρ_ (unop A.X)).inv","tactic":"rw [← unop_whiskerLeft, ← unop_comp, Mon_.mul_one]","premises":[{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"CategoryTheory.unop_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Opposite.lean","def_pos":[181,14],"def_end_pos":[181,30]},{"full_name":"Mon_.mul_one","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[37,2],"def_end_pos":[37,9]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ (ρ_ A.X).hom.unop = (ρ_ (unop A.X)).inv","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ A.mul.unop ≫ unop A.X ◁ A.mul.unop = A.mul.unop ≫ A.mul.unop ▷ unop A.X ≫ (α_ (unop A.X) (unop A.X) (unop A.X)).hom","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ ((α_ A.X A.X A.X).inv ≫ A.mul ▷ A.X ≫ A.mul).unop =\n    (A.mul ▷ A.X ≫ A.mul).unop ≫ (α_ (unop A.X) (unop A.X) (unop A.X)).hom","tactic":"rw [← unop_whiskerRight, ← unop_whiskerLeft, ← unop_comp_assoc, ← unop_comp,\n      Mon_.mul_assoc_flip]","premises":[{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"CategoryTheory.unop_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Opposite.lean","def_pos":[181,14],"def_end_pos":[181,30]},{"full_name":"CategoryTheory.unop_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Opposite.lean","def_pos":[186,14],"def_end_pos":[186,31]},{"full_name":"Mon_.mul_assoc_flip","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[77,8],"def_end_pos":[77,22]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ ((α_ A.X A.X A.X).inv ≫ A.mul ▷ A.X ≫ A.mul).unop =\n    (A.mul ▷ A.X ≫ A.mul).unop ≫ (α_ (unop A.X) (unop A.X) (unop A.X)).hom","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","commit":"","full_name":"GromovHausdorff.HD_candidatesBDist_le","start":[310,0],"end":[332,22],"file_path":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ HD (candidatesBDist X Y) ≤ diam univ + 1 + diam univ","state_after":"case refine_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nx : X\n⊢ ⨅ y, (candidatesBDist X Y) (inl x, inr y) ≤ diam univ + 1 + diam univ\n\ncase refine_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y✝ z t : X ⊕ Y\ny : Y\n⊢ ⨅ x, (candidatesBDist X Y) (inl x, inr y) ≤ diam univ + 1 + diam univ","tactic":"refine max_le (ciSup_le fun x => ?_) (ciSup_le fun y => ?_)","premises":[{"full_name":"ciSup_le","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[703,8],"def_end_pos":[703,16]},{"full_name":"max_le","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[66,8],"def_end_pos":[66,14]}]}]}
{"url":"Mathlib/LinearAlgebra/FiniteDimensional.lean","commit":"","full_name":"Submodule.finrank_add_finrank_le_of_disjoint","start":[76,0],"end":[80,30],"file_path":"Mathlib/LinearAlgebra/FiniteDimensional.lean","tactics":[{"state_before":"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns t : Submodule K V\nhdisjoint : Disjoint s t\n⊢ finrank K ↥s + finrank K ↥t ≤ finrank K V","state_after":"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns t : Submodule K V\nhdisjoint : Disjoint s t\n⊢ finrank K ↥(s ⊔ t) ≤ finrank K V","tactic":"rw [← Submodule.finrank_sup_add_finrank_inf_eq s t, hdisjoint.eq_bot, finrank_bot, add_zero]","premises":[{"full_name":"Disjoint.eq_bot","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[122,8],"def_end_pos":[122,23]},{"full_name":"Submodule.finrank_sup_add_finrank_inf_eq","def_path":"Mathlib/LinearAlgebra/FiniteDimensional.lean","def_pos":[52,8],"def_end_pos":[52,38]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"finrank_bot","def_path":"Mathlib/LinearAlgebra/Dimension/Finite.lean","def_pos":[370,8],"def_end_pos":[370,19]}]},{"state_before":"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns t : Submodule K V\nhdisjoint : Disjoint s t\n⊢ finrank K ↥(s ⊔ t) ≤ finrank K V","state_after":"no goals","tactic":"exact Submodule.finrank_le _","premises":[{"full_name":"Submodule.finrank_le","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[374,8],"def_end_pos":[374,28]}]}]}
{"url":"Mathlib/Combinatorics/Additive/ETransform.lean","commit":"","full_name":"Finset.mulETransformRight.card","start":[138,0],"end":[141,76],"file_path":"Mathlib/Combinatorics/Additive/ETransform.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Group α\ne : α\nx : Finset α × Finset α\n⊢ x.2.card + (e⁻¹ • x.2).card = 2 * x.2.card","state_after":"no goals","tactic":"rw [card_smul_finset, two_mul]","premises":[{"full_name":"Finset.card_smul_finset","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[1669,8],"def_end_pos":[1669,24]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]}]}]}
{"url":"Mathlib/SetTheory/Game/Nim.lean","commit":"","full_name":"SetTheory.PGame.moveRight_nim","start":[104,0],"end":[104,98],"file_path":"Mathlib/SetTheory/Game/Nim.lean","tactics":[{"state_before":"o : Ordinal.{u_1}\ni : ↑(Set.Iio o)\n⊢ (nim o).moveRight (toRightMovesNim i) = nim ↑i","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","commit":"","full_name":"SimpleGraph.CliqueFreeOn.of_succ","start":[464,0],"end":[469,69],"file_path":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\ns s₁ s₂ : Set α\nt : Finset α\na b : α\nm n : ℕ\nhs : G.CliqueFreeOn s (n + 1)\nha : a ∈ s\n⊢ G.CliqueFreeOn (s ∩ G.neighborSet a) n","state_after":"no goals","tactic":"classical\n  refine fun t hts ht => hs ?_ (ht.insert fun b hb => (hts hb).2)\n  push_cast\n  exact Set.insert_subset_iff.2 ⟨ha, hts.trans Set.inter_subset_left⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.insert_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[912,8],"def_end_pos":[912,25]},{"full_name":"Set.inter_subset_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[761,8],"def_end_pos":[761,25]},{"full_name":"SimpleGraph.IsNClique.insert","def_path":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","def_pos":[220,8],"def_end_pos":[220,24]}]}]}
{"url":"Mathlib/Order/SuccPred/Basic.lean","commit":"","full_name":"Order.Ico_pred_right_eq_insert","start":[757,0],"end":[758,82],"file_path":"Mathlib/Order/SuccPred/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\na b : α\ninst✝ : NoMinOrder α\nh : a ≤ b\n⊢ Ioc (pred a) b = insert a (Ioc a b)","state_after":"no goals","tactic":"simp_rw [← Ioi_inter_Iic, Ioi_pred_eq_insert, insert_inter_of_mem (mem_Iic.2 h)]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Order.Ioi_pred_eq_insert","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[754,8],"def_end_pos":[754,26]},{"full_name":"Set.Ioi_inter_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[512,8],"def_end_pos":[512,21]},{"full_name":"Set.insert_inter_of_mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1583,8],"def_end_pos":[1583,27]},{"full_name":"Set.mem_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[114,8],"def_end_pos":[114,15]}]}]}
{"url":"Mathlib/CategoryTheory/Iso.lean","commit":"","full_name":"CategoryTheory.Iso.cancel_iso_inv_right_assoc","start":[499,0],"end":[502,43],"file_path":"Mathlib/CategoryTheory/Iso.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y✝ Z✝ W X X' Y Z : C\nf : W ⟶ X\ng : X ⟶ Y\nf' : W ⟶ X'\ng' : X' ⟶ Y\nh : Z ≅ Y\n⊢ f ≫ g ≫ h.inv = f' ≫ g' ≫ h.inv ↔ f ≫ g = f' ≫ g'","state_after":"no goals","tactic":"simp only [← Category.assoc, cancel_mono]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Kuratowski.lean","commit":"","full_name":"LipschitzOnWith.extend_lp_infty","start":[120,0],"end":[153,24],"file_path":"Mathlib/Topology/MetricSpace/Kuratowski.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : LipschitzOnWith K f s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","tactic":"rw [LipschitzOnWith.coordinate] at hfl","premises":[{"full_name":"LipschitzOnWith.coordinate","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[1127,8],"def_end_pos":[1127,34]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\nthis : ∀ (i : ι), ∃ g, LipschitzWith K g ∧ EqOn (fun x => ↑(f x) i) g s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","tactic":"have (i : ι) : ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s :=\n    LipschitzOnWith.extend_real (hfl i)","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"LipschitzOnWith.extend_real","def_path":"Mathlib/Topology/MetricSpace/Lipschitz.lean","def_pos":[331,8],"def_end_pos":[331,35]},{"full_name":"LipschitzWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[54,4],"def_end_pos":[54,17]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.EqOn","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[229,4],"def_end_pos":[229,8]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\nthis : ∀ (i : ι), ∃ g, LipschitzWith K g ∧ EqOn (fun x => ↑(f x) i) g s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\ng : ι → α → ℝ\nhgl : ∀ (i : ι), LipschitzWith K (g i)\nhgeq : ∀ (i : ι), EqOn (fun x => ↑(f x) i) (g i) s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","tactic":"choose g hgl hgeq using this","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\ng : ι → α → ℝ\nhgl : ∀ (i : ι), LipschitzWith K (g i)\nhgeq : ∀ (i : ι), EqOn (fun x => ↑(f x) i) (g i) s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","state_after":"case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\ng : ι → α → ℝ\nhgl : ∀ (i : ι), LipschitzWith K (g i)\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) ∅\nhgeq : ∀ (i : ι), EqOn (fun x => ↑(f x) i) (g i) ∅\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g ∅\n\ncase inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Type u_1\nf : α → ↥(lp (fun i => ℝ) ⊤)\nK : ℝ≥0\nhfl : ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s\ng : ι → α → ℝ\nhgl : ∀ (i : ι), LipschitzWith K (g i)\nhgeq : ∀ (i : ι), EqOn (fun x => ↑(f x) i) (g i) s\na₀ : α\nha₀_in_s : a₀ ∈ s\n⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s","tactic":"rcases s.eq_empty_or_nonempty with rfl | ⟨a₀, ha₀_in_s⟩","premises":[{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]}]}]}
{"url":"Mathlib/Topology/Instances/Int.lean","commit":"","full_name":"Int.ball_eq_Ioo","start":[54,0],"end":[55,54],"file_path":"Mathlib/Topology/Instances/Int.lean","tactics":[{"state_before":"x : ℤ\nr : ℝ\n⊢ ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉","state_after":"no goals","tactic":"rw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]","premises":[{"full_name":"Int.preimage_Ioo","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[1194,8],"def_end_pos":[1194,20]},{"full_name":"Int.preimage_ball","def_path":"Mathlib/Topology/Instances/Int.lean","def_pos":[50,8],"def_end_pos":[50,21]},{"full_name":"Real.ball_eq_Ioo","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[1162,8],"def_end_pos":[1162,24]}]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Normed.lean","commit":"","full_name":"summable_powerSeries_of_norm_lt_one","start":[603,0],"end":[607,71],"file_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nz : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i\nhz : ‖z‖ < 1\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, f i * 1 ^ i","state_after":"no goals","tactic":"simp [h]","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nz : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i\nhz : ‖z‖ < 1\n⊢ ‖z‖ < ‖1‖","state_after":"no goals","tactic":"simp [hz]","premises":[]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Basic.lean","commit":"","full_name":"NNReal.tendsto_const_div_atTop_nhds_zero_nat","start":[55,0],"end":[57,79],"file_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nC : ℝ≥0\n⊢ Tendsto (fun n => C / ↑n) atTop (𝓝 0)","state_after":"no goals","tactic":"simpa using tendsto_const_nhds.mul NNReal.tendsto_inverse_atTop_nhds_zero_nat","premises":[{"full_name":"Filter.Tendsto.mul","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[116,8],"def_end_pos":[116,26]},{"full_name":"NNReal.tendsto_inverse_atTop_nhds_zero_nat","def_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","def_pos":[48,8],"def_end_pos":[48,50]},{"full_name":"tendsto_const_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[844,8],"def_end_pos":[844,26]}]}]}
{"url":"Mathlib/Algebra/Order/Module/Defs.lean","commit":"","full_name":"le_inv_smul_iff_of_pos","start":[740,0],"end":[741,99],"file_path":"Mathlib/Algebra/Order/Module/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁵ : GroupWithZero α\ninst✝⁴ : Preorder α\ninst✝³ : Preorder β\ninst✝² : MulAction α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : 0 < a\n⊢ b₁ ≤ a⁻¹ • b₂ ↔ a • b₁ ≤ b₂","state_after":"no goals","tactic":"rw [← smul_le_smul_iff_of_pos_left ha, smul_inv_smul₀ ha.ne']","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"smul_inv_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[39,8],"def_end_pos":[39,22]},{"full_name":"smul_le_smul_iff_of_pos_left","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[288,6],"def_end_pos":[288,34]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","commit":"","full_name":"EisensteinSeries.eisensteinSeries_tendstoLocallyUniformlyOn","start":[183,0],"end":[194,31],"file_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","tactics":[{"state_before":"z : ℍ\nk : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ TendstoLocallyUniformlyOn (fun s => (fun z => ∑ x ∈ s, eisSummand k (↑x) z) ∘ ↑ofComplex)\n    ((eisensteinSeries_SIF a k).toFun ∘ ↑ofComplex) Filter.atTop {z | 0 < z.im}","state_after":"z : ℍ\nk : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ TendstoLocallyUniformlyOn (fun s => (fun z => ∑ x ∈ s, eisSummand k (↑x) z) ∘ ↑ofComplex)\n    ((eisensteinSeries_SIF a k).toFun ∘ ↑ofComplex) Filter.atTop (Subtype.val '' Set.univ)","tactic":"rw [← Subtype.coe_image_univ {z : ℂ | 0 < z.im}]","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.im","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[31,2],"def_end_pos":[31,4]},{"full_name":"Subtype.coe_image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[1184,8],"def_end_pos":[1184,22]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"z : ℍ\nk : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ TendstoLocallyUniformlyOn (fun s => (fun z => ∑ x ∈ s, eisSummand k (↑x) z) ∘ ↑ofComplex)\n    ((eisensteinSeries_SIF a k).toFun ∘ ↑ofComplex) Filter.atTop (Subtype.val '' Set.univ)","state_after":"z : ℍ\nk : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ TendstoLocallyUniformlyOn (fun n z => ∑ x ∈ n, eisSummand k (↑x) z) (eisensteinSeries_SIF a k).toFun Filter.atTop ⊤\n\nz : ℍ\nk : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ MapsTo (↑(OpenEmbedding.toPartialHomeomorph UpperHalfPlane.coe openEmbedding_coe).symm)\n    (OpenEmbedding.toPartialHomeomorph UpperHalfPlane.coe openEmbedding_coe).target ⊤","tactic":"apply TendstoLocallyUniformlyOn.comp (s := ⊤) _ _ _ (PartialHomeomorph.continuousOn_symm _)","premises":[{"full_name":"PartialHomeomorph.continuousOn_symm","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[97,8],"def_end_pos":[97,25]},{"full_name":"TendstoLocallyUniformlyOn.comp","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[634,8],"def_end_pos":[634,38]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]}]}
{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"ContinuousLinearEquiv.integrable_comp_iff","start":[1421,0],"end":[1425,64],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁷ : MeasurableSpace δ\ninst✝⁶ : NormedAddCommGroup β\ninst✝⁵ : NormedAddCommGroup γ\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nH : Type u_7\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nφ : α → H\nL : H ≃L[𝕜] E\nh : Integrable (fun a => L (φ a)) μ\n⊢ Integrable φ μ","state_after":"no goals","tactic":"simpa using ContinuousLinearMap.integrable_comp (L.symm : E →L[𝕜] H) h","premises":[{"full_name":"ContinuousLinearEquiv.symm","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1773,14],"def_end_pos":[1773,18]},{"full_name":"ContinuousLinearMap","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[224,10],"def_end_pos":[224,29]},{"full_name":"ContinuousLinearMap.integrable_comp","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1415,8],"def_end_pos":[1415,43]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]}]}
{"url":".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean","commit":"","full_name":"SatisfiesM.seq","start":[89,0],"end":[96,61],"file_path":".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean","tactics":[{"state_before":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx : m α\nhf : SatisfiesM p₁ f\nhx : SatisfiesM p₂ x\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\n⊢ SatisfiesM q (Seq.seq f fun x_1 => x)","state_after":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ SatisfiesM q (Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x)","tactic":"match f, x, hf, hx with | _, _, ⟨f, rfl⟩, ⟨x, rfl⟩ => ?_","premises":[{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ SatisfiesM q (Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x)","state_after":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Subtype.val <$>\n      Seq.seq\n        ((fun x x_1 =>\n            match x with\n            | ⟨a, h₁⟩ =>\n              match x_1 with\n              | ⟨b, h₂⟩ => ⟨a b, ⋯⟩) <$>\n          f)\n        fun x_1 => x) =\n    Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x","tactic":"refine ⟨(fun ⟨a, h₁⟩ ⟨b, h₂⟩ => ⟨a b, H h₁ h₂⟩) <$> f <*> x, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Functor.map","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2749,2],"def_end_pos":[2749,5]},{"full_name":"Seq.seq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2762,2],"def_end_pos":[2762,5]},{"full_name":"Unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[142,7],"def_end_pos":[142,11]}]},{"state_before":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Subtype.val <$>\n      Seq.seq\n        ((fun x x_1 =>\n            match x with\n            | ⟨a, h₁⟩ =>\n              match x_1 with\n              | ⟨b, h₂⟩ => ⟨a b, ⋯⟩) <$>\n          f)\n        fun x_1 => x) =\n    Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x","state_after":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (pure Subtype.val) fun x_1 =>\n      Seq.seq (Seq.seq (pure fun x x_2 => ⟨x.val x_2.val, ⋯⟩) fun x => f) fun x_2 => x) =\n    Seq.seq (Seq.seq (pure Subtype.val) fun x => f) fun x_1 => Seq.seq (pure Subtype.val) fun x_2 => x","tactic":"simp only [← pure_seq]","premises":[{"full_name":"LawfulApplicative.pure_seq","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[49,2],"def_end_pos":[49,10]}]},{"state_before":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (pure Subtype.val) fun x_1 =>\n      Seq.seq (Seq.seq (pure fun x x_2 => ⟨x.val x_2.val, ⋯⟩) fun x => f) fun x_2 => x) =\n    Seq.seq (Seq.seq (pure Subtype.val) fun x => f) fun x_1 => Seq.seq (pure Subtype.val) fun x_2 => x","state_after":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (Seq.seq (pure (Function.comp Subtype.val ∘ fun x x_1 => ⟨x.val x_1.val, ⋯⟩)) fun x => f) fun x_1 => x) =\n    Seq.seq ((fun h => h Subtype.val) <$> Function.comp <$> Seq.seq (pure Subtype.val) fun x => f) fun x_1 => x","tactic":"simp [SatisfiesM, seq_assoc]","premises":[{"full_name":"LawfulApplicative.seq_assoc","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[52,2],"def_end_pos":[52,11]},{"full_name":"SatisfiesM","def_path":".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean","def_pos":[35,4],"def_end_pos":[35,14]}]},{"state_before":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (Seq.seq (pure (Function.comp Subtype.val ∘ fun x x_1 => ⟨x.val x_1.val, ⋯⟩)) fun x => f) fun x_1 => x) =\n    Seq.seq ((fun h => h Subtype.val) <$> Function.comp <$> Seq.seq (pure Subtype.val) fun x => f) fun x_1 => x","state_after":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (Seq.seq (pure (Function.comp Subtype.val ∘ fun x x_1 => ⟨x.val x_1.val, ⋯⟩)) fun x => f) fun x_1 => x) =\n    Seq.seq\n      (Seq.seq (pure fun h => h Subtype.val) fun x =>\n        Seq.seq (pure Function.comp) fun x => Seq.seq (pure Subtype.val) fun x => f)\n      fun x_1 => x","tactic":"simp only [← pure_seq]","premises":[{"full_name":"LawfulApplicative.pure_seq","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[49,2],"def_end_pos":[49,10]}]},{"state_before":"m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (Seq.seq (pure (Function.comp Subtype.val ∘ fun x x_1 => ⟨x.val x_1.val, ⋯⟩)) fun x => f) fun x_1 => x) =\n    Seq.seq\n      (Seq.seq (pure fun h => h Subtype.val) fun x =>\n        Seq.seq (pure Function.comp) fun x => Seq.seq (pure Subtype.val) fun x => f)\n      fun x_1 => x","state_after":"no goals","tactic":"simp [seq_assoc, Function.comp_def]","premises":[{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"LawfulApplicative.seq_assoc","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[52,2],"def_end_pos":[52,11]}]}]}
{"url":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","commit":"","full_name":"MvPFunctor.comp.get_map","start":[124,0],"end":[126,5],"file_path":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","tactics":[{"state_before":"n m : ℕ\nP : MvPFunctor.{u} n\nQ : Fin2 n → MvPFunctor.{u} m\nα β : TypeVec.{u} m\nf : α ⟹ β\nx : ↑(P.comp Q) α\n⊢ get (f <$$> x) = (fun i x => f <$$> x) <$$> get x","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/FieldTheory/PerfectClosure.lean","commit":"","full_name":"PerfectClosure.frobenius_mk","start":[382,0],"end":[386,22],"file_path":"Mathlib/FieldTheory/PerfectClosure.lean","tactics":[{"state_before":"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ (frobenius (PerfectClosure K p) p) (mk K p x) = mk K p (x.1, x.2 ^ p)","state_after":"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ mk K p x ^ p = mk K p (x.1, x.2 ^ p)","tactic":"simp only [frobenius_def]","premises":[{"full_name":"frobenius_def","def_path":"Mathlib/Algebra/CharP/ExpChar.lean","def_pos":[281,8],"def_end_pos":[281,21]}]},{"state_before":"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ mk K p x ^ p = mk K p (x.1, x.2 ^ p)","state_after":"no goals","tactic":"exact mk_pow K p x p","premises":[{"full_name":"PerfectClosure.mk_pow","def_path":"Mathlib/FieldTheory/PerfectClosure.lean","def_pos":[336,8],"def_end_pos":[336,14]}]}]}
{"url":"Mathlib/Analysis/Calculus/MeanValue.lean","commit":"","full_name":"norm_image_sub_le_of_norm_deriv_le_segment'","start":[339,0],"end":[348,80],"file_path":"Mathlib/Analysis/Calculus/MeanValue.lean","tactics":[{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nf' : ℝ → E\nC : ℝ\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C\n⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)","state_after":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nf' : ℝ → E\nC : ℝ\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C\nx : ℝ\nhx : x ∈ Ico a b\n⊢ HasDerivWithinAt f (f' x) (Ici x) x","tactic":"refine\n    norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt)\n      (fun x hx => ?_) bound","premises":[{"full_name":"HasDerivWithinAt.continuousWithinAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[643,8],"def_end_pos":[643,43]},{"full_name":"norm_image_sub_le_of_norm_deriv_right_le_segment","def_path":"Mathlib/Analysis/Calculus/MeanValue.lean","def_pos":[324,8],"def_end_pos":[324,56]}]},{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nf' : ℝ → E\nC : ℝ\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C\nx : ℝ\nhx : x ∈ Ico a b\n⊢ HasDerivWithinAt f (f' x) (Ici x) x","state_after":"no goals","tactic":"exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx)","premises":[{"full_name":"HasDerivWithinAt.mono_of_mem","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[321,8],"def_end_pos":[321,36]},{"full_name":"Icc_mem_nhdsWithin_Ici","def_path":"Mathlib/Topology/Order/OrderClosed.lean","def_pos":[507,8],"def_end_pos":[507,30]},{"full_name":"Set.Ico_subset_Icc_self","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[422,8],"def_end_pos":[422,27]}]}]}
{"url":"Mathlib/Data/Finset/Lattice.lean","commit":"","full_name":"Finset.prodMk_sup'_sup'","start":[767,0],"end":[774,86],"file_path":"Mathlib/Data/Finset/Lattice.lean","tactics":[{"state_before":"F : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nhs : s.Nonempty\nht : t.Nonempty\nf : ι → α\ng : κ → β\ni : α × β\n⊢ (s.sup' hs f, t.sup' ht g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i","state_after":"case intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nht : t.Nonempty\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\n⊢ (s.sup' ⋯ f, t.sup' ht g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i","tactic":"obtain ⟨a, ha⟩ := hs","premises":[]},{"state_before":"case intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nht : t.Nonempty\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\n⊢ (s.sup' ⋯ f, t.sup' ht g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i","state_after":"case intro.intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\nb : κ\nhb : b ∈ t\n⊢ (s.sup' ⋯ f, t.sup' ⋯ g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i","tactic":"obtain ⟨b, hb⟩ := ht","premises":[]},{"state_before":"case intro.intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\nb : κ\nhb : b ∈ t\n⊢ (s.sup' ⋯ f, t.sup' ⋯ g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i","state_after":"case intro.intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\nb : κ\nhb : b ∈ t\n⊢ ((∀ b ∈ s, f b ≤ i.1) ∧ ∀ b ∈ t, g b ≤ i.2) ↔ ∀ (a : ι) (b : κ), a ∈ s → b ∈ t → f a ≤ i.1 ∧ g b ≤ i.2","tactic":"simp only [Prod.map, sup'_le_iff, mem_product, and_imp, Prod.forall, Prod.le_def]","premises":[{"full_name":"Finset.mem_product","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[50,8],"def_end_pos":[50,19]},{"full_name":"Finset.sup'_le_iff","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[723,8],"def_end_pos":[723,19]},{"full_name":"Prod.forall","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[28,8],"def_end_pos":[28,16]},{"full_name":"Prod.le_def","def_path":"Mathlib/Order/Basic.lean","def_pos":[1126,8],"def_end_pos":[1126,14]},{"full_name":"Prod.map","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1190,4],"def_end_pos":[1190,12]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]}]},{"state_before":"case intro.intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\nb : κ\nhb : b ∈ t\n⊢ ((∀ b ∈ s, f b ≤ i.1) ∧ ∀ b ∈ t, g b ≤ i.2) ↔ ∀ (a : ι) (b : κ), a ∈ s → b ∈ t → f a ≤ i.1 ∧ g b ≤ i.2","state_after":"no goals","tactic":"exact ⟨by aesop, fun h ↦ ⟨fun i hi ↦ (h _ _ hi hb).1, fun j hj ↦ (h _ _ ha hj).2⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Algebra/Homology/QuasiIso.lean","commit":"","full_name":"ChainComplex.quasiIsoAt₀_iff","start":[111,0],"end":[115,45],"file_path":"Mathlib/Algebra/Homology/QuasiIso.lean","tactics":[{"state_before":"ι : Type u_1\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasZeroMorphisms C\nc : ComplexShape ι\nK✝ L✝ M K' L' : HomologicalComplex C c\nK L : ChainComplex C ℕ\nf : K ⟶ L\ninst✝³ : HasHomology K 0\ninst✝² : HasHomology L 0\ninst✝¹ : (sc' K 1 0 0).HasHomology\ninst✝ : (sc' L 1 0 0).HasHomology\n⊢ (ComplexShape.down ℕ).prev 0 = 1","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"ι : Type u_1\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasZeroMorphisms C\nc : ComplexShape ι\nK✝ L✝ M K' L' : HomologicalComplex C c\nK L : ChainComplex C ℕ\nf : K ⟶ L\ninst✝³ : HasHomology K 0\ninst✝² : HasHomology L 0\ninst✝¹ : (sc' K 1 0 0).HasHomology\ninst✝ : (sc' L 1 0 0).HasHomology\n⊢ (ComplexShape.down ℕ).next 0 = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/DFinsupp/Basic.lean","commit":"","full_name":"DFinsupp.sum_finset_sum_index","start":[1845,0],"end":[1853,84],"file_path":"Mathlib/Data/DFinsupp/Basic.lean","tactics":[{"state_before":"ι : Type u\nγ✝ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\nγ : Type w\nα : Type x\ninst✝² : (i : ι) → AddCommMonoid (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\ns : Finset α\ng : α → Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_zero : ∀ (i : ι), h i 0 = 1\nh_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ * h i b₂\n⊢ ∏ i ∈ s, (g i).prod h = (∑ i ∈ s, g i).prod h","state_after":"no goals","tactic":"classical\n  exact Finset.induction_on s (by simp [prod_zero_index])\n        (by simp (config := { contextual := true }) [prod_add_index, h_zero, h_add])","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"DFinsupp.prod_add_index","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[1600,8],"def_end_pos":[1600,22]},{"full_name":"DFinsupp.prod_zero_index","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[1527,8],"def_end_pos":[1527,23]},{"full_name":"Finset.induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1076,18],"def_end_pos":[1076,30]}]}]}
{"url":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","commit":"","full_name":"MeasurableSet.isClopenable","start":[282,0],"end":[290,48],"file_path":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","tactics":[{"state_before":"α : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\n⊢ IsClopenable s","state_after":"α : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\n⊢ ∀ {s : Set α}, MeasurableSet s → IsClopenable s","tactic":"revert s","premises":[]},{"state_before":"α : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\n⊢ ∀ {s : Set α}, MeasurableSet s → IsClopenable s","state_after":"case h_open\nα : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\n⊢ ∀ (U : Set α), IsOpen U → IsClopenable U\n\ncase h_compl\nα : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\n⊢ ∀ (t : Set α), MeasurableSet t → IsClopenable t → IsClopenable tᶜ\n\ncase h_union\nα : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\n⊢ ∀ (f : ℕ → Set α),\n    Pairwise (Disjoint on f) →\n      (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), IsClopenable (f i)) → IsClopenable (⋃ i, f i)","tactic":"apply MeasurableSet.induction_on_open","premises":[{"full_name":"MeasurableSet.induction_on_open","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[222,8],"def_end_pos":[222,39]}]}]}
{"url":"Mathlib/Computability/DFA.lean","commit":"","full_name":"DFA.reindex_apply_step","start":[205,0],"end":[219,24],"file_path":"Mathlib/Computability/DFA.lean","tactics":[{"state_before":"α : Type u\nσ : Type v\nM✝ : DFA α σ\nα' : Type u_1\nσ' : Type u_2\ng : σ ≃ σ'\nM : DFA α σ\n⊢ (fun M => { step := fun s a => g.symm (M.step (g s) a), start := g.symm M.start, accept := ⇑g ⁻¹' M.accept })\n      ((fun M => { step := fun s a => g (M.step (g.symm s) a), start := g M.start, accept := ⇑g.symm ⁻¹' M.accept })\n        M) =\n    M","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u\nσ : Type v\nM✝ : DFA α σ\nα' : Type u_1\nσ' : Type u_2\ng : σ ≃ σ'\nM : DFA α σ'\n⊢ (fun M => { step := fun s a => g (M.step (g.symm s) a), start := g M.start, accept := ⇑g.symm ⁻¹' M.accept })\n      ((fun M => { step := fun s a => g.symm (M.step (g s) a), start := g.symm M.start, accept := ⇑g ⁻¹' M.accept })\n        M) =\n    M","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/List/Sort.lean","commit":"","full_name":"List.mergeSort_nil","start":[484,0],"end":[485,69],"file_path":"Mathlib/Data/List/Sort.lean","tactics":[{"state_before":"α : Type u\nr : α → α → Prop\ninst✝ : DecidableRel r\n⊢ mergeSort r [] = []","state_after":"no goals","tactic":"rw [List.mergeSort]","premises":[{"full_name":"List.mergeSort","def_path":"Mathlib/Data/List/Sort.lean","def_pos":[395,4],"def_end_pos":[395,13]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.rel_zero_right","start":[2387,0],"end":[2388,84],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr : α → β → Prop\np : γ → δ → Prop\na : Multiset α\n⊢ Rel r a 0 ↔ a = 0","state_after":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr : α → β → Prop\np : γ → δ → Prop\na : Multiset α\n⊢ (a = 0 ∧ 0 = 0 ∨ ∃ a_1 b as bs, r a_1 b ∧ Rel r as bs ∧ a = a_1 ::ₘ as ∧ 0 = b ::ₘ bs) ↔ a = 0","tactic":"rw [rel_iff]","premises":[{"full_name":"Multiset.rel_iff","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[2337,2],"def_end_pos":[2337,8]}]},{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr : α → β → Prop\np : γ → δ → Prop\na : Multiset α\n⊢ (a = 0 ∧ 0 = 0 ∨ ∃ a_1 b as bs, r a_1 b ∧ Rel r as bs ∧ a = a_1 ::ₘ as ∧ 0 = b ::ₘ bs) ↔ a = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/GroupTheory/Perm/List.lean","commit":"","full_name":"List.formPerm_apply_nthLe","start":[225,0],"end":[229,30],"file_path":"Mathlib/GroupTheory/Perm/List.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\nl : List α\nx : α\nxs : List α\nh : xs.Nodup\nn : ℕ\nhn : n < xs.length\n⊢ xs.formPerm (xs.nthLe n hn) = xs.nthLe ((n + 1) % xs.length) ⋯","state_after":"no goals","tactic":"apply formPerm_apply_get _ h","premises":[{"full_name":"List.formPerm_apply_get","def_path":"Mathlib/GroupTheory/Perm/List.lean","def_pos":[219,8],"def_end_pos":[219,26]}]}]}
{"url":"Mathlib/Algebra/CharP/ExpChar.lean","commit":"","full_name":"ExpChar.eq","start":[56,0],"end":[63,23],"file_path":"Mathlib/Algebra/CharP/ExpChar.lean","tactics":[{"state_before":"R : Type u\ninst✝ : Semiring R\np q : ℕ\nhp : ExpChar R p\nhq : ExpChar R q\n⊢ p = q","state_after":"case zero\nR : Type u\ninst✝ : Semiring R\nq : ℕ\nhq : ExpChar R q\nhp : CharZero R\n⊢ 1 = q\n\ncase prime\nR : Type u\ninst✝ : Semiring R\np q : ℕ\nhq : ExpChar R q\nhp' : Nat.Prime p\nhp : CharP R p\n⊢ p = q","tactic":"cases' hp with hp _ hp' hp","premises":[]}]}
{"url":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","commit":"","full_name":"IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd","start":[147,0],"end":[157,44],"file_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ v.intValuationDef r ≤ ↑(ofAdd (-↑n)) ↔ v.asIdeal ^ n ∣ Ideal.span {r}","state_after":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ (if r = 0 then 0 else ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors)))) ≤\n      ↑(ofAdd (-↑n)) ↔\n    v.asIdeal ^ n ∣ Ideal.span {r}","tactic":"rw [intValuationDef]","premises":[{"full_name":"IsDedekindDomain.HeightOneSpectrum.intValuationDef","def_path":"Mathlib/RingTheory/DedekindDomain/AdicValuation.lean","def_pos":[76,4],"def_end_pos":[76,19]}]},{"state_before":"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ (if r = 0 then 0 else ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors)))) ≤\n      ↑(ofAdd (-↑n)) ↔\n    v.asIdeal ^ n ∣ Ideal.span {r}","state_after":"case pos\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\nn : ℕ\nhr : r = 0\n⊢ 0 ≤ ↑(ofAdd (-↑n)) ↔ v.asIdeal ^ n ∣ Ideal.span {r}\n\ncase neg\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\nn : ℕ\nhr : ¬r = 0\n⊢ ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors))) ≤ ↑(ofAdd (-↑n)) ↔\n    v.asIdeal ^ n ∣ Ideal.span {r}","tactic":"split_ifs with hr","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Module/LinearMap/Polynomial.lean","commit":"","full_name":"LinearMap.polyCharpolyAux_map_eval","start":[308,0],"end":[313,67],"file_path":"Mathlib/Algebra/Module/LinearMap/Polynomial.lean","tactics":[{"state_before":"R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Free R M\nx : ι → R\n⊢ Polynomial.map (MvPolynomial.eval x) (φ.polyCharpolyAux b bₘ) =\n    charpoly (φ (b.repr.symm (Finsupp.equivFunOnFinite.symm x)))","state_after":"no goals","tactic":"simp only [← polyCharpolyAux_map_eq_charpoly φ b bₘ, LinearEquiv.apply_symm_apply,\n    Finsupp.equivFunOnFinite, Equiv.coe_fn_symm_mk, Finsupp.coe_mk]","premises":[{"full_name":"Equiv.coe_fn_symm_mk","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[220,16],"def_end_pos":[220,30]},{"full_name":"Finsupp.coe_mk","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[131,8],"def_end_pos":[131,14]},{"full_name":"Finsupp.equivFunOnFinite","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[194,4],"def_end_pos":[194,20]},{"full_name":"LinearEquiv.apply_symm_apply","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[342,8],"def_end_pos":[342,24]},{"full_name":"LinearMap.polyCharpolyAux_map_eq_charpoly","def_path":"Mathlib/Algebra/Module/LinearMap/Polynomial.lean","def_pos":[296,6],"def_end_pos":[296,37]}]}]}
{"url":"Mathlib/GroupTheory/Finiteness.lean","commit":"","full_name":"AddMonoid.fg_iff_addSubmonoid_fg","start":[137,0],"end":[140,98],"file_path":"Mathlib/GroupTheory/Finiteness.lean","tactics":[{"state_before":"M : Type u_1\nN✝ : Type u_2\ninst✝¹ : Monoid M\ninst✝ : AddMonoid N✝\nN : Submonoid M\n⊢ FG ↥N ↔ N.FG","state_after":"M : Type u_1\nN✝ : Type u_2\ninst✝¹ : Monoid M\ninst✝ : AddMonoid N✝\nN : Submonoid M\n⊢ FG ↥N ↔ (Submonoid.map N.subtype ⊤).FG","tactic":"conv_rhs => rw [← N.range_subtype, MonoidHom.mrange_eq_map]","premises":[{"full_name":"MonoidHom.mrange_eq_map","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[763,8],"def_end_pos":[763,21]},{"full_name":"Submonoid.range_subtype","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[989,8],"def_end_pos":[989,21]}]},{"state_before":"M : Type u_1\nN✝ : Type u_2\ninst✝¹ : Monoid M\ninst✝ : AddMonoid N✝\nN : Submonoid M\n⊢ FG ↥N ↔ (Submonoid.map N.subtype ⊤).FG","state_after":"no goals","tactic":"exact ⟨fun h => h.out.map N.subtype, fun h => ⟨h.map_injective N.subtype Subtype.coe_injective⟩⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Monoid.FG.out","def_path":"Mathlib/GroupTheory/Finiteness.lean","def_pos":[74,2],"def_end_pos":[74,5]},{"full_name":"Submonoid.FG.map","def_path":"Mathlib/GroupTheory/Finiteness.lean","def_pos":[119,8],"def_end_pos":[119,24]},{"full_name":"Submonoid.FG.map_injective","def_path":"Mathlib/GroupTheory/Finiteness.lean","def_pos":[126,8],"def_end_pos":[126,34]},{"full_name":"Submonoid.subtype","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[544,4],"def_end_pos":[544,11]},{"full_name":"Subtype.coe_injective","def_path":"Mathlib/Data/Subtype.lean","def_pos":[102,8],"def_end_pos":[102,21]}]}]}
{"url":"Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean","commit":"","full_name":"liouville_liouvilleNumber","start":[179,0],"end":[192,84],"file_path":"Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean","tactics":[{"state_before":"m : ℕ\nhm : 2 ≤ m\n⊢ Liouville (liouvilleNumber ↑m)","state_after":"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\n⊢ Liouville (liouvilleNumber ↑m)","tactic":"have mZ1 : 1 < (m : ℤ) := by norm_cast","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]}]},{"state_before":"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\n⊢ Liouville (liouvilleNumber ↑m)","state_after":"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\n⊢ Liouville (liouvilleNumber ↑m)","tactic":"have m1 : 1 < (m : ℝ) := by norm_cast","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\n⊢ Liouville (liouvilleNumber ↑m)","state_after":"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn : ℕ\n⊢ ∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n","tactic":"intro n","premises":[]},{"state_before":"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn : ℕ\n⊢ ∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n","state_after":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ ∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n","tactic":"rcases partialSum_eq_rat (zero_lt_two.trans_le hm) n with ⟨p, hp⟩","premises":[{"full_name":"LiouvilleNumber.partialSum_eq_rat","def_path":"Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean","def_pos":[162,8],"def_end_pos":[162,25]},{"full_name":"zero_lt_two","def_path":"Mathlib/Algebra/Order/Monoid/NatCast.lean","def_pos":[62,14],"def_end_pos":[62,25]}]},{"state_before":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ ∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n","state_after":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ liouvilleNumber ↑m ≠ ↑↑p / ↑(↑m ^ n !) ∧ |liouvilleNumber ↑m - ↑↑p / ↑(↑m ^ n !)| < 1 / ↑(↑m ^ n !) ^ n","tactic":"refine ⟨p, m ^ n !, one_lt_pow mZ1 n.factorial_ne_zero, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Nat.factorial","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[29,4],"def_end_pos":[29,13]},{"full_name":"Nat.factorial_ne_zero","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[60,8],"def_end_pos":[60,25]},{"full_name":"one_lt_pow","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[95,8],"def_end_pos":[95,18]}]},{"state_before":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ liouvilleNumber ↑m ≠ ↑↑p / ↑(↑m ^ n !) ∧ |liouvilleNumber ↑m - ↑↑p / ↑(↑m ^ n !)| < 1 / ↑(↑m ^ n !) ^ n","state_after":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n","tactic":"push_cast","premises":[]},{"state_before":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n","state_after":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\n⊢ liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n","tactic":"rw [Nat.cast_pow] at hp","premises":[{"full_name":"Nat.cast_pow","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[82,6],"def_end_pos":[82,14]}]},{"state_before":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\n⊢ liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n","state_after":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n    |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n","tactic":"rw [← partialSum_add_remainder m1 n, ← hp]","premises":[{"full_name":"LiouvilleNumber.partialSum_add_remainder","def_path":"Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean","def_pos":[90,8],"def_end_pos":[90,32]}]},{"state_before":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n    |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n","state_after":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n    |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n","tactic":"have hpos := remainder_pos m1 n","premises":[{"full_name":"LiouvilleNumber.remainder_pos","def_path":"Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean","def_pos":[82,8],"def_end_pos":[82,21]}]},{"state_before":"case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n    |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n","state_after":"no goals","tactic":"simpa [abs_of_pos hpos, hpos.ne'] using @remainder_lt n m (by assumption_mod_cast)","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"LiouvilleNumber.remainder_lt","def_path":"Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean","def_pos":[154,8],"def_end_pos":[154,20]},{"full_name":"abs_of_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[79,2],"def_end_pos":[79,13]}]}]}
{"url":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","commit":"","full_name":"MeasureTheory.JordanDecomposition.real_smul_posPart_neg","start":[133,0],"end":[135,71],"file_path":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).posPart = (-r).toNNReal • j.negPart","state_after":"no goals","tactic":"rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.JordanDecomposition.neg_posPart","def_path":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","def_pos":[95,8],"def_end_pos":[95,19]},{"full_name":"MeasureTheory.JordanDecomposition.real_smul_def","def_path":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","def_pos":[110,8],"def_end_pos":[110,21]},{"full_name":"MeasureTheory.JordanDecomposition.smul_negPart","def_path":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","def_pos":[107,8],"def_end_pos":[107,20]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Rayleigh.lean","commit":"","full_name":"IsSelfAdjoint.hasEigenvector_of_isLocalExtrOn","start":[168,0],"end":[175,46],"file_path":"Mathlib/Analysis/InnerProductSpace/Rayleigh.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ HasEigenvector (↑T) (↑(T.rayleighQuotient x₀)) x₀","state_after":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ x₀ ∈ eigenspace ↑T ↑(T.rayleighQuotient x₀)","tactic":"refine ⟨?_, hx₀⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]},{"state_before":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ x₀ ∈ eigenspace ↑T ↑(T.rayleighQuotient x₀)","state_after":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ ↑T x₀ = ↑(T.rayleighQuotient x₀) • x₀","tactic":"rw [Module.End.mem_eigenspace_iff]","premises":[{"full_name":"Module.End.mem_eigenspace_iff","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[97,8],"def_end_pos":[97,26]}]},{"state_before":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ ↑T x₀ = ↑(T.rayleighQuotient x₀) • x₀","state_after":"no goals","tactic":"exact hT.eq_smul_self_of_isLocalExtrOn hextr","premises":[{"full_name":"IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn","def_path":"Mathlib/Analysis/InnerProductSpace/Rayleigh.lean","def_pos":[161,8],"def_end_pos":[161,37]}]}]}
{"url":"Mathlib/Probability/Kernel/Basic.lean","commit":"","full_name":"ProbabilityTheory.Kernel.lintegral_const","start":[471,0],"end":[473,64],"file_path":"Mathlib/Probability/Kernel/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nf : β → ℝ≥0∞\nμ : Measure β\na : α\n⊢ ∫⁻ (x : β), f x ∂(const α μ) a = ∫⁻ (x : β), f x ∂μ","state_after":"no goals","tactic":"rw [const_apply]","premises":[{"full_name":"ProbabilityTheory.Kernel.const_apply","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[439,8],"def_end_pos":[439,19]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean","commit":"","full_name":"Polynomial.IsEisensteinAt.coeff_mem","start":[198,0],"end":[202,26],"file_path":"Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : f.IsEisensteinAt 𝓟\nn : ℕ\nhn : n ≠ f.natDegree\n⊢ f.coeff n ∈ 𝓟","state_after":"case inl\nR : Type u\ninst✝ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : f.IsEisensteinAt 𝓟\nn : ℕ\nhn : n ≠ f.natDegree\nh₁ : n < f.natDegree\n⊢ f.coeff n ∈ 𝓟\n\ncase inr\nR : Type u\ninst✝ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : f.IsEisensteinAt 𝓟\nn : ℕ\nhn : n ≠ f.natDegree\nh₂ : n > f.natDegree\n⊢ f.coeff n ∈ 𝓟","tactic":"cases' ne_iff_lt_or_gt.1 hn with h₁ h₂","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"ne_iff_lt_or_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[305,8],"def_end_pos":[305,23]}]}]}
{"url":"Mathlib/Order/Part.lean","commit":"","full_name":"Monotone.partBind","start":[21,0],"end":[25,49],"file_path":"Mathlib/Order/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Preorder α\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhg : Monotone g\n⊢ Monotone fun x => (f x).bind (g x)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Preorder α\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhg : Monotone g\nx y : α\nh : x ≤ y\na : γ\n⊢ a ∈ (fun x => (f x).bind (g x)) x → a ∈ (fun x => (f x).bind (g x)) y","tactic":"rintro x y h a","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Preorder α\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhg : Monotone g\nx y : α\nh : x ≤ y\na : γ\n⊢ a ∈ (fun x => (f x).bind (g x)) x → a ∈ (fun x => (f x).bind (g x)) y","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Preorder α\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhg : Monotone g\nx y : α\nh : x ≤ y\na : γ\n⊢ ∀ x_1 ∈ f x, a ∈ g x x_1 → ∃ a_3 ∈ f y, a ∈ g y a_3","tactic":"simp only [and_imp, exists_prop, Part.bind_eq_bind, Part.mem_bind_iff, exists_imp]","premises":[{"full_name":"Part.bind_eq_bind","def_path":"Mathlib/Data/Part.lean","def_pos":[516,8],"def_end_pos":[516,20]},{"full_name":"Part.mem_bind_iff","def_path":"Mathlib/Data/Part.lean","def_pos":[413,8],"def_end_pos":[413,20]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"exists_imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[200,8],"def_end_pos":[200,18]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Preorder α\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhg : Monotone g\nx y : α\nh : x ≤ y\na : γ\n⊢ ∀ x_1 ∈ f x, a ∈ g x x_1 → ∃ a_3 ∈ f y, a ∈ g y a_3","state_after":"no goals","tactic":"exact fun b hb ha ↦ ⟨b, hf h _ hb, hg h _ _ ha⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/CategoryTheory/Elements.lean","commit":"","full_name":"CategoryTheory.CategoryOfElements.structuredArrowEquivalence_unitIso_inv","start":[170,0],"end":[177,69],"file_path":"Mathlib/CategoryTheory/Elements.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nX : F.Elements\n⊢ (𝟭 F.Elements).obj X = (toStructuredArrow F ⋙ fromStructuredArrow F).obj X","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Algebra/Lie/IdealOperations.lean","commit":"","full_name":"LieSubmodule.lie_eq_bot_iff","start":[128,0],"end":[132,19],"file_path":"Mathlib/Algebra/Lie/IdealOperations.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅I, N⁆ = ⊥ ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0","state_after":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ (∀ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}, m = 0) ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0","tactic":"rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_eq_bot_iff]","premises":[{"full_name":"LieSubmodule.lieIdeal_oper_eq_span","def_path":"Mathlib/Algebra/Lie/IdealOperations.lean","def_pos":[52,8],"def_end_pos":[52,29]},{"full_name":"LieSubmodule.lieSpan_eq_bot_iff","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[698,8],"def_end_pos":[698,26]}]},{"state_before":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ (∀ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}, m = 0) ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0","state_after":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ (∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0) → ∀ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}, m = 0","tactic":"refine ⟨fun h x hx m hm => h ⁅x, m⁆ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩","premises":[{"full_name":"Bracket.bracket","def_path":"Mathlib/Data/Bracket.lean","def_pos":[35,2],"def_end_pos":[35,9]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ (∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0) → ∀ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}, m = 0","state_after":"case intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nh : ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ⁅↑⟨x, hx⟩, ↑⟨n, hn⟩⁆ = 0","tactic":"rintro h - ⟨⟨x, hx⟩, ⟨⟨n, hn⟩, rfl⟩⟩","premises":[]},{"state_before":"case intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nh : ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ = 0\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ⁅↑⟨x, hx⟩, ↑⟨n, hn⟩⁆ = 0","state_after":"no goals","tactic":"exact h x hx n hn","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Caratheodory.lean","commit":"","full_name":"Caratheodory.minCardFinsetOfMemConvexHull_card_le_card","start":[117,0],"end":[119,50],"file_path":"Mathlib/Analysis/Convex/Caratheodory.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ (convexHull 𝕜) s\nt : Finset E\nht₁ : ↑t ⊆ s\nht₂ : x ∈ (convexHull 𝕜) ↑t\n⊢ t ∈ {t | ↑t ⊆ s ∧ x ∈ (convexHull 𝕜) ↑t}","state_after":"no goals","tactic":"exact ⟨ht₁, ht₂⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]}]}
{"url":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","commit":"","full_name":"Real.fderiv_fourierChar_neg_bilinear_left_apply","start":[137,0],"end":[142,6],"file_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","tactics":[{"state_before":"V : Type u_1\nW : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nv y : V\nw : W\n⊢ (fderiv ℝ (fun v => ↑(𝐞 (-(L v) w))) v) y = -2 * ↑π * I * ↑((L y) w) * ↑(𝐞 (-(L v) w))","state_after":"V : Type u_1\nW : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nv y : V\nw : W\n⊢ 2 * ↑π * I * ↑(𝐞 (-(L v) w)) * ↑((L y) w) = 2 * ↑π * I * ↑((L y) w) * ↑(𝐞 (-(L v) w))","tactic":"simp only [(hasFDerivAt_fourierChar_neg_bilinear_left L v w).fderiv, neg_mul,\n    ContinuousLinearMap.coe_smul', ContinuousLinearMap.coe_comp', Pi.smul_apply,\n    Function.comp_apply, ContinuousLinearMap.flip_apply, ofRealCLM_apply, smul_eq_mul, neg_inj]","premises":[{"full_name":"Complex.ofRealCLM_apply","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[364,8],"def_end_pos":[364,23]},{"full_name":"ContinuousLinearMap.coe_comp'","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[699,8],"def_end_pos":[699,17]},{"full_name":"ContinuousLinearMap.coe_smul'","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[536,8],"def_end_pos":[536,17]},{"full_name":"ContinuousLinearMap.flip_apply","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean","def_pos":[163,8],"def_end_pos":[163,18]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"HasFDerivAt.fderiv","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[513,18],"def_end_pos":[513,36]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"Real.hasFDerivAt_fourierChar_neg_bilinear_left","def_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","def_pos":[132,6],"def_end_pos":[132,47]},{"full_name":"neg_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[294,2],"def_end_pos":[294,13]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"V : Type u_1\nW : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nv y : V\nw : W\n⊢ 2 * ↑π * I * ↑(𝐞 (-(L v) w)) * ↑((L y) w) = 2 * ↑π * I * ↑((L y) w) * ↑(𝐞 (-(L v) w))","state_after":"no goals","tactic":"ring","premises":[]}]}
{"url":"Mathlib/NumberTheory/ArithmeticFunction.lean","commit":"","full_name":"ArithmeticFunction.intCoe_one","start":[195,0],"end":[199,18],"file_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : AddGroupWithOne R\n⊢ ↑1 = 1","state_after":"case h\nR : Type u_1\ninst✝ : AddGroupWithOne R\nn : ℕ\n⊢ ↑1 n = 1 n","tactic":"ext n","premises":[]},{"state_before":"case h\nR : Type u_1\ninst✝ : AddGroupWithOne R\nn : ℕ\n⊢ ↑1 n = 1 n","state_after":"no goals","tactic":"simp [one_apply]","premises":[{"full_name":"ArithmeticFunction.one_apply","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[131,8],"def_end_pos":[131,17]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Content.lean","commit":"","full_name":"MeasureTheory.Content.sup_le","start":[102,0],"end":[105,21],"file_path":"Mathlib/MeasureTheory/Measure/Content.lean","tactics":[{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ (fun s => ↑(μ.toFun s)) (K₁ ⊔ K₂) ≤ (fun s => ↑(μ.toFun s)) K₁ + (fun s => ↑(μ.toFun s)) K₂","state_after":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ ↑(μ.toFun (K₁ ⊔ K₂)) ≤ ↑(μ.toFun K₁) + ↑(μ.toFun K₂)","tactic":"simp only [apply_eq_coe_toFun]","premises":[{"full_name":"MeasureTheory.Content.apply_eq_coe_toFun","def_path":"Mathlib/MeasureTheory/Measure/Content.lean","def_pos":[91,8],"def_end_pos":[91,26]}]},{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ ↑(μ.toFun (K₁ ⊔ K₂)) ≤ ↑(μ.toFun K₁) + ↑(μ.toFun K₂)","state_after":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ μ.toFun (K₁ ⊔ K₂) ≤ μ.toFun K₁ + μ.toFun K₂","tactic":"norm_cast","premises":[]},{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ μ.toFun (K₁ ⊔ K₂) ≤ μ.toFun K₁ + μ.toFun K₂","state_after":"no goals","tactic":"exact μ.sup_le' _ _","premises":[{"full_name":"MeasureTheory.Content.sup_le'","def_path":"Mathlib/MeasureTheory/Measure/Content.lean","def_pos":[73,2],"def_end_pos":[73,9]}]}]}
{"url":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","commit":"","full_name":"measurable_from_prod_countable'","start":[707,0],"end":[718,73],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : Countable β\nx✝ : MeasurableSpace γ\nf : α × β → γ\nhf : ∀ (y : β), Measurable fun x => f (x, y)\nh'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)\ns : Set γ\nhs : MeasurableSet s\n⊢ MeasurableSet (f ⁻¹' s)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : Countable β\nx✝ : MeasurableSpace γ\nf : α × β → γ\nhf : ∀ (y : β), Measurable fun x => f (x, y)\nh'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)\ns : Set γ\nhs : MeasurableSet s\nthis : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y\n⊢ MeasurableSet (f ⁻¹' s)","tactic":"have : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ (measurableAtom y : Set β) := by\n    ext1 ⟨x, y⟩\n    simp only [mem_preimage, mem_iUnion, mem_prod]\n    refine ⟨fun h ↦ ⟨y, h, mem_measurableAtom_self y⟩, ?_⟩\n    rintro ⟨y', hy's, hy'⟩\n    rwa [h'f y' y x hy']","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]},{"full_name":"Set.mem_prod","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[184,8],"def_end_pos":[184,16]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"measurableAtom","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[561,4],"def_end_pos":[561,18]},{"full_name":"mem_measurableAtom_self","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[564,14],"def_end_pos":[564,37]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : Countable β\nx✝ : MeasurableSpace γ\nf : α × β → γ\nhf : ∀ (y : β), Measurable fun x => f (x, y)\nh'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)\ns : Set γ\nhs : MeasurableSet s\nthis : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y\n⊢ MeasurableSet (f ⁻¹' s)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : Countable β\nx✝ : MeasurableSpace γ\nf : α × β → γ\nhf : ∀ (y : β), Measurable fun x => f (x, y)\nh'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)\ns : Set γ\nhs : MeasurableSet s\nthis : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y\n⊢ MeasurableSet (⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y)","tactic":"rw [this]","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : Countable β\nx✝ : MeasurableSpace γ\nf : α × β → γ\nhf : ∀ (y : β), Measurable fun x => f (x, y)\nh'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)\ns : Set γ\nhs : MeasurableSet s\nthis : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y\n⊢ MeasurableSet (⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y)","state_after":"no goals","tactic":"exact .iUnion (fun y ↦ (hf y hs).prod (.measurableAtom_of_countable y))","premises":[{"full_name":"MeasurableSet.iUnion","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[106,18],"def_end_pos":[106,38]},{"full_name":"MeasurableSet.measurableAtom_of_countable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[580,6],"def_end_pos":[580,47]},{"full_name":"MeasurableSet.prod","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[680,18],"def_end_pos":[680,36]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.op_distinguished","start":[412,0],"end":[415,29],"file_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ (triangleOpEquivalence C).functor.obj (Opposite.op T) ∈ distinguishedTriangles","state_after":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ ∃ T',\n    ∃ (_ : T' ∈ distinguishedTriangles),\n      Nonempty\n        ((triangleOpEquivalence C).functor.obj (Opposite.op T) ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))","tactic":"rw [mem_distTriang_op_iff']","premises":[{"full_name":"CategoryTheory.Pretriangulated.mem_distTriang_op_iff'","def_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","def_pos":[407,6],"def_end_pos":[407,28]}]},{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ ∃ T',\n    ∃ (_ : T' ∈ distinguishedTriangles),\n      Nonempty\n        ((triangleOpEquivalence C).functor.obj (Opposite.op T) ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))","state_after":"no goals","tactic":"exact ⟨T, hT, ⟨Iso.refl _⟩⟩","premises":[{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]}]}
{"url":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","commit":"","full_name":"contMDiff_neg_sphere","start":[458,0],"end":[464,28],"file_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","tactics":[{"state_before":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ F H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : SmoothManifoldWithCorners I M\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\n⊢ ContMDiff 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) ⊤ fun x => -x","state_after":"case hf\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ F H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : SmoothManifoldWithCorners I M\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\n⊢ ContMDiff 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) 𝓘(ℝ, E) ⊤ fun x => -↑x","tactic":"apply ContMDiff.codRestrict_sphere","premises":[{"full_name":"ContMDiff.codRestrict_sphere","def_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","def_pos":[435,8],"def_end_pos":[435,36]}]},{"state_before":"case hf\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ F H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : SmoothManifoldWithCorners I M\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\n⊢ ContMDiff 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) 𝓘(ℝ, E) ⊤ fun x => -↑x","state_after":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ F H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : SmoothManifoldWithCorners I M\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\n⊢ ContMDiff 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) 𝓘(ℝ, E) ⊤ Subtype.val","tactic":"apply contDiff_neg.contMDiff.comp _","premises":[{"full_name":"ContMDiff.comp","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Basic.lean","def_pos":[112,8],"def_end_pos":[112,22]},{"full_name":"contDiff_neg","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1208,8],"def_end_pos":[1208,20]}]},{"state_before":"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ F H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : SmoothManifoldWithCorners I M\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\n⊢ ContMDiff 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) 𝓘(ℝ, E) ⊤ Subtype.val","state_after":"no goals","tactic":"exact contMDiff_coe_sphere","premises":[{"full_name":"contMDiff_coe_sphere","def_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","def_pos":[413,8],"def_end_pos":[413,28]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","commit":"","full_name":"Finset.sum_affineCombinationSingleWeights","start":[599,0],"end":[603,94],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\nh : i ∈ s\n⊢ ∑ j ∈ s, affineCombinationSingleWeights k i j = 1","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\nh : i ∈ s\n⊢ ∑ j ∈ s, affineCombinationSingleWeights k i j = affineCombinationSingleWeights k i i","tactic":"rw [← affineCombinationSingleWeights_apply_self k i]","premises":[{"full_name":"Finset.affineCombinationSingleWeights_apply_self","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[592,8],"def_end_pos":[592,49]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\nh : i ∈ s\n⊢ ∑ j ∈ s, affineCombinationSingleWeights k i j = affineCombinationSingleWeights k i i","state_after":"no goals","tactic":"exact sum_eq_single_of_mem i h fun j _ hj => affineCombinationSingleWeights_apply_of_ne k hj","premises":[{"full_name":"Finset.affineCombinationSingleWeights_apply_of_ne","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[596,8],"def_end_pos":[596,50]},{"full_name":"Finset.sum_eq_single_of_mem","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[854,2],"def_end_pos":[854,13]}]}]}
{"url":"Mathlib/Algebra/GroupPower/IterateHom.lean","commit":"","full_name":"SemiconjBy.function_semiconj_mul_right_swap","start":[147,0],"end":[149,88],"file_path":"Mathlib/Algebra/GroupPower/IterateHom.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nH : Type u_4\ninst✝ : Semigroup G\na b c : G\nh : SemiconjBy a b c\nj : G\n⊢ (fun x => x * a) ((fun x => x * c) j) = (fun x => x * b) ((fun x => x * a) j)","state_after":"no goals","tactic":"simp_rw [mul_assoc, ← h.eq]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"SemiconjBy.eq","def_path":"Mathlib/Algebra/Group/Semiconj/Defs.lean","def_pos":[45,18],"def_end_pos":[45,20]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
{"url":"Mathlib/Analysis/Calculus/LHopital.lean","commit":"","full_name":"deriv.lhopital_zero_nhds_left","start":[375,0],"end":[388,69],"file_path":"Mathlib/Analysis/Calculus/LHopital.lean","tactics":[{"state_before":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","state_after":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\nhdg : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","tactic":"have hdg : ∀ᶠ x in 𝓝[<] a, DifferentiableAt ℝ g x :=\n    hg'.mp (eventually_of_forall fun _ hg' =>\n      by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))","premises":[{"full_name":"DifferentiableAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[171,4],"def_end_pos":[171,20]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.mp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[998,8],"def_end_pos":[998,21]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"by_contradiction","def_path":"Mathlib/Logic/Basic.lean","def_pos":[171,8],"def_end_pos":[171,24]},{"full_name":"deriv_zero_of_not_differentiableAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[222,8],"def_end_pos":[222,42]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]}]},{"state_before":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\nhdg : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","state_after":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\nhdg : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt f (deriv f x) x\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","tactic":"have hdf' : ∀ᶠ x in 𝓝[<] a, HasDerivAt f (deriv f x) x :=\n    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)","premises":[{"full_name":"DifferentiableAt.hasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[366,8],"def_end_pos":[366,35]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.mp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[998,8],"def_end_pos":[998,21]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"Set.Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[143,4],"def_end_pos":[143,9]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]}]},{"state_before":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\nhdg : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt f (deriv f x) x\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","state_after":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\nhdg : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt f (deriv f x) x\nhdg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt g (deriv g x) x\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","tactic":"have hdg' : ∀ᶠ x in 𝓝[<] a, HasDerivAt g (deriv g x) x :=\n    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)","premises":[{"full_name":"DifferentiableAt.hasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[366,8],"def_end_pos":[366,35]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.mp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[998,8],"def_end_pos":[998,21]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"Set.Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[143,4],"def_end_pos":[143,9]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]}]},{"state_before":"a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[<] a) l\nhdg : ∀ᶠ (x : ℝ) in 𝓝[<] a, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt f (deriv f x) x\nhdg' : ∀ᶠ (x : ℝ) in 𝓝[<] a, HasDerivAt g (deriv g x) x\n⊢ Tendsto (fun x => f x / g x) (𝓝[<] a) l","state_after":"no goals","tactic":"exact HasDerivAt.lhopital_zero_nhds_left hdf' hdg' hg' hfa hga hdiv","premises":[{"full_name":"HasDerivAt.lhopital_zero_nhds_left","def_path":"Mathlib/Analysis/Calculus/LHopital.lean","def_pos":[286,8],"def_end_pos":[286,31]}]}]}
{"url":"Mathlib/RingTheory/WittVector/Basic.lean","commit":"","full_name":"_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_zero","start":[170,0],"end":[171,22],"file_path":"Mathlib/RingTheory/WittVector/Basic.lean","tactics":[{"state_before":"p : ℕ\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type u_4\nβ : Type u_5\nx y : 𝕎 R\n⊢ WittVector.ghostFun 0 = 0","state_after":"no goals","tactic":"ghost_fun_tac 0, ![]","premises":[{"full_name":"Add.add","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1310,2],"def_end_pos":[1310,5]},{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.uncurry","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[189,4],"def_end_pos":[189,11]},{"full_name":"HAdd.hAdd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1180,2],"def_end_pos":[1180,6]},{"full_name":"HMul.hMul","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1199,2],"def_end_pos":[1199,6]},{"full_name":"HPow.hPow","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1237,2],"def_end_pos":[1237,6]},{"full_name":"HSMul.hSMul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[78,2],"def_end_pos":[78,7]},{"full_name":"HSub.hSub","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1190,2],"def_end_pos":[1190,6]},{"full_name":"Matrix.vecEmpty","def_path":"Mathlib/Data/Fin/VecNotation.lean","def_pos":[48,4],"def_end_pos":[48,12]},{"full_name":"Mul.mul","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1320,2],"def_end_pos":[1320,5]},{"full_name":"MvPolynomial.aeval_bind₁","def_path":"Mathlib/Algebra/MvPolynomial/Monad.lean","def_pos":[251,8],"def_end_pos":[251,19]},{"full_name":"MvPolynomial.aeval_rename","def_path":"Mathlib/Algebra/MvPolynomial/Rename.lean","def_pos":[173,8],"def_end_pos":[173,20]},{"full_name":"Neg.neg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1329,2],"def_end_pos":[1329,5]},{"full_name":"OfNat.ofNat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1105,2],"def_end_pos":[1105,7]},{"full_name":"One.one","def_path":"Mathlib/Algebra/Group/ZeroOne.lean","def_pos":[25,2],"def_end_pos":[25,5]},{"full_name":"Pow.pow","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1358,2],"def_end_pos":[1358,5]},{"full_name":"SMul.smul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[101,2],"def_end_pos":[101,6]},{"full_name":"Sub.sub","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1315,2],"def_end_pos":[1315,5]},{"full_name":"WittVector.eval","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[159,4],"def_end_pos":[159,8]},{"full_name":"WittVector.peval","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[146,4],"def_end_pos":[146,9]},{"full_name":"WittVector.wittNSMul","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[116,4],"def_end_pos":[116,13]},{"full_name":"WittVector.wittOne","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[108,4],"def_end_pos":[108,11]},{"full_name":"WittVector.wittZSMul","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[120,4],"def_end_pos":[120,13]},{"full_name":"WittVector.wittZero","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[104,4],"def_end_pos":[104,12]},{"full_name":"Zero.zero","def_path":"Mathlib/Algebra/Group/ZeroOne.lean","def_pos":[13,2],"def_end_pos":[13,6]},{"full_name":"_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun","def_path":"Mathlib/RingTheory/WittVector/Basic.lean","def_pos":[140,12],"def_end_pos":[140,20]},{"full_name":"wittStructureInt_prop","def_path":"Mathlib/RingTheory/WittVector/StructurePolynomial.lean","def_pos":[290,8],"def_end_pos":[290,29]}]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Degrees.lean","commit":"","full_name":"MvPolynomial.degreeOf_mul_X_ne","start":[264,0],"end":[273,86],"file_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ni j : σ\nf : MvPolynomial σ R\nh : i ≠ j\n⊢ degreeOf i (f * X j) = degreeOf i f","state_after":"no goals","tactic":"classical\n  repeat' rw [degreeOf_eq_sup (R := R) i]\n  rw [support_mul_X]\n  simp only [Finset.sup_map]\n  congr\n  ext\n  simp only [Finsupp.single, Nat.one_ne_zero, add_right_eq_self, addRightEmbedding_apply, coe_mk,\n    Pi.add_apply, comp_apply, ite_eq_right_iff, Finsupp.coe_add, Pi.single_eq_of_ne h]","premises":[{"full_name":"Finset.sup_map","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[66,8],"def_end_pos":[66,15]},{"full_name":"Finsupp.coe_add","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[888,25],"def_end_pos":[888,32]},{"full_name":"Finsupp.coe_mk","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[131,8],"def_end_pos":[131,14]},{"full_name":"Finsupp.single","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[228,4],"def_end_pos":[228,10]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"MvPolynomial.degreeOf_eq_sup","def_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","def_pos":[216,8],"def_end_pos":[216,23]},{"full_name":"MvPolynomial.support_mul_X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[657,8],"def_end_pos":[657,21]},{"full_name":"Nat.one_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[702,18],"def_end_pos":[702,29]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"Pi.single_eq_of_ne","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[271,2],"def_end_pos":[271,13]},{"full_name":"addRightEmbedding_apply","def_path":"Mathlib/Algebra/Group/Embedding.lean","def_pos":[31,23],"def_end_pos":[31,28]},{"full_name":"add_right_eq_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[232,2],"def_end_pos":[232,13]},{"full_name":"ite_eq_right_iff","def_path":".lake/packages/lean4/src/lean/Init/ByCases.lean","def_pos":[51,16],"def_end_pos":[51,32]}]}]}
{"url":"Mathlib/RingTheory/Idempotents.lean","commit":"","full_name":"OrthogonalIdempotents.lift_of_isNilpotent_ker_aux","start":[169,0],"end":[183,74],"file_path":"Mathlib/RingTheory/Idempotents.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nI : Type u_3\ne✝ : I → R\nhe✝ : OrthogonalIdempotents e✝\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn : ℕ\ne : Fin n → S\nhe : OrthogonalIdempotents e\nhe' : ∀ (i : Fin n), e i ∈ f.range\n⊢ ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e","state_after":"case zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nI : Type u_3\ne✝ : I → R\nhe✝ : OrthogonalIdempotents e✝\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne : Fin 0 → S\nhe : OrthogonalIdempotents e\nhe' : ∀ (i : Fin 0), e i ∈ f.range\n⊢ ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e\n\ncase succ\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nI : Type u_3\ne✝ : I → R\nhe✝ : OrthogonalIdempotents e✝\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn : ℕ\nIH :\n  ∀ {e : Fin n → S},\n    OrthogonalIdempotents e → (∀ (i : Fin n), e i ∈ f.range) → ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e\ne : Fin (n + 1) → S\nhe : OrthogonalIdempotents e\nhe' : ∀ (i : Fin (n + 1)), e i ∈ f.range\n⊢ ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e","tactic":"induction' n with n IH","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean","commit":"","full_name":"CategoryTheory.FunctorToTypes.binaryProductIso_inv_comp_snd","start":[107,0],"end":[110,49],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nF G : C ⥤ Type w\n⊢ (binaryProductIso F G).inv ≫ Limits.prod.snd = prod.snd","state_after":"no goals","tactic":"simp [binaryProductIso, binaryProductLimitCone]","premises":[{"full_name":"CategoryTheory.FunctorToTypes.binaryProductIso","def_path":"Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean","def_pos":[86,18],"def_end_pos":[86,34]},{"full_name":"CategoryTheory.FunctorToTypes.binaryProductLimitCone","def_path":"Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean","def_pos":[82,4],"def_end_pos":[82,26]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Nat.lean","commit":"","full_name":"Nat.card_uIcc","start":[85,0],"end":[87,99],"file_path":"Mathlib/Order/Interval/Finset/Nat.lean","tactics":[{"state_before":"a b c : ℕ\n⊢ a ⊔ b + 1 - a ⊓ b = (↑b - ↑a).natAbs + 1","state_after":"no goals","tactic":"rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega","premises":[{"full_name":"Int.natCast_inj","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[98,19],"def_end_pos":[98,30]},{"full_name":"Int.ofNat_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[302,21],"def_end_pos":[302,30]},{"full_name":"inf_eq_min","def_path":"Mathlib/Order/Lattice.lean","def_pos":[668,8],"def_end_pos":[668,18]},{"full_name":"sup_eq_max","def_path":"Mathlib/Order/Lattice.lean","def_pos":[665,8],"def_end_pos":[665,18]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean","commit":"","full_name":"IsCoprime.exists_SL2_col","start":[403,0],"end":[412,8],"file_path":"Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\na b : R\nhab : IsCoprime a b\nj : Fin 2\n⊢ ∃ g, ↑g 0 j = a ∧ ↑g 1 j = b","state_after":"case intro.intro\nR : Type u_1\ninst✝ : CommRing R\na b : R\nj : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ ∃ g, ↑g 0 j = a ∧ ↑g 1 j = b","tactic":"obtain ⟨u, v, h⟩ := hab","premises":[]}]}
{"url":"Mathlib/Computability/PartrecCode.lean","commit":"","full_name":"Nat.Partrec.Code.eval_prec_zero","start":[476,0],"end":[481,19],"file_path":"Mathlib/Computability/PartrecCode.lean","tactics":[{"state_before":"cf cg : Code\na : ℕ\n⊢ (cf.prec cg).eval (Nat.pair a 0) = cf.eval a","state_after":"cf cg : Code\na : ℕ\n⊢ Nat.rec (cf.eval (a, 0).1)\n      (fun y IH => do\n        let i ← IH\n        cg.eval (Nat.pair (a, 0).1 (Nat.pair y i)))\n      (a, 0).2 =\n    cf.eval a","tactic":"rw [eval, Nat.unpaired, Nat.unpair_pair]","premises":[{"full_name":"Nat.Partrec.Code.eval","def_path":"Mathlib/Computability/PartrecCode.lean","def_pos":[460,4],"def_end_pos":[460,8]},{"full_name":"Nat.unpair_pair","def_path":"Mathlib/Data/Nat/Pairing.lean","def_pos":[56,8],"def_end_pos":[56,19]},{"full_name":"Nat.unpaired","def_path":"Mathlib/Computability/Primrec.lean","def_pos":[59,4],"def_end_pos":[59,12]}]},{"state_before":"cf cg : Code\na : ℕ\n⊢ Nat.rec (cf.eval (a, 0).1)\n      (fun y IH => do\n        let i ← IH\n        cg.eval (Nat.pair (a, 0).1 (Nat.pair y i)))\n      (a, 0).2 =\n    cf.eval a","state_after":"cf cg : Code\na : ℕ\n⊢ Nat.rec (cf.eval a)\n      (fun y IH => do\n        let i ← IH\n        cg.eval (Nat.pair a (Nat.pair y i)))\n      0 =\n    cf.eval a","tactic":"simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only []","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.Simp.Config.arith","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[182,2],"def_end_pos":[182,7]},{"full_name":"Lean.Meta.Simp.Config.autoUnfold","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[187,2],"def_end_pos":[187,12]},{"full_name":"Lean.Meta.Simp.Config.beta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[157,2],"def_end_pos":[157,6]},{"full_name":"Lean.Meta.Simp.Config.contextual","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[136,2],"def_end_pos":[136,12]},{"full_name":"Lean.Meta.Simp.Config.decide","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[180,2],"def_end_pos":[180,8]},{"full_name":"Lean.Meta.Simp.Config.dsimp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[193,2],"def_end_pos":[193,7]},{"full_name":"Lean.Meta.Simp.Config.eta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[162,2],"def_end_pos":[162,5]},{"full_name":"Lean.Meta.Simp.Config.etaStruct","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[167,2],"def_end_pos":[167,11]},{"full_name":"Lean.Meta.Simp.Config.failIfUnchanged","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[198,2],"def_end_pos":[198,17]},{"full_name":"Lean.Meta.Simp.Config.ground","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[205,2],"def_end_pos":[205,8]},{"full_name":"Lean.Meta.Simp.Config.implicitDefEqProofs","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[228,2],"def_end_pos":[228,21]},{"full_name":"Lean.Meta.Simp.Config.index","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"Lean.Meta.Simp.Config.iota","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[171,2],"def_end_pos":[171,6]},{"full_name":"Lean.Meta.Simp.Config.maxDischargeDepth","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[131,2],"def_end_pos":[131,19]},{"full_name":"Lean.Meta.Simp.Config.maxSteps","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[126,2],"def_end_pos":[126,10]},{"full_name":"Lean.Meta.Simp.Config.memoize","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[140,2],"def_end_pos":[140,9]},{"full_name":"Lean.Meta.Simp.Config.singlePass","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[146,2],"def_end_pos":[146,12]},{"full_name":"Lean.Meta.Simp.Config.unfoldPartialApp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[210,2],"def_end_pos":[210,18]},{"full_name":"Lean.Meta.Simp.Config.zeta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[152,2],"def_end_pos":[152,6]},{"full_name":"Lean.Meta.Simp.Config.zetaDelta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[215,2],"def_end_pos":[215,11]},{"full_name":"Lean.Meta.Simp.neutralConfig","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[238,4],"def_end_pos":[238,17]}]},{"state_before":"cf cg : Code\na : ℕ\n⊢ Nat.rec (cf.eval a)\n      (fun y IH => do\n        let i ← IH\n        cg.eval (Nat.pair a (Nat.pair y i)))\n      0 =\n    cf.eval a","state_after":"no goals","tactic":"rw [Nat.rec_zero]","premises":[{"full_name":"Nat.rec_zero","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[674,6],"def_end_pos":[674,14]}]}]}
{"url":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","commit":"","full_name":"Asymptotics.isBigO_iff_isBoundedUnder_le_div","start":[1188,0],"end":[1193,83],"file_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nh : ∀ᶠ (x : α) in l, g'' x ≠ 0\n⊢ f =O[l] g'' ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => ‖f x‖ / ‖g'' x‖","state_after":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nh : ∀ᶠ (x : α) in l, g'' x ≠ 0\n⊢ (∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g'' x‖) ↔ ∃ b, ∀ᶠ (a : α) in l, ‖f a‖ / ‖g'' a‖ ≤ b","tactic":"simp only [isBigO_iff, IsBoundedUnder, IsBounded, eventually_map]","premises":[{"full_name":"Asymptotics.isBigO_iff","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[104,8],"def_end_pos":[104,18]},{"full_name":"Filter.IsBounded","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[48,4],"def_end_pos":[48,13]},{"full_name":"Filter.IsBoundedUnder","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[53,4],"def_end_pos":[53,18]},{"full_name":"Filter.eventually_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1650,8],"def_end_pos":[1650,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nh : ∀ᶠ (x : α) in l, g'' x ≠ 0\n⊢ (∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g'' x‖) ↔ ∃ b, ∀ᶠ (a : α) in l, ‖f a‖ / ‖g'' a‖ ≤ b","state_after":"no goals","tactic":"exact\n    exists_congr fun c =>\n      eventually_congr <| h.mono fun x hx => (div_le_iff <| norm_pos_iff.2 hx).symm","premises":[{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.eventually_congr","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1019,8],"def_end_pos":[1019,24]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"div_le_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[51,8],"def_end_pos":[51,18]},{"full_name":"exists_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[210,8],"def_end_pos":[210,20]},{"full_name":"norm_pos_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1216,29],"def_end_pos":[1216,41]}]}]}
{"url":"Mathlib/Analysis/BoundedVariation.lean","commit":"","full_name":"LocallyBoundedVariationOn.ae_differentiableWithinAt","start":[827,0],"end":[832,42],"file_path":"Mathlib/Analysis/BoundedVariation.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁴ : LinearOrder α\nE : Type u_2\ninst✝³ : PseudoEMetricSpace E\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nf : ℝ → V\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : ℝ) ∂volume.restrict s, DifferentiableWithinAt ℝ f s x","state_after":"α : Type u_1\ninst✝⁴ : LinearOrder α\nE : Type u_2\ninst✝³ : PseudoEMetricSpace E\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nf : ℝ → V\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","tactic":"rw [ae_restrict_iff' hs]","premises":[{"full_name":"MeasureTheory.ae_restrict_iff'","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[547,8],"def_end_pos":[547,24]}]},{"state_before":"α : Type u_1\ninst✝⁴ : LinearOrder α\nE : Type u_2\ninst✝³ : PseudoEMetricSpace E\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nf : ℝ → V\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","state_after":"no goals","tactic":"exact h.ae_differentiableWithinAt_of_mem","premises":[{"full_name":"LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[815,8],"def_end_pos":[815,40]}]}]}
{"url":"Mathlib/RingTheory/WittVector/Basic.lean","commit":"","full_name":"WittVector.mapFun.nsmul","start":[109,0],"end":[109,96],"file_path":"Mathlib/RingTheory/WittVector/Basic.lean","tactics":[{"state_before":"p : ℕ\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type u_4\nβ : Type u_5\nf : R →+* S\nx✝ y : 𝕎 R\nn : ℕ\nx : 𝕎 R\n⊢ mapFun (⇑f) (n • x) = n • mapFun (⇑f) x","state_after":"no goals","tactic":"map_fun_tac","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"MvPolynomial.coe_eval₂Hom","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[969,8],"def_end_pos":[969,20]},{"full_name":"MvPolynomial.eval₂Hom_congr","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[972,8],"def_end_pos":[972,22]},{"full_name":"MvPolynomial.map_aeval","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[1363,8],"def_end_pos":[1363,17]},{"full_name":"RingHom.ext_int","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[367,8],"def_end_pos":[367,15]},{"full_name":"WittVector.add_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[314,8],"def_end_pos":[314,17]},{"full_name":"WittVector.mapFun","def_path":"Mathlib/RingTheory/WittVector/Basic.lean","def_pos":[65,4],"def_end_pos":[65,10]},{"full_name":"WittVector.mk","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[57,4],"def_end_pos":[57,17]},{"full_name":"WittVector.mul_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[322,8],"def_end_pos":[322,17]},{"full_name":"WittVector.neg_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[326,8],"def_end_pos":[326,17]},{"full_name":"WittVector.nsmul_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[329,8],"def_end_pos":[329,19]},{"full_name":"WittVector.peval","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[146,4],"def_end_pos":[146,9]},{"full_name":"WittVector.pow_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[337,8],"def_end_pos":[337,17]},{"full_name":"WittVector.sub_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[318,8],"def_end_pos":[318,17]},{"full_name":"WittVector.zero_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[295,8],"def_end_pos":[295,18]},{"full_name":"WittVector.zsmul_coeff","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[333,8],"def_end_pos":[333,19]},{"full_name":"algebraMap_int_eq","def_path":"Mathlib/Algebra/Algebra/Basic.lean","def_pos":[254,8],"def_end_pos":[254,25]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean","commit":"","full_name":"midpoint_eq_right_iff","start":[155,0],"end":[157,51],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean","tactics":[{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx✝ y✝ z x y : P\n⊢ midpoint R x y = y ↔ x = y","state_after":"no goals","tactic":"rw [midpoint_comm, midpoint_eq_left_iff, eq_comm]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"midpoint_comm","def_path":"Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean","def_pos":[67,8],"def_end_pos":[67,21]},{"full_name":"midpoint_eq_left_iff","def_path":"Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean","def_pos":[148,8],"def_end_pos":[148,28]}]}]}
{"url":"Mathlib/Probability/Density.lean","commit":"","full_name":"MeasureTheory.measurable_pdf","start":[154,0],"end":[157,30],"file_path":"Mathlib/Probability/Density.lean","tactics":[{"state_before":"Ω : Type u_1\nE : Type u_2\ninst✝ : MeasurableSpace E\nm : MeasurableSpace Ω\nX : Ω → E\nℙ : Measure Ω\nμ : autoParam (Measure E) _auto✝\n⊢ Measurable (pdf X ℙ μ)","state_after":"no goals","tactic":"exact measurable_rnDeriv _ _","premises":[{"full_name":"MeasureTheory.Measure.measurable_rnDeriv","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[95,8],"def_end_pos":[95,26]}]}]}
{"url":"Mathlib/Algebra/Group/Commute/Defs.lean","commit":"","full_name":"AddCommute.add_neg_cancel","start":[203,0],"end":[205,33],"file_path":"Mathlib/Algebra/Group/Commute/Defs.lean","tactics":[{"state_before":"G : Type u_1\nM : Type u_2\nS : Type u_3\ninst✝ : Group G\na b : G\nh : Commute a b\n⊢ a * b * a⁻¹ = b","state_after":"no goals","tactic":"rw [h.eq, mul_inv_cancel_right]","premises":[{"full_name":"Commute.eq","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[52,18],"def_end_pos":[52,20]},{"full_name":"mul_inv_cancel_right","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1067,8],"def_end_pos":[1067,28]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"IsOpen.mul_right","start":[1155,0],"end":[1158,43],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\nhs : IsOpen s\n⊢ IsOpen (s * t)","state_after":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\nhs : IsOpen s\n⊢ IsOpen (⋃ a ∈ t, op a • s)","tactic":"rw [← iUnion_op_smul_set]","premises":[{"full_name":"Set.iUnion_op_smul_set","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[315,8],"def_end_pos":[315,26]}]},{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\nhs : IsOpen s\n⊢ IsOpen (⋃ a ∈ t, op a • s)","state_after":"no goals","tactic":"exact isOpen_biUnion fun a _ => hs.smul _","premises":[{"full_name":"IsOpen.smul","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[220,8],"def_end_pos":[220,19]},{"full_name":"isOpen_biUnion","def_path":"Mathlib/Topology/Basic.lean","def_pos":[102,8],"def_end_pos":[102,22]}]}]}
{"url":"Mathlib/Topology/Algebra/FilterBasis.lean","commit":"","full_name":"GroupFilterBasis.nhds_one_hasBasis","start":[170,0],"end":[174,32],"file_path":"Mathlib/Topology/Algebra/FilterBasis.lean","tactics":[{"state_before":"G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ (𝓝 1).HasBasis (fun V => V ∈ B) id","state_after":"G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ toFilterBasis.filter.HasBasis (fun V => V ∈ B) id","tactic":"rw [B.nhds_one_eq]","premises":[{"full_name":"GroupFilterBasis.nhds_one_eq","def_path":"Mathlib/Topology/Algebra/FilterBasis.lean","def_pos":[158,8],"def_end_pos":[158,19]}]},{"state_before":"G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ toFilterBasis.filter.HasBasis (fun V => V ∈ B) id","state_after":"no goals","tactic":"exact B.toFilterBasis.hasBasis","premises":[{"full_name":"FilterBasis.hasBasis","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[299,18],"def_end_pos":[299,45]}]}]}
{"url":"Mathlib/CategoryTheory/Monad/Algebra.lean","commit":"","full_name":"CategoryTheory.Monad.algebraEquivOfIsoMonads_counitIso","start":[241,0],"end":[253,72],"file_path":"Mathlib/CategoryTheory/Monad/Algebra.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT T₁ T₂ : Monad C\nh : T₁ ≅ T₂\n⊢ 𝟙 T₁ = h.hom ≫ h.inv","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT T₁ T₂ : Monad C\nh : T₁ ≅ T₂\n⊢ h.inv ≫ h.hom = 𝟙 T₂","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Complex/ReImTopology.lean","commit":"","full_name":"Complex.frontier_setOf_re_le","start":[111,0],"end":[113,62],"file_path":"Mathlib/Analysis/Complex/ReImTopology.lean","tactics":[{"state_before":"a : ℝ\n⊢ frontier {z | z.re ≤ a} = {z | z.re = a}","state_after":"no goals","tactic":"simpa only [frontier_Iic] using frontier_preimage_re (Iic a)","premises":[{"full_name":"Complex.frontier_preimage_re","def_path":"Mathlib/Analysis/Complex/ReImTopology.lean","def_pos":[73,8],"def_end_pos":[73,28]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"frontier_Iic","def_path":"Mathlib/Topology/Order/DenselyOrdered.lean","def_pos":[144,8],"def_end_pos":[144,20]}]}]}
{"url":"Mathlib/Algebra/Order/Group/PosPart.lean","commit":"","full_name":"leOnePart_eq_one","start":[119,0],"end":[120,71],"file_path":"Mathlib/Algebra/Order/Group/PosPart.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Lattice α\ninst✝¹ : Group α\na : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\n⊢ a⁻ᵐ = 1 ↔ 1 ≤ a","state_after":"no goals","tactic":"simp [leOnePart_eq_one']","premises":[{"full_name":"leOnePart_eq_one'","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[96,6],"def_end_pos":[96,23]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","commit":"","full_name":"CochainComplex.mappingCone.inl_v_fst_v","start":[71,0],"end":[75,17],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{?u.16556, u_1} C\ninst✝³ : Category.{?u.16560, u_2} D\ninst✝² : Preadditive C\ninst✝¹ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\np q : ℤ\nhpq : q + 1 = p\n⊢ p + -1 = q","state_after":"no goals","tactic":"rw [← hpq, add_neg_cancel_right]","premises":[{"full_name":"add_neg_cancel_right","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1066,2],"def_end_pos":[1066,13]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{?u.16560, u_2} D\ninst✝² : Preadditive C\ninst✝¹ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\np q : ℤ\nhpq : q + 1 = p\n⊢ (inl φ).v p q ⋯ ≫ (↑(fst φ)).v q p hpq = 𝟙 (F.X p)","state_after":"no goals","tactic":"simp [inl, fst]","premises":[{"full_name":"CochainComplex.mappingCone.fst","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[60,18],"def_end_pos":[60,21]},{"full_name":"CochainComplex.mappingCone.inl","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[53,18],"def_end_pos":[53,21]}]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","commit":"","full_name":"cfcₙ_apply_of_not_continuousOn","start":[215,0],"end":[217,76],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : TopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : NonUnitalContinuousFunctionalCalculus R p\nf✝ g : R → R\na✝ : A\nhf✝ : autoParam (ContinuousOn f✝ (σₙ R a✝)) _auto✝\nhf0 : autoParam (f✝ 0 = 0) _auto✝\nhg : autoParam (ContinuousOn g (σₙ R a✝)) _auto✝\nhg0 : autoParam (g 0 = 0) _auto✝\nha : autoParam (p a✝) _auto✝\nf : R → R\na : A\nhf : ¬ContinuousOn f (σₙ R a)\n⊢ cfcₙ f a = 0","state_after":"no goals","tactic":"rw [cfcₙ_def, dif_neg (not_and_of_not_right _ (not_and_of_not_left _ hf))]","premises":[{"full_name":"cfcₙ_def","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","def_pos":[186,30],"def_end_pos":[186,34]},{"full_name":"dif_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[954,8],"def_end_pos":[954,15]},{"full_name":"not_and_of_not_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[48,8],"def_end_pos":[48,27]},{"full_name":"not_and_of_not_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[49,8],"def_end_pos":[49,28]}]}]}
{"url":"Mathlib/CategoryTheory/Elements.lean","commit":"","full_name":"CategoryTheory.CategoryOfElements.costructuredArrow_yoneda_equivalence_naturality","start":[257,0],"end":[268,20],"file_path":"Mathlib/CategoryTheory/Elements.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α","state_after":"case h_obj\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ ∀ (X : F₁.Elementsᵒᵖ),\n    ((map α).op ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X\n\ncase h_map\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nF₁ F₂ : Cᵒᵖ ⥤ Type v\nα : F₁ ⟶ F₂\n⊢ autoParam\n    (∀ (X Y : F₁.Elementsᵒᵖ) (f : X ⟶ Y),\n      ((map α).op ⋙ toCostructuredArrow F₂).map f =\n        eqToHom ⋯ ≫ (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫ eqToHom ⋯)\n    _auto✝","tactic":"fapply Functor.ext","premises":[{"full_name":"CategoryTheory.Functor.ext","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[179,8],"def_end_pos":[179,11]}]}]}
{"url":"Mathlib/Algebra/Group/Int.lean","commit":"","full_name":"Int.isUnit_ne_iff_eq_neg","start":[99,0],"end":[100,96],"file_path":"Mathlib/Algebra/Group/Int.lean","tactics":[{"state_before":"u v : ℤ\nhu : IsUnit u\nhv : IsUnit v\n⊢ u ≠ v ↔ u = -v","state_after":"no goals","tactic":"obtain rfl | rfl := isUnit_eq_one_or hu <;> obtain rfl | rfl := isUnit_eq_one_or hv <;> decide","premises":[{"full_name":"Int.isUnit_eq_one_or","def_path":"Mathlib/Algebra/Group/Int.lean","def_pos":[96,6],"def_end_pos":[96,22]}]}]}
{"url":"Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean","commit":"","full_name":"ContinuousLinearMapWOT.continuous_dual_apply","start":[171,0],"end":[172,60],"file_path":"Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : Module 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : E\ny : NormedSpace.Dual 𝕜 F\n⊢ Continuous fun A => y (A x)","state_after":"no goals","tactic":"refine (continuous_pi_iff.mp continuous_inducingFn) ⟨x, y⟩","premises":[{"full_name":"ContinuousLinearMapWOT.continuous_inducingFn","def_path":"Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean","def_pos":[168,6],"def_end_pos":[168,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"continuous_pi_iff","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1133,8],"def_end_pos":[1133,25]}]}]}
{"url":"Mathlib/Topology/MetricSpace/ThickenedIndicator.lean","commit":"","full_name":"thickenedIndicatorAux_zero","start":[85,0],"end":[93,17],"file_path":"Mathlib/Topology/MetricSpace/ThickenedIndicator.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : x ∉ thickening δ E\n⊢ thickenedIndicatorAux δ E x = 0","state_after":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ thickenedIndicatorAux δ E x = 0","tactic":"rw [thickening, mem_setOf_eq, not_lt] at x_out","premises":[{"full_name":"Metric.thickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[49,4],"def_end_pos":[49,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ thickenedIndicatorAux δ E x = 0","state_after":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ = 0","tactic":"unfold thickenedIndicatorAux","premises":[{"full_name":"thickenedIndicatorAux","def_path":"Mathlib/Topology/MetricSpace/ThickenedIndicator.lean","def_pos":[52,4],"def_end_pos":[52,25]}]},{"state_before":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ = 0","state_after":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥","tactic":"apply le_antisymm _ bot_le","premises":[{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥","state_after":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - ENNReal.ofReal δ / ENNReal.ofReal δ\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥","tactic":"have key := tsub_le_tsub\n    (@rfl _ (1 : ℝ≥0∞)).le (ENNReal.div_le_div x_out (@rfl _ (ENNReal.ofReal δ : ℝ≥0∞)).le)","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.div_le_div","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[332,28],"def_end_pos":[332,38]},{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"tsub_le_tsub","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[113,18],"def_end_pos":[113,30]}]},{"state_before":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - ENNReal.ofReal δ / ENNReal.ofReal δ\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥","state_after":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥","tactic":"rw [ENNReal.div_self (ne_of_gt (ENNReal.ofReal_pos.mpr δ_pos)) ofReal_ne_top] at key","premises":[{"full_name":"ENNReal.div_self","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[375,18],"def_end_pos":[375,26]},{"full_name":"ENNReal.ofReal_ne_top","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[286,16],"def_end_pos":[286,29]},{"full_name":"ENNReal.ofReal_pos","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[192,8],"def_end_pos":[192,18]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]},{"state_before":"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥","state_after":"no goals","tactic":"simpa using key","premises":[]}]}
{"url":"Mathlib/Order/Interval/Multiset.lean","commit":"","full_name":"Multiset.Ico_filter_lt_of_le_right","start":[207,0],"end":[210,5],"file_path":"Mathlib/Order/Interval/Multiset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na b c : α\ninst✝ : DecidablePred fun x => x < c\nhcb : c ≤ b\n⊢ filter (fun x => x < c) (Ico a b) = Ico a c","state_after":"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na b c : α\ninst✝ : DecidablePred fun x => x < c\nhcb : c ≤ b\n⊢ (Finset.Ico a c).val = Ico a c","tactic":"rw [Ico, ← Finset.filter_val, Finset.Ico_filter_lt_of_le_right hcb]","premises":[{"full_name":"Finset.Ico_filter_lt_of_le_right","def_path":"Mathlib/Order/Interval/Finset/Basic.lean","def_pos":[270,8],"def_end_pos":[270,33]},{"full_name":"Finset.filter_val","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2148,8],"def_end_pos":[2148,18]},{"full_name":"Multiset.Ico","def_path":"Mathlib/Order/Interval/Multiset.lean","def_pos":[48,4],"def_end_pos":[48,7]}]},{"state_before":"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na b c : α\ninst✝ : DecidablePred fun x => x < c\nhcb : c ≤ b\n⊢ (Finset.Ico a c).val = Ico a c","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/RingTheory/ChainOfDivisors.lean","commit":"","full_name":"factor_orderIso_map_one_eq_bot","start":[212,0],"end":[219,17],"file_path":"Mathlib/RingTheory/ChainOfDivisors.lean","tactics":[{"state_before":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\n⊢ ↑(d ⟨1, ⋯⟩) = 1","state_after":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\n⊢ ↑(d ⟨1, ⋯⟩) = 1","tactic":"letI : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot bot_le","premises":[{"full_name":"Associates","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[729,7],"def_end_pos":[729,17]},{"full_name":"OrderBot","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[172,6],"def_end_pos":[172,14]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"Subtype.orderBot","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[607,17],"def_end_pos":[607,25]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]}]},{"state_before":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\n⊢ ↑(d ⟨1, ⋯⟩) = 1","state_after":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot ⋯\n⊢ ↑(d ⟨1, ⋯⟩) = 1","tactic":"letI : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot bot_le","premises":[{"full_name":"Associates","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[729,7],"def_end_pos":[729,17]},{"full_name":"OrderBot","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[172,6],"def_end_pos":[172,14]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"Subtype.orderBot","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[607,17],"def_end_pos":[607,25]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]}]},{"state_before":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot ⋯\n⊢ ↑(d ⟨1, ⋯⟩) = 1","state_after":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot ⋯\n⊢ d ⊥ = ⊥","tactic":"simp only [← Associates.bot_eq_one, Subtype.mk_bot, bot_le, Subtype.coe_eq_bot_iff]","premises":[{"full_name":"Associates.bot_eq_one","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[783,8],"def_end_pos":[783,18]},{"full_name":"Subtype.coe_eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[641,8],"def_end_pos":[641,22]},{"full_name":"Subtype.mk_bot","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[627,8],"def_end_pos":[627,14]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]}]},{"state_before":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot ⋯\n⊢ d ⊥ = ⊥","state_after":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot ⋯\nthis : BotHomClass ({ l // l ≤ m } ≃o { l // l ≤ n }) { l // l ≤ m } { l // l ≤ n } := OrderIsoClass.toBotHomClass\n⊢ d ⊥ = ⊥","tactic":"letI : BotHomClass ({ l // l ≤ m } ≃o { l // l ≤ n }) _ _ := OrderIsoClass.toBotHomClass","premises":[{"full_name":"BotHomClass","def_path":"Mathlib/Order/Hom/Bounded.lean","def_pos":[69,6],"def_end_pos":[69,17]},{"full_name":"OrderIso","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[97,7],"def_end_pos":[97,15]},{"full_name":"OrderIsoClass.toBotHomClass","def_path":"Mathlib/Order/Hom/Bounded.lean","def_pos":[119,27],"def_end_pos":[119,54]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]}]},{"state_before":"M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝ : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l ≤ m } ≃o { l // l ≤ n }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot ⋯\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot ⋯\nthis : BotHomClass ({ l // l ≤ m } ≃o { l // l ≤ n }) { l // l ≤ m } { l // l ≤ n } := OrderIsoClass.toBotHomClass\n⊢ d ⊥ = ⊥","state_after":"no goals","tactic":"exact map_bot d","premises":[{"full_name":"BotHomClass.map_bot","def_path":"Mathlib/Order/Hom/Bounded.lean","def_pos":[71,2],"def_end_pos":[71,9]}]}]}
{"url":"Mathlib/Computability/Primrec.lean","commit":"","full_name":"Primrec.nat_mod","start":[699,0],"end":[702,44],"file_path":"Mathlib/Computability/Primrec.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nσ : Type u_5\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nm n : ℕ\n⊢ (m, n).1 - (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2 = m % n","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nσ : Type u_5\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nm n : ℕ\n⊢ (m, n).1 = m % n + (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2","tactic":"apply Nat.sub_eq_of_eq_add","premises":[{"full_name":"Nat.sub_eq_of_eq_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[920,18],"def_end_pos":[920,34]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nσ : Type u_5\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nm n : ℕ\n⊢ (m, n).1 = m % n + (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2","state_after":"no goals","tactic":"simp [add_comm (m % n), Nat.div_add_mod]","premises":[{"full_name":"Nat.div_add_mod","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[186,8],"def_end_pos":[186,19]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Abelian.lean","commit":"","full_name":"CategoryTheory.ShortComplex.LeftHomologyData.ofAbelian_i","start":[71,0],"end":[103,16],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Abelian.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\n⊢ S.LeftHomologyData","tactic":"let γ := kernel.ι S.g ≫ cokernel.π S.f","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.cokernel.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[678,7],"def_end_pos":[678,17]},{"full_name":"CategoryTheory.Limits.kernel.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[244,7],"def_end_pos":[244,15]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\n⊢ S.LeftHomologyData","tactic":"let f' := kernel.lift S.g S.f S.zero","premises":[{"full_name":"CategoryTheory.Limits.kernel.lift","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[260,7],"def_end_pos":[260,18]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]},{"full_name":"CategoryTheory.ShortComplex.zero","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[41,2],"def_end_pos":[41,6]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\n⊢ S.LeftHomologyData","tactic":"have hf' : f' = kernel.lift γ f' (by simp [γ, f']) ≫ kernel.ι γ := by rw [kernel.lift_ι]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.kernel.lift","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[260,7],"def_end_pos":[260,18]},{"full_name":"CategoryTheory.Limits.kernel.lift_ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[264,8],"def_end_pos":[264,21]},{"full_name":"CategoryTheory.Limits.kernel.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[244,7],"def_end_pos":[244,15]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\n⊢ S.LeftHomologyData","tactic":"have wπ : f' ≫ cokernel.π (kernel.ι γ) = 0 := by\n    rw [hf']\n    simp only [assoc, cokernel.condition, comp_zero]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.cokernel.condition","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[686,8],"def_end_pos":[686,26]},{"full_name":"CategoryTheory.Limits.cokernel.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[678,7],"def_end_pos":[678,17]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.Limits.kernel.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[244,7],"def_end_pos":[244,15]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\n⊢ S.LeftHomologyData","tactic":"let e : Abelian.image S.f ≅ kernel γ :=\n    IsLimit.conePointUniqueUpToIso S.abelianImageToKernelIsKernel (limit.isLimit _)","premises":[{"full_name":"CategoryTheory.Abelian.image","def_path":"Mathlib/CategoryTheory/Abelian/Images.lean","def_pos":[41,17],"def_end_pos":[41,22]},{"full_name":"CategoryTheory.Iso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[49,10],"def_end_pos":[49,13]},{"full_name":"CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[133,4],"def_end_pos":[133,26]},{"full_name":"CategoryTheory.Limits.kernel","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[240,7],"def_end_pos":[240,13]},{"full_name":"CategoryTheory.Limits.limit.isLimit","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[156,4],"def_end_pos":[156,17]},{"full_name":"CategoryTheory.ShortComplex.abelianImageToKernelIsKernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/Abelian.lean","def_pos":[59,18],"def_end_pos":[59,46]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\nhe : e.hom ≫ kernel.ι γ = S.abelianImageToKernel\n⊢ S.LeftHomologyData","tactic":"have he : e.hom ≫ kernel.ι γ = S.abelianImageToKernel :=\n    IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingParallelPair.zero","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso_hom_comp","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[137,8],"def_end_pos":[137,39]},{"full_name":"CategoryTheory.Limits.WalkingParallelPair.zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[60,4],"def_end_pos":[60,8]},{"full_name":"CategoryTheory.Limits.kernel.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[244,7],"def_end_pos":[244,15]},{"full_name":"CategoryTheory.ShortComplex.abelianImageToKernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/Abelian.lean","def_pos":[40,18],"def_end_pos":[40,38]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\nhe : e.hom ≫ kernel.ι γ = S.abelianImageToKernel\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\nhe : e.hom ≫ kernel.ι γ = S.abelianImageToKernel\nfac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ\n⊢ S.LeftHomologyData","tactic":"have fac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ := by\n    rw [hf', he]\n    simp only [f', kernel.lift_ι, abelianImageToKernel, ← cancel_mono (kernel.ι S.g), assoc]","premises":[{"full_name":"CategoryTheory.Abelian.factorThruImage","def_path":"Mathlib/CategoryTheory/Abelian/Images.lean","def_pos":[49,17],"def_end_pos":[49,32]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Limits.kernel.lift_ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[264,8],"def_end_pos":[264,21]},{"full_name":"CategoryTheory.Limits.kernel.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[244,7],"def_end_pos":[244,15]},{"full_name":"CategoryTheory.ShortComplex.abelianImageToKernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/Abelian.lean","def_pos":[40,18],"def_end_pos":[40,38]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\nhe : e.hom ≫ kernel.ι γ = S.abelianImageToKernel\nfac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ\n⊢ S.LeftHomologyData","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\nhe : e.hom ≫ kernel.ι γ = S.abelianImageToKernel\nfac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ\nhπ : IsColimit (CokernelCofork.ofπ (cokernel.π (kernel.ι γ)) wπ)\n⊢ S.LeftHomologyData","tactic":"have hπ : IsColimit (CokernelCofork.ofπ _ wπ) :=\n    CokernelCofork.IsColimit.ofπ _ _\n    (fun x hx => cokernel.desc _ x (by\n      simpa only [← cancel_epi e.hom, ← cancel_epi (Abelian.factorThruImage S.f),\n        comp_zero, fac, assoc] using hx))\n    (fun x hx => cokernel.π_desc _ _ _)\n    (fun x hx b hb => coequalizer.hom_ext (by simp only [hb, cokernel.π_desc]))","premises":[{"full_name":"CategoryTheory.Abelian.factorThruImage","def_path":"Mathlib/CategoryTheory/Abelian/Images.lean","def_pos":[49,17],"def_end_pos":[49,32]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Limits.CokernelCofork.IsColimit.ofπ","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[563,4],"def_end_pos":[563,32]},{"full_name":"CategoryTheory.Limits.CokernelCofork.ofπ","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[524,7],"def_end_pos":[524,25]},{"full_name":"CategoryTheory.Limits.IsColimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[507,10],"def_end_pos":[507,19]},{"full_name":"CategoryTheory.Limits.coequalizer.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[896,8],"def_end_pos":[896,27]},{"full_name":"CategoryTheory.Limits.cokernel.desc","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[696,7],"def_end_pos":[696,20]},{"full_name":"CategoryTheory.Limits.cokernel.π_desc","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[700,8],"def_end_pos":[700,23]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\nhe : e.hom ≫ kernel.ι γ = S.abelianImageToKernel\nfac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ\nhπ : IsColimit (CokernelCofork.ofπ (cokernel.π (kernel.ι γ)) wπ)\n⊢ S.LeftHomologyData","state_after":"no goals","tactic":"exact\n    { K := kernel S.g,\n      H := Abelian.coimage (kernel.ι S.g ≫ cokernel.π S.f)\n      i := kernel.ι _,\n      π := cokernel.π _\n      wi := kernel.condition _\n      hi := kernelIsKernel _\n      wπ := wπ\n      hπ := hπ }","premises":[{"full_name":"CategoryTheory.Abelian.coimage","def_path":"Mathlib/CategoryTheory/Abelian/Images.lean","def_pos":[65,17],"def_end_pos":[65,24]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.cokernel.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[678,7],"def_end_pos":[678,17]},{"full_name":"CategoryTheory.Limits.kernel","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[240,7],"def_end_pos":[240,13]},{"full_name":"CategoryTheory.Limits.kernel.condition","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[251,8],"def_end_pos":[251,24]},{"full_name":"CategoryTheory.Limits.kernel.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[244,7],"def_end_pos":[244,15]},{"full_name":"CategoryTheory.Limits.kernelIsKernel","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[255,4],"def_end_pos":[255,18]},{"full_name":"CategoryTheory.ShortComplex.f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[37,2],"def_end_pos":[37,3]},{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"mul_eq_of_eq_div'","start":[900,0],"end":[902,57],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : CommGroup G\na b c d : G\nh : b = c / a\n⊢ a * b = c","state_after":"no goals","tactic":"rw [h, div_eq_mul_inv, mul_comm c, mul_inv_cancel_left]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_inv_cancel_left","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1063,8],"def_end_pos":[1063,27]}]}]}
{"url":"Mathlib/LinearAlgebra/TensorProduct/Pi.lean","commit":"","full_name":"TensorProduct.piScalarRight_symm_single","start":[89,0],"end":[92,22],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Pi.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nN : Type u_3\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx : N\ni : ι\n⊢ (piScalarRight R S N ι).symm (Pi.single i x) = x ⊗ₜ[R] Pi.single i 1","state_after":"no goals","tactic":"simp [piScalarRight]","premises":[{"full_name":"TensorProduct.piScalarRight","def_path":"Mathlib/LinearAlgebra/TensorProduct/Pi.lean","def_pos":[77,18],"def_end_pos":[77,31]}]}]}
{"url":"Mathlib/Algebra/Star/Subalgebra.lean","commit":"","full_name":"StarAlgHom.range_eq_map_top","start":[705,0],"end":[707,84],"file_path":"Mathlib/Algebra/Star/Subalgebra.lean","tactics":[{"state_before":"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarRing B\ninst✝³ : StarModule R B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F R A B\ninst✝ : StarAlgHomClass F R A B\nf g : F\nφ : A →⋆ₐ[R] B\nx : B\n⊢ x ∈ φ.range → x ∈ map φ ⊤","state_after":"case intro\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarRing B\ninst✝³ : StarModule R B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F R A B\ninst✝ : StarAlgHomClass F R A B\nf g : F\nφ : A →⋆ₐ[R] B\nx : B\na : A\nha : φ.toRingHom a = x\n⊢ x ∈ map φ ⊤","tactic":"rintro ⟨a, ha⟩","premises":[]},{"state_before":"case intro\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarRing B\ninst✝³ : StarModule R B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F R A B\ninst✝ : StarAlgHomClass F R A B\nf g : F\nφ : A →⋆ₐ[R] B\nx : B\na : A\nha : φ.toRingHom a = x\n⊢ x ∈ map φ ⊤","state_after":"no goals","tactic":"exact ⟨a, by simp, ha⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarRing B\ninst✝³ : StarModule R B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F R A B\ninst✝ : StarAlgHomClass F R A B\nf g : F\nφ : A →⋆ₐ[R] B\nx : B\n⊢ x ∈ map φ ⊤ → x ∈ φ.range","state_after":"case intro.intro\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarRing B\ninst✝³ : StarModule R B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F R A B\ninst✝ : StarAlgHomClass F R A B\nf g : F\nφ : A →⋆ₐ[R] B\nx : B\na : A\nha : ↑φ.toAlgHom a = x\n⊢ x ∈ φ.range","tactic":"rintro ⟨a, -, ha⟩","premises":[]},{"state_before":"case intro.intro\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarRing B\ninst✝³ : StarModule R B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F R A B\ninst✝ : StarAlgHomClass F R A B\nf g : F\nφ : A →⋆ₐ[R] B\nx : B\na : A\nha : ↑φ.toAlgHom a = x\n⊢ x ∈ φ.range","state_after":"no goals","tactic":"exact ⟨a, ha⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","commit":"","full_name":"FractionalIdeal.zero_le","start":[387,0],"end":[391,18],"file_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\n⊢ 0 ≤ I","state_after":"R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ x ∈ I","tactic":"intro x hx","premises":[]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ x ∈ I","state_after":"R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ 0 ∈ I","tactic":"rw [(mem_zero_iff _).mp hx]","premises":[{"full_name":"FractionalIdeal.mem_zero_iff","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[277,8],"def_end_pos":[277,20]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ 0 ∈ I","state_after":"no goals","tactic":"exact zero_mem I","premises":[{"full_name":"FractionalIdeal.zero_mem","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[185,6],"def_end_pos":[185,14]}]}]}
{"url":"Mathlib/Algebra/Algebra/Unitization.lean","commit":"","full_name":"Unitization.inr_star","start":[529,0],"end":[532,55],"file_path":"Mathlib/Algebra/Algebra/Unitization.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝² : AddMonoid R\ninst✝¹ : StarAddMonoid R\ninst✝ : Star A\na : A\n⊢ (↑(star a)).fst = (star ↑a).fst","state_after":"no goals","tactic":"simp only [fst_star, star_zero, fst_inr]","premises":[{"full_name":"Unitization.fst_inr","def_path":"Mathlib/Algebra/Algebra/Unitization.lean","def_pos":[113,8],"def_end_pos":[113,15]},{"full_name":"Unitization.fst_star","def_path":"Mathlib/Algebra/Algebra/Unitization.lean","def_pos":[517,8],"def_end_pos":[517,16]},{"full_name":"star_zero","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[243,8],"def_end_pos":[243,17]}]}]}
{"url":"Mathlib/Algebra/Order/Ring/Abs.lean","commit":"","full_name":"abs_le_of_sq_le_sq","start":[115,0],"end":[116,38],"file_path":"Mathlib/Algebra/Order/Ring/Abs.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedRing α\nn : ℕ\na b c : α\nh : a ^ 2 ≤ b ^ 2\nhb : 0 ≤ b\n⊢ |a| ≤ b","state_after":"no goals","tactic":"rwa [← abs_of_nonneg hb, ← sq_le_sq]","premises":[{"full_name":"abs_of_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[76,2],"def_end_pos":[76,13]},{"full_name":"sq_le_sq","def_path":"Mathlib/Algebra/Order/Ring/Abs.lean","def_pos":[102,6],"def_end_pos":[102,14]}]}]}
{"url":"Mathlib/Data/List/OfFn.lean","commit":"","full_name":"List.nthLe_ofFn","start":[78,0],"end":[81,14],"file_path":"Mathlib/Data/List/OfFn.lean","tactics":[{"state_before":"α : Type u\nn : ℕ\nf : Fin n → α\ni : Fin n\n⊢ (ofFn f).nthLe ↑i ⋯ = f i","state_after":"no goals","tactic":"simp [nthLe]","premises":[{"full_name":"List.nthLe","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[584,4],"def_end_pos":[584,9]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Basic.lean","commit":"","full_name":"Uniform.continuousOn_iff'_right","start":[1719,0],"end":[1721,52],"file_path":"Mathlib/Topology/UniformSpace/Basic.lean","tactics":[{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nf : β → α\ns : Set β\n⊢ ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α)","state_after":"no goals","tactic":"simp [ContinuousOn, continuousWithinAt_iff'_right]","premises":[{"full_name":"ContinuousOn","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[164,4],"def_end_pos":[164,16]},{"full_name":"Uniform.continuousWithinAt_iff'_right","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[1711,8],"def_end_pos":[1711,37]}]}]}
{"url":"Mathlib/Order/Interval/Set/Pi.lean","commit":"","full_name":"Set.pi_univ_Ioc_update_right","start":[86,0],"end":[94,5],"file_path":"Mathlib/Order/Interval/Set/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\n⊢ (univ.pi fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ univ.pi fun i => Ioc (x i) (y i)","state_after":"ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ (univ.pi fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ univ.pi fun i => Ioc (x i) (y i)","tactic":"have : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀) := by\n    rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, inter_left_comm,\n      inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"Set.Iic_subset_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[341,8],"def_end_pos":[341,22]},{"full_name":"Set.Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[62,4],"def_end_pos":[62,7]},{"full_name":"Set.Ioi_inter_Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[512,8],"def_end_pos":[512,21]},{"full_name":"Set.inter_eq_self_of_subset_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[781,8],"def_end_pos":[781,36]},{"full_name":"Set.inter_left_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[754,8],"def_end_pos":[754,23]}]},{"state_before":"ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ (univ.pi fun i => Ioc (x i) (update y i₀ m i)) = {z | z i₀ ≤ m} ∩ univ.pi fun i => Ioc (x i) (y i)","state_after":"ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ ({x_1 | x_1 i₀ ∈ Iic m ∩ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j => Ioc (x j) (y j)) =\n    {z | z i₀ ≤ m} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j => Ioc (x j) (y j)","tactic":"simp_rw [univ_pi_update i₀ y m fun i z ↦ Ioc (x i) z, ← pi_inter_compl ({i₀} : Set ι),\n    singleton_pi', ← inter_assoc, this]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[62,4],"def_end_pos":[62,7]},{"full_name":"Set.inter_assoc","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[745,8],"def_end_pos":[745,19]},{"full_name":"Set.pi_inter_compl","def_path":"Mathlib/Data/Set/Prod.lean","def_pos":[746,8],"def_end_pos":[746,22]},{"full_name":"Set.singleton_pi'","def_path":"Mathlib/Data/Set/Prod.lean","def_pos":[702,8],"def_end_pos":[702,21]},{"full_name":"Set.univ_pi_update","def_path":"Mathlib/Data/Set/Prod.lean","def_pos":[764,8],"def_end_pos":[764,22]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Preorder (α i)\nx✝ y✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nx y : (i : ι) → α i\ni₀ : ι\nm : α i₀\nhm : m ≤ y i₀\nthis : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀)\n⊢ ({x_1 | x_1 i₀ ∈ Iic m ∩ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j => Ioc (x j) (y j)) =\n    {z | z i₀ ≤ m} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ {i₀}ᶜ.pi fun j => Ioc (x j) (y j)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","commit":"","full_name":"CategoryTheory.Limits.coprod.map_desc","start":[744,0],"end":[748,14],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nX Y S T U V W : C\ninst✝¹ : HasBinaryCoproduct U W\ninst✝ : HasBinaryCoproduct T V\nf : U ⟶ S\ng : W ⟶ S\nh : T ⟶ U\nk : V ⟶ W\n⊢ map h k ≫ desc f g = desc (h ≫ f) (k ≫ g)","state_after":"no goals","tactic":"ext <;> simp","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Operator/Banach.lean","commit":"","full_name":"ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz","start":[529,0],"end":[536,81],"file_path":"Mathlib/Analysis/Normed/Operator/Banach.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\n⊢ Bijective ⇑f ↔ (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f","state_after":"case eq_top\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nh : Bijective ⇑f\n⊢ (LinearMap.range f).topologicalClosure = ⊤\n\ncase anti\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nh : Bijective ⇑f\n⊢ ∃ c, AntilipschitzWith c ⇑f\n\ncase surj\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nx✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f\nhd : (LinearMap.range f).topologicalClosure = ⊤\nc : ℝ≥0\nhf : AntilipschitzWith c ⇑f\n⊢ Surjective ⇑f","tactic":"refine ⟨fun h ↦ ⟨?eq_top, ?anti⟩, fun ⟨hd, c, hf⟩ ↦ ⟨hf.injective, ?surj⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"AntilipschitzWith.injective","def_path":"Mathlib/Topology/MetricSpace/Antilipschitz.lean","def_pos":[90,18],"def_end_pos":[90,27]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"case eq_top\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nh : Bijective ⇑f\n⊢ (LinearMap.range f).topologicalClosure = ⊤\n\ncase anti\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nh : Bijective ⇑f\n⊢ ∃ c, AntilipschitzWith c ⇑f\n\ncase surj\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nx✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f\nhd : (LinearMap.range f).topologicalClosure = ⊤\nc : ℝ≥0\nhf : AntilipschitzWith c ⇑f\n⊢ Surjective ⇑f","state_after":"case anti\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nh : Bijective ⇑f\n⊢ ∃ c, AntilipschitzWith c ⇑f\n\ncase surj\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nx✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f\nhd : (LinearMap.range f).topologicalClosure = ⊤\nc : ℝ≥0\nhf : AntilipschitzWith c ⇑f\n⊢ Surjective ⇑f","tactic":"case eq_top => simpa [SetLike.ext'_iff] using h.2.denseRange.closure_eq","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Function.Surjective.denseRange","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1536,8],"def_end_pos":[1536,38]},{"full_name":"SetLike.ext'_iff","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[157,8],"def_end_pos":[157,16]}]},{"state_before":"case anti\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nh : Bijective ⇑f\n⊢ ∃ c, AntilipschitzWith c ⇑f\n\ncase surj\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nx✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f\nhd : (LinearMap.range f).topologicalClosure = ⊤\nc : ℝ≥0\nhf : AntilipschitzWith c ⇑f\n⊢ Surjective ⇑f","state_after":"case surj\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nx✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f\nhd : (LinearMap.range f).topologicalClosure = ⊤\nc : ℝ≥0\nhf : AntilipschitzWith c ⇑f\n⊢ Surjective ⇑f","tactic":"case anti =>\n    refine ⟨_, ContinuousLinearEquiv.ofBijective f ?_ ?_ |>.antilipschitz⟩ <;>\n    simp only [LinearMap.range_eq_top, LinearMapClass.ker_eq_bot, h.1, h.2]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ContinuousLinearEquiv.antilipschitz","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean","def_pos":[293,18],"def_end_pos":[293,31]},{"full_name":"ContinuousLinearEquiv.ofBijective","def_path":"Mathlib/Analysis/Normed/Operator/Banach.lean","def_pos":[360,18],"def_end_pos":[360,29]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"LinearMap.range_eq_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[89,8],"def_end_pos":[89,20]},{"full_name":"LinearMapClass.ker_eq_bot","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[180,8],"def_end_pos":[180,40]}]},{"state_before":"case surj\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁹ : RingHomInvPair σ σ'\ninst✝⁸ : RingHomInvPair σ' σ\ninst✝⁷ : RingHomIsometric σ\ninst✝⁶ : RingHomIsometric σ'\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜' F\nf✝ : E →SL[σ] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →SL[σ] F\nx✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f\nhd : (LinearMap.range f).topologicalClosure = ⊤\nc : ℝ≥0\nhf : AntilipschitzWith c ⇑f\n⊢ Surjective ⇑f","state_after":"no goals","tactic":"case surj => rwa [← LinearMap.range_eq_top, ← closed_range_of_antilipschitz hf]","premises":[{"full_name":"ContinuousLinearMap.closed_range_of_antilipschitz","def_path":"Mathlib/Analysis/Normed/Operator/Banach.lean","def_pos":[524,6],"def_end_pos":[524,35]},{"full_name":"LinearMap.range_eq_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[89,8],"def_end_pos":[89,20]}]}]}
{"url":"Mathlib/LinearAlgebra/Trace.lean","commit":"","full_name":"LinearMap.trace_mul_comm","start":[90,0],"end":[95,75],"file_path":"Mathlib/LinearAlgebra/Trace.lean","tactics":[{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf g : M →ₗ[R] M\nH : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ (trace R M) (f * g) = (trace R M) (g * f)","state_after":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\nf g : M →ₗ[R] M\nH : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\ns : Finset M\nb : Basis { x // x ∈ s } R M\n⊢ (trace R M) (f * g) = (trace R M) (g * f)","tactic":"let ⟨s, ⟨b⟩⟩ := H","premises":[]},{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\nf g : M →ₗ[R] M\nH : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\ns : Finset M\nb : Basis { x // x ∈ s } R M\n⊢ (trace R M) (f * g) = (trace R M) (g * f)","state_after":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\nf g : M →ₗ[R] M\nH : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\ns : Finset M\nb : Basis { x // x ∈ s } R M\n⊢ ((toMatrix b b) f * (toMatrix b b) g).trace = ((toMatrix b b) g * (toMatrix b b) f).trace","tactic":"simp_rw [trace_eq_matrix_trace R b, LinearMap.toMatrix_mul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearMap.toMatrix_mul","def_path":"Mathlib/LinearAlgebra/Matrix/ToLin.lean","def_pos":[637,8],"def_end_pos":[637,30]},{"full_name":"LinearMap.trace_eq_matrix_trace","def_path":"Mathlib/LinearAlgebra/Trace.lean","def_pos":[85,8],"def_end_pos":[85,29]}]},{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\nf g : M →ₗ[R] M\nH : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\ns : Finset M\nb : Basis { x // x ∈ s } R M\n⊢ ((toMatrix b b) f * (toMatrix b b) g).trace = ((toMatrix b b) g * (toMatrix b b) f).trace","state_after":"no goals","tactic":"apply Matrix.trace_mul_comm","premises":[{"full_name":"Matrix.trace_mul_comm","def_path":"Mathlib/LinearAlgebra/Matrix/Trace.lean","def_pos":[151,8],"def_end_pos":[151,22]}]},{"state_before":"R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf g : M →ₗ[R] M\nH : ¬∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ (trace R M) (f * g) = (trace R M) (g * f)","state_after":"no goals","tactic":"rw [trace, dif_neg H, LinearMap.zero_apply, LinearMap.zero_apply]","premises":[{"full_name":"LinearMap.trace","def_path":"Mathlib/LinearAlgebra/Trace.lean","def_pos":[72,4],"def_end_pos":[72,9]},{"full_name":"LinearMap.zero_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[786,8],"def_end_pos":[786,18]},{"full_name":"dif_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[954,8],"def_end_pos":[954,15]}]}]}
{"url":"Mathlib/AlgebraicTopology/SimplicialObject.lean","commit":"","full_name":"CategoryTheory.CosimplicialObject.δ_comp_δ_self'","start":[462,0],"end":[466,20],"file_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : j = i.castSucc\n⊢ X.δ i ≫ X.δ j = X.δ i ≫ X.δ i.succ","state_after":"C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 2)\n⊢ X.δ i ≫ X.δ i.castSucc = X.δ i ≫ X.δ i.succ","tactic":"subst H","premises":[]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 2)\n⊢ X.δ i ≫ X.δ i.castSucc = X.δ i ≫ X.δ i.succ","state_after":"no goals","tactic":"rw [δ_comp_δ_self]","premises":[{"full_name":"CategoryTheory.CosimplicialObject.δ_comp_δ_self","def_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","def_pos":[457,8],"def_end_pos":[457,21]}]}]}
{"url":"Mathlib/Analysis/Convex/Cone/Basic.lean","commit":"","full_name":"ConvexCone.to_orderedSMul","start":[271,0],"end":[279,45],"file_path":"Mathlib/Analysis/Convex/Cone/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : OrderedAddCommGroup E\ninst✝ : Module 𝕜 E\nS : ConvexCone 𝕜 E\nh : ∀ (x y : E), x ≤ y ↔ y - x ∈ S\n⊢ ∀ ⦃a b : E⦄ ⦃c : 𝕜⦄, a < b → 0 < c → c • a ≤ c • b","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : OrderedAddCommGroup E\ninst✝ : Module 𝕜 E\nS : ConvexCone 𝕜 E\nh : ∀ (x y : E), x ≤ y ↔ y - x ∈ S\nx y : E\nz : 𝕜\nxy : x < y\nhz : 0 < z\n⊢ z • x ≤ z • y","tactic":"intro x y z xy hz","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : OrderedAddCommGroup E\ninst✝ : Module 𝕜 E\nS : ConvexCone 𝕜 E\nh : ∀ (x y : E), x ≤ y ↔ y - x ∈ S\nx y : E\nz : 𝕜\nxy : x < y\nhz : 0 < z\n⊢ z • x ≤ z • y","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : OrderedAddCommGroup E\ninst✝ : Module 𝕜 E\nS : ConvexCone 𝕜 E\nh : ∀ (x y : E), x ≤ y ↔ y - x ∈ S\nx y : E\nz : 𝕜\nxy : x < y\nhz : 0 < z\n⊢ z • (y - x) ∈ S","tactic":"rw [h (z • x) (z • y), ← smul_sub z y x]","premises":[{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : OrderedAddCommGroup E\ninst✝ : Module 𝕜 E\nS : ConvexCone 𝕜 E\nh : ∀ (x y : E), x ≤ y ↔ y - x ∈ S\nx y : E\nz : 𝕜\nxy : x < y\nhz : 0 < z\n⊢ z • (y - x) ∈ S","state_after":"no goals","tactic":"exact smul_mem S hz ((h x y).mp xy.le)","premises":[{"full_name":"ConvexCone.smul_mem","def_path":"Mathlib/Analysis/Convex/Cone/Basic.lean","def_pos":[95,8],"def_end_pos":[95,16]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]}]}
{"url":"Mathlib/Data/List/Rotate.lean","commit":"","full_name":"List.isRotated_reverse_iff","start":[454,0],"end":[456,35],"file_path":"Mathlib/Data/List/Rotate.lean","tactics":[{"state_before":"α : Type u\nl l' : List α\n⊢ l.reverse ~r l'.reverse ↔ l ~r l'","state_after":"no goals","tactic":"simp [isRotated_reverse_comm_iff]","premises":[{"full_name":"List.isRotated_reverse_comm_iff","def_path":"Mathlib/Data/List/Rotate.lean","def_pos":[449,8],"def_end_pos":[449,34]}]}]}
{"url":"Mathlib/Topology/Algebra/Monoid.lean","commit":"","full_name":"continuous_finprod_cond","start":[709,0],"end":[714,88],"file_path":"Mathlib/Topology/Algebra/Monoid.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\np : ι → Prop\nhc : ∀ (i : ι), p i → Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n⊢ Continuous fun x => ∏ᶠ (i : ι) (_ : p i), f i x","state_after":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\np : ι → Prop\nhc : ∀ (i : ι), p i → Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n⊢ Continuous fun x => ∏ᶠ (j : Subtype p), f (↑j) x","tactic":"simp only [← finprod_subtype_eq_finprod_cond]","premises":[{"full_name":"finprod_subtype_eq_finprod_cond","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[843,8],"def_end_pos":[843,39]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\np : ι → Prop\nhc : ∀ (i : ι), p i → Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n⊢ Continuous fun x => ∏ᶠ (j : Subtype p), f (↑j) x","state_after":"no goals","tactic":"exact continuous_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective)","premises":[{"full_name":"LocallyFinite.comp_injective","def_path":"Mathlib/Topology/LocallyFinite.lean","def_pos":[47,8],"def_end_pos":[47,22]},{"full_name":"Subtype.coe_injective","def_path":"Mathlib/Data/Subtype.lean","def_pos":[102,8],"def_end_pos":[102,21]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"continuous_finprod","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[702,8],"def_end_pos":[702,26]}]}]}
{"url":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","commit":"","full_name":"seminormFromConst_isPowMul","start":[174,0],"end":[185,90],"file_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\n⊢ seminormFromConst' hf1 hc hpm (x ^ m) = seminormFromConst' hf1 hc hpm x ^ m","state_after":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\n⊢ ⋯.choose = ⋯.choose ^ m","tactic":"simp only [seminormFromConst']","premises":[{"full_name":"seminormFromConst'","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","def_pos":[102,4],"def_end_pos":[102,22]}]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\n⊢ ⋯.choose = ⋯.choose ^ m","state_after":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\n⊢ ⋯.choose = ⋯.choose ^ m","tactic":"have hlim : Tendsto (fun n ↦ seminormFromConst_seq c f (x ^ m) (m * n)) atTop\n      (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m))) := by\n    apply (seminormFromConst_isLimit hf1 hc hpm (x ^ m)).comp\n      (tendsto_atTop_atTop_of_monotone (fun _ _ hnk ↦ mul_le_mul_left' hnk m) _)\n    rintro n; use n; exact le_mul_of_one_le_left' hm","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.comp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2652,8],"def_end_pos":[2652,20]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Filter.tendsto_atTop_atTop_of_monotone","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1214,8],"def_end_pos":[1214,39]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"le_mul_of_one_le_left'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[346,8],"def_end_pos":[346,30]},{"full_name":"mul_le_mul_left'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[62,8],"def_end_pos":[62,24]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"seminormFromConst'","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","def_pos":[102,4],"def_end_pos":[102,22]},{"full_name":"seminormFromConst_isLimit","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","def_pos":[108,8],"def_end_pos":[108,33]},{"full_name":"seminormFromConst_seq","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","def_pos":[51,4],"def_end_pos":[51,25]}]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\n⊢ ⋯.choose = ⋯.choose ^ m","state_after":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\n⊢ Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (⋯.choose ^ m))","tactic":"apply tendsto_nhds_unique hlim","premises":[{"full_name":"tendsto_nhds_unique","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1386,8],"def_end_pos":[1386,27]}]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\n⊢ Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (⋯.choose ^ m))","state_after":"case h.e'_3\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\n⊢ (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) = fun x_1 => seminormFromConst_seq c f x x_1 ^ m","tactic":"convert (seminormFromConst_isLimit hf1 hc hpm x).pow m using 1","premises":[{"full_name":"Filter.Tendsto.pow","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[531,8],"def_end_pos":[531,26]},{"full_name":"seminormFromConst_isLimit","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","def_pos":[108,8],"def_end_pos":[108,33]}]},{"state_before":"case h.e'_3\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\n⊢ (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) = fun x_1 => seminormFromConst_seq c f x x_1 ^ m","state_after":"case h.e'_3.h\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\nn : ℕ\n⊢ seminormFromConst_seq c f (x ^ m) (m * n) = seminormFromConst_seq c f x n ^ m","tactic":"ext n","premises":[]},{"state_before":"case h.e'_3.h\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n => seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' hf1 hc hpm (x ^ m)))\nn : ℕ\n⊢ seminormFromConst_seq c f (x ^ m) (m * n) = seminormFromConst_seq c f x n ^ m","state_after":"no goals","tactic":"simp only [seminormFromConst_seq, div_pow, ← hpm _ hm, ← pow_mul, mul_pow, mul_comm m n]","premises":[{"full_name":"div_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[589,6],"def_end_pos":[589,13]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]},{"full_name":"seminormFromConst_seq","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean","def_pos":[51,4],"def_end_pos":[51,25]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","commit":"","full_name":"Bimod.isoOfIso_inv_hom","start":[137,0],"end":[157,50],"file_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ Q.actLeft ≫ f.inv = X.X ◁ f.inv ≫ P.actLeft","state_after":"no goals","tactic":"rw [← cancel_mono f.hom, Category.assoc, Category.assoc, Iso.inv_hom_id, Category.comp_id,\n          f_left_act_hom, ← Category.assoc, ← MonoidalCategory.whiskerLeft_comp, Iso.inv_hom_id,\n          MonoidalCategory.whiskerLeft_id, Category.id_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[170,2],"def_end_pos":[170,16]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ Q.actRight ≫ f.inv = f.inv ▷ Y.X ≫ P.actRight","state_after":"no goals","tactic":"rw [← cancel_mono f.hom, Category.assoc, Category.assoc, Iso.inv_hom_id, Category.comp_id,\n          f_right_act_hom, ← Category.assoc, ← comp_whiskerRight, Iso.inv_hom_id,\n          MonoidalCategory.id_whiskerRight, Category.id_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[172,2],"def_end_pos":[172,17]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ { hom := f.hom, left_act_hom := ⋯, right_act_hom := ⋯ } ≫ { hom := f.inv, left_act_hom := ⋯, right_act_hom := ⋯ } =\n    𝟙 P","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ ({ hom := f.hom, left_act_hom := ⋯, right_act_hom := ⋯ } ≫\n        { hom := f.inv, left_act_hom := ⋯, right_act_hom := ⋯ }).hom =\n    (𝟙 P).hom","tactic":"ext","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ ({ hom := f.hom, left_act_hom := ⋯, right_act_hom := ⋯ } ≫\n        { hom := f.inv, left_act_hom := ⋯, right_act_hom := ⋯ }).hom =\n    (𝟙 P).hom","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ f.hom ≫ f.inv = 𝟙 P.X","tactic":"dsimp","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ f.hom ≫ f.inv = 𝟙 P.X","state_after":"no goals","tactic":"rw [Iso.hom_inv_id]","premises":[{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ { hom := f.inv, left_act_hom := ⋯, right_act_hom := ⋯ } ≫ { hom := f.hom, left_act_hom := ⋯, right_act_hom := ⋯ } =\n    𝟙 Q","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ ({ hom := f.inv, left_act_hom := ⋯, right_act_hom := ⋯ } ≫\n        { hom := f.hom, left_act_hom := ⋯, right_act_hom := ⋯ }).hom =\n    (𝟙 Q).hom","tactic":"ext","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ ({ hom := f.inv, left_act_hom := ⋯, right_act_hom := ⋯ } ≫\n        { hom := f.hom, left_act_hom := ⋯, right_act_hom := ⋯ }).hom =\n    (𝟙 Q).hom","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ f.inv ≫ f.hom = 𝟙 Q.X","tactic":"dsimp","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\nX Y : Mon_ C\nP Q : Bimod X Y\nf : P.X ≅ Q.X\nf_left_act_hom : P.actLeft ≫ f.hom = X.X ◁ f.hom ≫ Q.actLeft\nf_right_act_hom : P.actRight ≫ f.hom = f.hom ▷ Y.X ≫ Q.actRight\n⊢ f.inv ≫ f.hom = 𝟙 Q.X","state_after":"no goals","tactic":"rw [Iso.inv_hom_id]","premises":[{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]}]}]}
{"url":"Mathlib/NumberTheory/PythagoreanTriples.lean","commit":"","full_name":"PythagoreanTriple.mul","start":[67,0],"end":[73,34],"file_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","tactics":[{"state_before":"x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\n⊢ k * x * (k * x) + k * y * (k * y) = k ^ 2 * (x * x + y * y)","state_after":"no goals","tactic":"ring","premises":[]},{"state_before":"x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\n⊢ k ^ 2 * (x * x + y * y) = k ^ 2 * (z * z)","state_after":"no goals","tactic":"rw [h.eq]","premises":[{"full_name":"PythagoreanTriple.eq","def_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","def_pos":[61,8],"def_end_pos":[61,10]}]},{"state_before":"x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\n⊢ k ^ 2 * (z * z) = k * z * (k * z)","state_after":"no goals","tactic":"ring","premises":[]}]}
{"url":"Mathlib/Data/Bool/Basic.lean","commit":"","full_name":"Bool.eq_false_eq_not_eq_true","start":[33,0],"end":[33,81],"file_path":"Mathlib/Data/Bool/Basic.lean","tactics":[{"state_before":"b : Bool\n⊢ (¬b = true) = (b = false)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","commit":"","full_name":"Ideal.dvd_iff_le","start":[582,0],"end":[595,68],"file_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\n⊢ I ∣ J","state_after":"case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : I = ⊥\n⊢ I ∣ J\n\ncase neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\n⊢ I ∣ J","tactic":"by_cases hI : I = ⊥","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\n⊢ I ∣ J","state_after":"case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\n⊢ I ∣ J","tactic":"have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.coeIdeal_ne_zero","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[302,8],"def_end_pos":[302,24]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\n⊢ I ∣ J","state_after":"case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\n⊢ I ∣ J","tactic":"have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=\n      le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))\n        (le_of_eq (inv_mul_cancel hI'))","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.coeIdeal_le_coeIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[269,8],"def_end_pos":[269,28]},{"full_name":"FractionalIdeal.mul_left_mono","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[534,8],"def_end_pos":[534,21]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"inv_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[50,8],"def_end_pos":[50,22]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\n⊢ I ∣ J","state_after":"case neg.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ I ∣ J","tactic":"obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this","premises":[{"full_name":"FractionalIdeal.le_one_iff_exists_coeIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[618,8],"def_end_pos":[618,34]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case neg.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ I ∣ J","state_after":"case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ J = I * H","tactic":"use H","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ J = I * H","state_after":"case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ ↑J = ↑(I * H)","tactic":"refine coeIdeal_injective (show (J : FractionalIdeal A⁰ (FractionRing A)) = ↑(I * H) from ?_)","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.coeIdeal_injective","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[291,8],"def_end_pos":[291,26]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ ↑J = ↑(I * H)","state_after":"no goals","tactic":"rw [coeIdeal_mul, hH, ← mul_assoc, mul_inv_cancel hI', one_mul]","premises":[{"full_name":"FractionalIdeal.coeIdeal_mul","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[530,8],"def_end_pos":[530,20]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_inv_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[195,14],"def_end_pos":[195,28]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","commit":"","full_name":"Mon_.whiskerRight_hom","start":[466,0],"end":[469,26],"file_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y Z : Mon_ C\nf : Y ⟶ Z\n⊢ (X ◁ f).hom = X.X ◁ f.hom","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y Z : Mon_ C\nf : Y ⟶ Z\n⊢ (X ◁ f).hom = 𝟙 X.X ⊗ f.hom","tactic":"rw [← id_tensorHom]","premises":[{"full_name":"CategoryTheory.MonoidalCategory.id_tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[222,8],"def_end_pos":[222,20]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y Z : Mon_ C\nf : Y ⟶ Z\n⊢ (X ◁ f).hom = 𝟙 X.X ⊗ f.hom","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","commit":"","full_name":"cfcₙ_star_id","start":[383,0],"end":[384,34],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : TopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : NonUnitalContinuousFunctionalCalculus R p\nf g : R → R\na : A\nhf : autoParam (ContinuousOn f (σₙ R a)) _auto✝\nhf0 : autoParam (f 0 = 0) _auto✝\nhg : autoParam (ContinuousOn g (σₙ R a)) _auto✝\nhg0 : autoParam (g 0 = 0) _auto✝\nha : autoParam (p a) _auto✝\n⊢ cfcₙ (fun x => star x) a = star a","state_after":"no goals","tactic":"rw [cfcₙ_star _ a, cfcₙ_id' R a]","premises":[{"full_name":"cfcₙ_id'","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","def_pos":[252,6],"def_end_pos":[252,14]},{"full_name":"cfcₙ_star","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","def_pos":[362,6],"def_end_pos":[362,15]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","commit":"","full_name":"CategoryTheory.Limits.epi_of_epi_image","start":[421,0],"end":[424,16],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝² : HasImage f\ninst✝¹ : Epi (image.ι f)\ninst✝ : Epi (factorThruImage f)\n⊢ Epi f","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝² : HasImage f\ninst✝¹ : Epi (image.ι f)\ninst✝ : Epi (factorThruImage f)\n⊢ Epi (factorThruImage f ≫ image.ι f)","tactic":"rw [← image.fac f]","premises":[{"full_name":"CategoryTheory.Limits.image.fac","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[299,8],"def_end_pos":[299,17]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝² : HasImage f\ninst✝¹ : Epi (image.ι f)\ninst✝ : Epi (factorThruImage f)\n⊢ Epi (factorThruImage f ≫ image.ι f)","state_after":"no goals","tactic":"apply epi_comp","premises":[{"full_name":"CategoryTheory.epi_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[275,8],"def_end_pos":[275,16]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Real.tan_pi","start":[865,0],"end":[866,63],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"⊢ tan π = 0","state_after":"no goals","tactic":"rw [tan_periodic.eq, tan_zero]","premises":[{"full_name":"Function.Periodic.eq","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[205,18],"def_end_pos":[205,29]},{"full_name":"Real.tan_periodic","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[862,8],"def_end_pos":[862,20]},{"full_name":"Real.tan_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[770,8],"def_end_pos":[770,16]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Circumcenter.lean","commit":"","full_name":"Affine.Simplex.circumcenter_reindex","start":[375,0],"end":[378,98],"file_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ (s.reindex e).circumcenter = s.circumcenter","state_after":"no goals","tactic":"simp_rw [circumcenter, circumsphere_reindex]","premises":[{"full_name":"Affine.Simplex.circumcenter","def_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","def_pos":[254,4],"def_end_pos":[254,16]},{"full_name":"Affine.Simplex.circumsphere_reindex","def_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","def_pos":[369,8],"def_end_pos":[369,28]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
{"url":"Mathlib/CategoryTheory/Subobject/FactorThru.lean","commit":"","full_name":"CategoryTheory.Subobject.factorThru_eq_zero","start":[126,0],"end":[135,8],"file_path":"Mathlib/CategoryTheory/Subobject/FactorThru.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\nf : X ⟶ Y\nh : P.Factors f\n⊢ P.factorThru f h = 0 ↔ f = 0","state_after":"case mp\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\nf : X ⟶ Y\nh : P.Factors f\n⊢ P.factorThru f h = 0 → f = 0\n\ncase mpr\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\nf : X ⟶ Y\nh : P.Factors f\n⊢ f = 0 → P.factorThru f h = 0","tactic":"fconstructor","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Covering/Differentiation.lean","commit":"","full_name":"VitaliFamily.measure_limRatioMeas_zero","start":[490,0],"end":[510,46],"file_path":"Mathlib/MeasureTheory/Covering/Differentiation.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\n⊢ ρ {x | v.limRatioMeas hρ x = 0} = 0","state_after":"α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\n⊢ ∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = 0}] x, ρ u = 0","tactic":"refine measure_null_of_locally_null _ fun x _ => ?_","premises":[{"full_name":"MeasureTheory.measure_null_of_locally_null","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[135,8],"def_end_pos":[135,36]}]},{"state_before":"α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\n⊢ ∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = 0}] x, ρ u = 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\n⊢ ∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = 0}] x, ρ u = 0","tactic":"obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ :=\n    Measure.exists_isOpen_measure_lt_top μ x","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"IsOpen","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[82,4],"def_end_pos":[82,10]},{"full_name":"MeasureTheory.Measure.exists_isOpen_measure_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[1098,8],"def_end_pos":[1098,44]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\n⊢ ∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = 0}] x, ρ u = 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\n⊢ ∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = 0}] x, ρ u = 0","tactic":"let s := {x : α | v.limRatioMeas hρ x = 0} ∩ o","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"VitaliFamily.limRatioMeas","def_path":"Mathlib/MeasureTheory/Covering/Differentiation.lean","def_pos":[414,18],"def_end_pos":[414,30]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\n⊢ ∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = 0}] x, ρ u = 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\n⊢ ρ s ≤ 0","tactic":"refine ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm ?_ bot_le⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]},{"full_name":"inter_mem_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[130,8],"def_end_pos":[130,28]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\n⊢ ρ s ≤ 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\n⊢ ρ s ≤ 0","tactic":"have μs : μ s ≠ ∞ := ((measure_mono inter_subset_right).trans_lt μo).ne","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Set.inter_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[764,8],"def_end_pos":[764,26]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\n⊢ ρ s ≤ 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\n⊢ ρ s ≤ 0","tactic":"have A : ∀ q : ℝ≥0, 0 < q → ρ s ≤ q * μ s := by\n    intro q hq\n    apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ\n    intro y hy\n    have : v.limRatioMeas hρ y = 0 := hy.1\n    simp only [this, mem_setOf_eq, hq, ENNReal.coe_pos]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"ENNReal.coe_pos","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[338,27],"def_end_pos":[338,34]},{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"VitaliFamily.limRatioMeas","def_path":"Mathlib/MeasureTheory/Covering/Differentiation.lean","def_pos":[414,18],"def_end_pos":[414,30]},{"full_name":"VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt","def_path":"Mathlib/MeasureTheory/Covering/Differentiation.lean","def_pos":[429,8],"def_end_pos":[429,48]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\n⊢ ρ s ≤ 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 (↑0 * μ s))\n⊢ ρ s ≤ 0","tactic":"have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞) * μ s) (𝓝[>] (0 : ℝ≥0)) (𝓝 ((0 : ℝ≥0) * μ s)) := by\n    apply ENNReal.Tendsto.mul_const _ (Or.inr μs)\n    rw [ENNReal.tendsto_coe]\n    exact nhdsWithin_le_nhds","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.Tendsto.mul_const","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[344,18],"def_end_pos":[344,35]},{"full_name":"ENNReal.tendsto_coe","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[67,8],"def_end_pos":[67,19]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"Set.Ioi","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[70,4],"def_end_pos":[70,7]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"nhdsWithin_le_nhds","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[162,8],"def_end_pos":[162,26]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 (↑0 * μ s))\n⊢ ρ s ≤ 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 0)\n⊢ ρ s ≤ 0","tactic":"simp only [zero_mul, ENNReal.coe_zero] at B","premises":[{"full_name":"ENNReal.coe_zero","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[220,27],"def_end_pos":[220,35]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 0)\n⊢ ρ s ≤ 0","state_after":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 0)\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 0, ρ s ≤ ↑c * μ s","tactic":"apply ge_of_tendsto B","premises":[{"full_name":"ge_of_tendsto","def_path":"Mathlib/Topology/Order/OrderClosed.lean","def_pos":[349,8],"def_end_pos":[349,21]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 0)\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 0, ρ s ≤ ↑c * μ s","state_after":"no goals","tactic":"filter_upwards [self_mem_nhdsWithin] using A","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"self_mem_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[124,8],"def_end_pos":[124,27]}]}]}
{"url":"Mathlib/Algebra/Group/Submonoid/MulOpposite.lean","commit":"","full_name":"AddSubmonoid.op_closure","start":[135,0],"end":[139,42],"file_path":"Mathlib/Algebra/Group/Submonoid/MulOpposite.lean","tactics":[{"state_before":"ι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\n⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)","state_after":"ι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\n⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}","tactic":"simp_rw [closure, op_sInf, Set.preimage_setOf_eq, Submonoid.unop_coe]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.preimage_setOf_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[92,8],"def_end_pos":[92,25]},{"full_name":"Submonoid.closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[329,4],"def_end_pos":[329,11]},{"full_name":"Submonoid.op_sInf","def_path":"Mathlib/Algebra/Group/Submonoid/MulOpposite.lean","def_pos":[114,8],"def_end_pos":[114,15]},{"full_name":"Submonoid.unop_coe","def_path":"Mathlib/Algebra/Group/Submonoid/MulOpposite.lean","def_pos":[34,23],"def_end_pos":[34,28]}]},{"state_before":"ι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\n⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}","state_after":"case e_a.h\nι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\na : Submonoid Mᵐᵒᵖ\n⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}","tactic":"congr with a","premises":[]},{"state_before":"case e_a.h\nι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\na : Submonoid Mᵐᵒᵖ\n⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}","state_after":"no goals","tactic":"exact MulOpposite.unop_surjective.forall","premises":[{"full_name":"Function.Surjective.forall","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[164,18],"def_end_pos":[164,35]},{"full_name":"MulOpposite.unop_surjective","def_path":"Mathlib/Algebra/Opposites.lean","def_pos":[129,8],"def_end_pos":[129,23]}]}]}
{"url":"Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean","commit":"","full_name":"CoalgebraCat.toComonObj_comul","start":[38,0],"end":[45,76],"file_path":"Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.ofHom Coalgebra.counit ▷ ModuleCat.of R ↑X.toModuleCat =\n    (λ_ (ModuleCat.of R ↑X.toModuleCat)).inv","state_after":"no goals","tactic":"simpa only [ModuleCat.of_coe] using Coalgebra.rTensor_counit_comp_comul","premises":[{"full_name":"Coalgebra.rTensor_counit_comp_comul","def_path":"Mathlib/RingTheory/Coalgebra/Basic.lean","def_pos":[52,2],"def_end_pos":[52,27]},{"full_name":"ModuleCat.of_coe","def_path":"Mathlib/Algebra/Category/ModuleCat/Basic.lean","def_pos":[172,16],"def_end_pos":[172,22]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.counit =\n    (ρ_ (ModuleCat.of R ↑X.toModuleCat)).inv","state_after":"no goals","tactic":"simpa only [ModuleCat.of_coe] using Coalgebra.lTensor_counit_comp_comul","premises":[{"full_name":"Coalgebra.lTensor_counit_comp_comul","def_path":"Mathlib/RingTheory/Coalgebra/Basic.lean","def_pos":[54,2],"def_end_pos":[54,27]},{"full_name":"ModuleCat.of_coe","def_path":"Mathlib/Algebra/Category/ModuleCat/Basic.lean","def_pos":[172,16],"def_end_pos":[172,22]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul =\n    ModuleCat.ofHom Coalgebra.comul ≫\n      ModuleCat.ofHom Coalgebra.comul ▷ ModuleCat.of R ↑X.toModuleCat ≫\n        (α_ (ModuleCat.of R ↑X.toModuleCat) (ModuleCat.of R ↑X.toModuleCat) (ModuleCat.of R ↑X.toModuleCat)).hom","state_after":"R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul =\n    ModuleCat.ofHom Coalgebra.comul ≫\n      ModuleCat.ofHom Coalgebra.comul ▷ X.toModuleCat ≫ (α_ X.toModuleCat X.toModuleCat X.toModuleCat).hom","tactic":"simp_rw [ModuleCat.of_coe]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"ModuleCat.of_coe","def_path":"Mathlib/Algebra/Category/ModuleCat/Basic.lean","def_pos":[172,16],"def_end_pos":[172,22]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul =\n    ModuleCat.ofHom Coalgebra.comul ≫\n      ModuleCat.ofHom Coalgebra.comul ▷ X.toModuleCat ≫ (α_ X.toModuleCat X.toModuleCat X.toModuleCat).hom","state_after":"no goals","tactic":"exact Coalgebra.coassoc.symm","premises":[{"full_name":"Coalgebra.coassoc","def_path":"Mathlib/RingTheory/Coalgebra/Basic.lean","def_pos":[50,2],"def_end_pos":[50,9]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","commit":"","full_name":"Matrix.charmatrix_apply_ne","start":[55,0],"end":[58,31],"file_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nm : Type u_3\nn : Type u_4\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM₁₁ : Matrix m m R\nM₁₂ : Matrix m n R\nM₂₁ : Matrix n m R\nM₂₂ M : Matrix n n R\ni j : n\nh : i ≠ j\n⊢ M.charmatrix i j = -C (M i j)","state_after":"no goals","tactic":"simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, diagonal_apply_ne _ h,\n    map_apply, sub_eq_neg_self]","premises":[{"full_name":"Matrix.charmatrix","def_path":"Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean","def_pos":[43,4],"def_end_pos":[43,14]},{"full_name":"Matrix.diagonal_apply_ne","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[368,8],"def_end_pos":[368,25]},{"full_name":"Matrix.map_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Matrix.scalar_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1119,8],"def_end_pos":[1119,20]},{"full_name":"Matrix.sub_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[257,8],"def_end_pos":[257,17]},{"full_name":"RingHom.mapMatrix_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1344,2],"def_end_pos":[1344,7]},{"full_name":"sub_eq_neg_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[602,2],"def_end_pos":[602,13]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"compact_open_separated_mul_right","start":[1433,0],"end":[1455,54],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∃ V ∈ 𝓝 1, K * V ⊆ U","state_after":"case refine_1\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∃ V ∈ 𝓝 1, ∅ * V ⊆ U\n\ncase refine_2\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∀ ⦃s t : Set G⦄, s ⊆ t → (∃ V ∈ 𝓝 1, t * V ⊆ U) → ∃ V ∈ 𝓝 1, s * V ⊆ U\n\ncase refine_3\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∀ ⦃s t : Set G⦄, (∃ V ∈ 𝓝 1, s * V ⊆ U) → (∃ V ∈ 𝓝 1, t * V ⊆ U) → ∃ V ∈ 𝓝 1, (s ∪ t) * V ⊆ U\n\ncase refine_4\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ V ∈ 𝓝 1, t * V ⊆ U","tactic":"refine hK.induction_on ?_ ?_ ?_ ?_","premises":[{"full_name":"IsCompact.induction_on","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[68,8],"def_end_pos":[68,30]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Plus.lean","commit":"","full_name":"CategoryTheory.GrothendieckTopology.plusMap_toPlus","start":[223,0],"end":[243,23],"file_path":"Mathlib/CategoryTheory/Sites/Plus.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\n⊢ J.plusMap (J.toPlus P) = J.toPlus (J.plusObj P)","state_after":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\n⊢ (J.plusMap (J.toPlus P)).app X = (J.toPlus (J.plusObj P)).app X","tactic":"ext X : 2","premises":[]},{"state_before":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\n⊢ (J.plusMap (J.toPlus P)).app X = (J.toPlus (J.plusObj P)).app X","state_after":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\n⊢ colimit.ι (J.diagram P (unop X)) S ≫ (J.plusMap (J.toPlus P)).app X =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.toPlus (J.plusObj P)).app X","tactic":"refine colimit.hom_ext (fun S => ?_)","premises":[{"full_name":"CategoryTheory.Limits.colimit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[757,8],"def_end_pos":[757,23]}]},{"state_before":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\n⊢ colimit.ι (J.diagram P (unop X)) S ≫ (J.plusMap (J.toPlus P)).app X =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.toPlus (J.plusObj P)).app X","state_after":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\n⊢ colimit.ι (J.diagram P (unop X)) S ≫\n      colimMap\n        (J.diagramNatTrans\n          { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)) =\n    colimit.ι (J.diagram P (unop X)) S ≫\n      ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)","tactic":"dsimp only [plusMap, toPlus]","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.plusMap","def_path":"Mathlib/CategoryTheory/Sites/Plus.lean","def_pos":[141,4],"def_end_pos":[141,11]},{"full_name":"CategoryTheory.GrothendieckTopology.toPlus","def_path":"Mathlib/CategoryTheory/Sites/Plus.lean","def_pos":[187,4],"def_end_pos":[187,10]}]},{"state_before":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\n⊢ colimit.ι (J.diagram P (unop X)) S ≫\n      colimMap\n        (J.diagramNatTrans\n          { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)) =\n    colimit.ι (J.diagram P (unop X)) S ≫\n      ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)","state_after":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ colimit.ι (J.diagram P (unop X)) S ≫\n      colimMap\n        (J.diagramNatTrans\n          { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)) =\n    colimit.ι (J.diagram P (unop X)) S ≫\n      ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)","tactic":"let e : S.unop ⟶ ⊤ := homOfLE (OrderTop.le_top _)","premises":[{"full_name":"CategoryTheory.homOfLE","def_path":"Mathlib/CategoryTheory/Category/Preorder.lean","def_pos":[65,4],"def_end_pos":[65,11]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]},{"full_name":"OrderTop.le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[44,2],"def_end_pos":[44,8]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ colimit.ι (J.diagram P (unop X)) S ≫\n      colimMap\n        (J.diagramNatTrans\n          { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)) =\n    colimit.ι (J.diagram P (unop X)) S ≫\n      ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)","state_after":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n            (unop X)).app\n        S ≫\n      colimit.ι (J.diagram (J.plusObj P) (unop X)) S =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      colimit.ι (J.diagram (J.plusObj P) (unop X)) (op (unop S))","tactic":"rw [ι_colimMap, ← colimit.w _ e.op, ← Category.assoc, ← Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.colimit.w","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[664,8],"def_end_pos":[664,17]},{"full_name":"CategoryTheory.Limits.ι_colimMap","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[705,8],"def_end_pos":[705,18]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n            (unop X)).app\n        S ≫\n      colimit.ι (J.diagram (J.plusObj P) (unop X)) S =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      colimit.ι (J.diagram (J.plusObj P) (unop X)) (op (unop S))","state_after":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)).app\n      S =\n    (colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n      (J.diagram (J.plusObj P) (unop X)).map e.op","tactic":"congr 1","premises":[]},{"state_before":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)).app\n      S =\n    (colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n      (J.diagram (J.plusObj P) (unop X)).map e.op","state_after":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n            (unop X)).app\n        S ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I","tactic":"refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)","premises":[{"full_name":"CategoryTheory.Limits.Multiequalizer.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[732,8],"def_end_pos":[732,15]}]},{"state_before":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n            (unop X)).app\n        S ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I","state_after":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ Multiequalizer.ι ((unop S).index P) I ≫\n      { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }.app (op I.Y) =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I","tactic":"erw [Multiequalizer.lift_ι]","premises":[{"full_name":"CategoryTheory.Limits.Multiequalizer.lift_ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[726,8],"def_end_pos":[726,14]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ Multiequalizer.ι ((unop S).index P) I ≫\n      { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }.app (op I.Y) =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I","state_after":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op","tactic":"simp only [unop_op, op_unop, diagram_map, Category.assoc, limit.lift_π,\n    Multifork.ofι_π_app]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.GrothendieckTopology.diagram_map","def_path":"Mathlib/CategoryTheory/Sites/Plus.lean","def_pos":[39,2],"def_end_pos":[39,7]},{"full_name":"CategoryTheory.Limits.Multifork.ofι_π_app","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[317,2],"def_end_pos":[317,7]},{"full_name":"CategoryTheory.Limits.limit.lift_π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[169,8],"def_end_pos":[169,20]},{"full_name":"Opposite.op_unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[61,8],"def_end_pos":[61,15]},{"full_name":"Opposite.unop_op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[64,8],"def_end_pos":[64,15]}]},{"state_before":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op","state_after":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op","tactic":"let ee : (J.pullback (I.map e).f).obj S.unop ⟶ ⊤ := homOfLE (OrderTop.le_top _)","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.Cover.Arrow.f","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[451,2],"def_end_pos":[451,3]},{"full_name":"CategoryTheory.GrothendieckTopology.Cover.Arrow.map","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[486,4],"def_end_pos":[486,13]},{"full_name":"CategoryTheory.GrothendieckTopology.pullback","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[644,4],"def_end_pos":[644,12]},{"full_name":"CategoryTheory.homOfLE","def_path":"Mathlib/CategoryTheory/Category/Preorder.lean","def_pos":[65,4],"def_end_pos":[65,11]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]},{"full_name":"OrderTop.le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[44,2],"def_end_pos":[44,8]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op","state_after":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ ((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫\n      colimit.ι (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))) =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n        ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫\n      colimit.ι (J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y)))\n        ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)","tactic":"erw [← colimit.w _ ee.op, ι_colimMap_assoc, colimit.ι_pre, diagramPullback_app,\n    ← Category.assoc, ← Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.GrothendieckTopology.diagramPullback_app","def_path":"Mathlib/CategoryTheory/Sites/Plus.lean","def_pos":[47,2],"def_end_pos":[47,7]},{"full_name":"CategoryTheory.Limits.colimit.w","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[664,8],"def_end_pos":[664,17]},{"full_name":"CategoryTheory.Limits.colimit.ι_pre","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[869,8],"def_end_pos":[869,21]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ ((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫\n      colimit.ι (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))) =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n        ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫\n      colimit.ι (J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y)))\n        ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)","state_after":"case w.h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ (Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n      ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯","tactic":"congr 1","premises":[]},{"state_before":"case w.h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ (Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n      ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯","state_after":"case w.h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\nII : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L\n⊢ ((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n        ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II","tactic":"refine Multiequalizer.hom_ext _ _ _ (fun II => ?_)","premises":[{"full_name":"CategoryTheory.Limits.Multiequalizer.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[732,8],"def_end_pos":[732,15]}]},{"state_before":"case w.h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\nII : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L\n⊢ ((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n        ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II","state_after":"case h.e'_2.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\nII : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L\ne_1✝ :\n  ((J.diagram P (unop X)).obj S ⟶ ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II) =\n    (multiequalizer ((unop S).index P) ⟶\n      ((unop S).index P).right\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } })\n⊢ ((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II =\n    Multiequalizer.ι ((unop S).index P)\n        (((unop S).index P).fstTo\n          { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }) ≫\n      ((unop S).index P).fst\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }\n\ncase h.e'_3.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\nII : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L\ne_1✝ :\n  ((J.diagram P (unop X)).obj S ⟶ ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II) =\n    (multiequalizer ((unop S).index P) ⟶\n      ((unop S).index P).right\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } })\n⊢ Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n        ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II =\n    Multiequalizer.ι ((unop S).index P)\n        (((unop S).index P).sndTo\n          { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }) ≫\n      ((unop S).index P).snd\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }","tactic":"convert Multiequalizer.condition (S.unop.index P)\n    (Cover.Relation.mk I II.base { g₁ := II.f, g₂ := 𝟙 _ }) using 1","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.GrothendieckTopology.Cover.Arrow.base","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[501,4],"def_end_pos":[501,14]},{"full_name":"CategoryTheory.GrothendieckTopology.Cover.Arrow.f","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[451,2],"def_end_pos":[451,3]},{"full_name":"CategoryTheory.GrothendieckTopology.Cover.index","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[604,4],"def_end_pos":[604,9]},{"full_name":"CategoryTheory.Limits.Multiequalizer.condition","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[716,8],"def_end_pos":[716,17]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]}]},{"state_before":"case h.e'_2.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\nII : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L\ne_1✝ :\n  ((J.diagram P (unop X)).obj S ⟶ ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II) =\n    (multiequalizer ((unop S).index P) ⟶\n      ((unop S).index P).right\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } })\n⊢ ((Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op) ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II =\n    Multiequalizer.ι ((unop S).index P)\n        (((unop S).index P).fstTo\n          { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }) ≫\n      ((unop S).index P).fst\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }\n\ncase h.e'_3.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\nII : ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).L\ne_1✝ :\n  ((J.diagram P (unop X)).obj S ⟶ ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II) =\n    (multiequalizer ((unop S).index P) ⟶\n      ((unop S).index P).right\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } })\n⊢ Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n        ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯ ≫\n      Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II =\n    Multiequalizer.ι ((unop S).index P)\n        (((unop S).index P).sndTo\n          { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }) ≫\n      ((unop S).index P).snd\n        { fst := I, snd := Cover.Arrow.base II, r := { Z := II.Y, g₁ := II.f, g₂ := 𝟙 II.Y, w := ⋯ } }","state_after":"no goals","tactic":"all_goals dsimp; simp","premises":[]}]}
{"url":"Mathlib/Data/List/Infix.lean","commit":"","full_name":"List.prefix_take_le_iff","start":[189,0],"end":[203,62],"file_path":"Mathlib/Data/List/Infix.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nL : List (List (Option α))\nhm : m < L.length\n⊢ take m L <+: take n L ↔ m ≤ n","state_after":"α : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nL : List (List (Option α))\nhm : m < L.length\n⊢ take m L = take (min m L.length) (take n L) ↔ m ≤ n","tactic":"simp only [prefix_iff_eq_take, length_take]","premises":[{"full_name":"List.length_take","def_path":".lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean","def_pos":[23,16],"def_end_pos":[23,27]},{"full_name":"List.prefix_iff_eq_take","def_path":"Mathlib/Data/List/Infix.lean","def_pos":[134,8],"def_end_pos":[134,26]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nL : List (List (Option α))\nhm : m < L.length\n⊢ take m L = take (min m L.length) (take n L) ↔ m ≤ n","state_after":"no goals","tactic":"induction m generalizing L n with\n  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]\n  | succ m IH =>\n    cases L with\n    | nil => simp_all\n    | cons l ls =>\n      cases n with\n      | zero =>\n        simp\n      | succ n =>\n        simp only [length_cons, succ_eq_add_one, Nat.add_lt_add_iff_right] at hm\n        simp [← @IH n ls hm, Nat.min_eq_left, Nat.le_of_lt hm]","premises":[{"full_name":"List.cons","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2289,4],"def_end_pos":[2289,8]},{"full_name":"List.length_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[70,16],"def_end_pos":[70,27]},{"full_name":"List.nil","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2286,4],"def_end_pos":[2286,7]},{"full_name":"List.take","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[715,4],"def_end_pos":[715,8]},{"full_name":"Nat.add_lt_add_iff_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[58,18],"def_end_pos":[58,38]},{"full_name":"Nat.le_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[343,18],"def_end_pos":[343,26]},{"full_name":"Nat.min_eq_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/MinMax.lean","def_pos":[27,18],"def_end_pos":[27,29]},{"full_name":"Nat.succ_eq_add_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[154,16],"def_end_pos":[154,31]},{"full_name":"Nat.zero_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1663,8],"def_end_pos":[1663,19]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]},{"full_name":"min_eq_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[92,8],"def_end_pos":[92,19]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Uniform.lean","commit":"","full_name":"dist_self_mul_right","start":[186,0],"end":[188,54],"file_path":"Mathlib/Analysis/Normed/Group/Uniform.lean","tactics":[{"state_before":"𝓕 : Type u_1\nα : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ dist a (a * b) = ‖b‖","state_after":"no goals","tactic":"rw [← dist_one_left, ← dist_mul_left a 1 b, mul_one]","premises":[{"full_name":"dist_mul_left","def_path":"Mathlib/Topology/MetricSpace/IsometricSMul.lean","def_pos":[258,8],"def_end_pos":[258,21]},{"full_name":"dist_one_left","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[368,8],"def_end_pos":[368,21]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
{"url":"Mathlib/AlgebraicTopology/SimplexCategory.lean","commit":"","full_name":"SimplexCategory.eq_id_of_epi","start":[731,0],"end":[739,16],"file_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","tactics":[{"state_before":"x : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ i = 𝟙 x","state_after":"x : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ IsIso i","tactic":"suffices IsIso i by\n    haveI := this\n    apply eq_id_of_isIso","premises":[{"full_name":"CategoryTheory.IsIso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[229,6],"def_end_pos":[229,11]},{"full_name":"SimplexCategory.eq_id_of_isIso","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[616,8],"def_end_pos":[616,22]}]},{"state_before":"x : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ IsIso i","state_after":"case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Function.Bijective (Hom.toOrderHom i).toFun","tactic":"apply isIso_of_bijective","premises":[{"full_name":"SimplexCategory.isIso_of_bijective","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[587,8],"def_end_pos":[587,26]}]},{"state_before":"case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Function.Bijective (Hom.toOrderHom i).toFun","state_after":"case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Function.Bijective ⇑(Hom.toOrderHom i)","tactic":"dsimp","premises":[]},{"state_before":"case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Function.Bijective ⇑(Hom.toOrderHom i)","state_after":"case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Epi i","tactic":"rw [Fintype.bijective_iff_surjective_and_card i.toOrderHom, ← epi_iff_surjective,\n    eq_self_iff_true, and_true_iff]","premises":[{"full_name":"Fintype.bijective_iff_surjective_and_card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[605,8],"def_end_pos":[605,41]},{"full_name":"SimplexCategory.Hom.toOrderHom","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[96,4],"def_end_pos":[96,14]},{"full_name":"SimplexCategory.epi_iff_surjective","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[516,8],"def_end_pos":[516,26]},{"full_name":"and_true_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[93,8],"def_end_pos":[93,20]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]}]},{"state_before":"case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Epi i","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Normed.lean","commit":"","full_name":"Monotone.cauchySeq_alternating_series_of_tendsto_zero","start":[653,0],"end":[658,87],"file_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ x ∈ Finset.range n, f x * (-1) ^ x","tactic":"simp_rw [mul_comm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ x ∈ Finset.range n, f x * (-1) ^ x","state_after":"no goals","tactic":"exact hfa.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le","premises":[{"full_name":"Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded","def_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","def_pos":[620,8],"def_end_pos":[620,64]},{"full_name":"norm_sum_neg_one_pow_le","def_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","def_pos":[649,8],"def_end_pos":[649,31]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","commit":"","full_name":"String.Iterator.ValidFor.toEnd'","start":[566,0],"end":[568,80],"file_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","tactics":[{"state_before":"it : Iterator\n⊢ ValidFor it.s.data.reverse [] it.toEnd","state_after":"it : Iterator\n⊢ ValidFor it.s.data.reverse [] { s := it.s, i := it.s.endPos }","tactic":"simp only [Iterator.toEnd]","premises":[{"full_name":"String.Iterator.toEnd","def_path":".lake/packages/lean4/src/lean/Init/Data/String/Basic.lean","def_pos":[630,4],"def_end_pos":[630,9]}]},{"state_before":"it : Iterator\n⊢ ValidFor it.s.data.reverse [] { s := it.s, i := it.s.endPos }","state_after":"no goals","tactic":"exact .of_eq _ (by simp [List.reverseAux_eq]) (by simp [endPos, utf8ByteSize])","premises":[{"full_name":"List.reverseAux_eq","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1498,8],"def_end_pos":[1498,21]},{"full_name":"String.Iterator.ValidFor.of_eq","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[493,8],"def_end_pos":[493,13]},{"full_name":"String.endPos","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2473,14],"def_end_pos":[2473,27]},{"full_name":"String.utf8ByteSize","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2441,4],"def_end_pos":[2441,23]}]}]}
{"url":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","commit":"","full_name":"NumberField.canonicalEmbedding.latticeBasis_apply","start":[136,0],"end":[140,48],"file_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ni : Free.ChooseBasisIndex ℤ (𝓞 K)\n⊢ (latticeBasis K) i = (canonicalEmbedding K) ((integralBasis K) i)","state_after":"no goals","tactic":"simp only [latticeBasis, integralBasis_apply, coe_basisOfLinearIndependentOfCardEqFinrank,\n    Function.comp_apply, Equiv.apply_symm_apply]","premises":[{"full_name":"Equiv.apply_symm_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[239,16],"def_end_pos":[239,32]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"NumberField.canonicalEmbedding.latticeBasis","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","def_pos":[108,18],"def_end_pos":[108,30]},{"full_name":"NumberField.integralBasis_apply","def_path":"Mathlib/NumberTheory/NumberField/Basic.lean","def_pos":[261,8],"def_end_pos":[261,27]},{"full_name":"coe_basisOfLinearIndependentOfCardEqFinrank","def_path":"Mathlib/LinearAlgebra/FiniteDimensional.lean","def_pos":[240,8],"def_end_pos":[240,51]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","commit":"","full_name":"SimpleGraph.subgraphOfAdj_verts","start":[79,0],"end":[90,11],"file_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","tactics":[{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nv✝ w✝ : V\nh : (fun a b => s(v, w) = s(a, b)) v✝ w✝\n⊢ G.Adj v✝ w✝","state_after":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nv✝ w✝ : V\nh : (fun a b => s(v, w) = s(a, b)) v✝ w✝\n⊢ s(v, w) ∈ G.edgeSet","tactic":"rw [← G.mem_edgeSet, ← h]","premises":[{"full_name":"SimpleGraph.mem_edgeSet","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[424,8],"def_end_pos":[424,19]}]},{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nv✝ w✝ : V\nh : (fun a b => s(v, w) = s(a, b)) v✝ w✝\n⊢ s(v, w) ∈ G.edgeSet","state_after":"no goals","tactic":"exact hvw","premises":[]},{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\na b : V\nh : (fun a b => s(v, w) = s(a, b)) a b\n⊢ a ∈ {v, w}","state_after":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\na b : V\nh : (a ∈ s(v, w)) = (a ∈ s(a, b))\n⊢ a ∈ {v, w}","tactic":"apply_fun fun e ↦ a ∈ e at h","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]}]},{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\na b : V\nh : (a ∈ s(v, w)) = (a ∈ s(a, b))\n⊢ a ∈ {v, w}","state_after":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\na b : V\nh : a = v ∨ a = w\n⊢ a ∈ {v, w}","tactic":"simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h","premises":[{"full_name":"Sym2.mem_iff","def_path":"Mathlib/Data/Sym/Sym2.lean","def_pos":[324,8],"def_end_pos":[324,15]},{"full_name":"eq_iff_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1378,16],"def_end_pos":[1378,26]},{"full_name":"iff_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[127,16],"def_end_pos":[127,24]},{"full_name":"true_or","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[120,16],"def_end_pos":[120,23]}]},{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\na b : V\nh : a = v ∨ a = w\n⊢ a ∈ {v, w}","state_after":"no goals","tactic":"exact h","premises":[]}]}
{"url":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","commit":"","full_name":"Polynomial.scaleRoots_mul","start":[170,0],"end":[173,32],"file_path":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝³ : Semiring S\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Field K\np : R[X]\nr s : R\n⊢ p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s","state_after":"case a\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝³ : Semiring S\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Field K\np : R[X]\nr s : R\nn✝ : ℕ\n⊢ (p.scaleRoots (r * s)).coeff n✝ = ((p.scaleRoots r).scaleRoots s).coeff n✝","tactic":"ext","premises":[]},{"state_before":"case a\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝³ : Semiring S\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Field K\np : R[X]\nr s : R\nn✝ : ℕ\n⊢ (p.scaleRoots (r * s)).coeff n✝ = ((p.scaleRoots r).scaleRoots s).coeff n✝","state_after":"no goals","tactic":"simp [mul_pow, mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]}]}]}
{"url":"Mathlib/Topology/DiscreteSubset.lean","commit":"","full_name":"Continuous.discrete_of_tendsto_cofinite_cocompact","start":[50,0],"end":[58,83],"file_path":"Mathlib/Topology/DiscreteSubset.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\n⊢ DiscreteTopology X","state_after":"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\n⊢ IsOpen {x}","tactic":"refine singletons_open_iff_discrete.mp (fun x ↦ ?_)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"singletons_open_iff_discrete","def_path":"Mathlib/Topology/Order.lean","def_pos":[307,8],"def_end_pos":[307,36]}]},{"state_before":"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\n⊢ IsOpen {x}","state_after":"case intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\n⊢ IsOpen {x}","tactic":"obtain ⟨K : Set Y, hK : IsCompact K, hK' : K ∈ 𝓝 (f x)⟩ := exists_compact_mem_nhds (f x)","premises":[{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"WeaklyLocallyCompactSpace.exists_compact_mem_nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[274,2],"def_end_pos":[274,25]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"case intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\n⊢ IsOpen {x}","state_after":"case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\nU : Set Y\nhU₁ : U ⊆ K\nhU₂ : IsOpen U\nhU₃ : f x ∈ U\n⊢ IsOpen {x}","tactic":"obtain ⟨U : Set Y, hU₁ : U ⊆ K, hU₂ : IsOpen U, hU₃ : f x ∈ U⟩ := mem_nhds_iff.mp hK'","premises":[{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsOpen","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[82,4],"def_end_pos":[82,10]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"mem_nhds_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[716,8],"def_end_pos":[716,20]}]},{"state_before":"case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\nU : Set Y\nhU₁ : U ⊆ K\nhU₂ : IsOpen U\nhU₃ : f x ∈ U\n⊢ IsOpen {x}","state_after":"case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\nU : Set Y\nhU₁ : U ⊆ K\nhU₂ : IsOpen U\nhU₃ : f x ∈ U\nhU₄ : (f ⁻¹' U).Finite\n⊢ IsOpen {x}","tactic":"have hU₄ : Set.Finite (f⁻¹' U) :=\n    Finite.subset (tendsto_cofinite_cocompact_iff.mp hf K hK) (preimage_mono hU₁)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.Finite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[56,14],"def_end_pos":[56,20]},{"full_name":"Set.Finite.subset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[633,8],"def_end_pos":[633,21]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Set.preimage_mono","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[56,8],"def_end_pos":[56,21]},{"full_name":"tendsto_cofinite_cocompact_iff","def_path":"Mathlib/Topology/DiscreteSubset.lean","def_pos":[44,6],"def_end_pos":[44,36]}]},{"state_before":"case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\nU : Set Y\nhU₁ : U ⊆ K\nhU₂ : IsOpen U\nhU₃ : f x ∈ U\nhU₄ : (f ⁻¹' U).Finite\n⊢ IsOpen {x}","state_after":"no goals","tactic":"exact isOpen_singleton_of_finite_mem_nhds _ ((hU₂.preimage hf').mem_nhds hU₃) hU₄","premises":[{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"IsOpen.preimage","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1327,8],"def_end_pos":[1327,23]},{"full_name":"isOpen_singleton_of_finite_mem_nhds","def_path":"Mathlib/Topology/Separation.lean","def_pos":[858,8],"def_end_pos":[858,43]}]}]}
{"url":"Mathlib/Topology/NoetherianSpace.lean","commit":"","full_name":"TopologicalSpace.NoetherianSpace.finite_irreducibleComponents","start":[192,0],"end":[200,41],"file_path":"Mathlib/Topology/NoetherianSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\n⊢ (irreducibleComponents α).Finite","state_after":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\nS : Set (Set α)\nhSf : S.Finite\nhSc : ∀ t ∈ S, IsClosed t\nhSi : ∀ t ∈ S, IsIrreducible t\nhSU : Set.univ = ⋃₀ S\n⊢ (irreducibleComponents α).Finite","tactic":"obtain ⟨S : Set (Set α), hSf, hSc, hSi, hSU⟩ :=\n    NoetherianSpace.exists_finite_set_isClosed_irreducible isClosed_univ (α := α)","premises":[{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible","def_path":"Mathlib/Topology/NoetherianSpace.lean","def_pos":[176,8],"def_end_pos":[176,62]},{"full_name":"isClosed_univ","def_path":"Mathlib/Topology/Basic.lean","def_pos":[157,16],"def_end_pos":[157,29]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\nS : Set (Set α)\nhSf : S.Finite\nhSc : ∀ t ∈ S, IsClosed t\nhSi : ∀ t ∈ S, IsIrreducible t\nhSU : Set.univ = ⋃₀ S\n⊢ (irreducibleComponents α).Finite","state_after":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\nS : Set (Set α)\nhSf : S.Finite\nhSc : ∀ t ∈ S, IsClosed t\nhSi : ∀ t ∈ S, IsIrreducible t\nhSU : Set.univ = ⋃₀ S\ns : Set α\nhs : s ∈ irreducibleComponents α\n⊢ s ∈ S","tactic":"refine hSf.subset fun s hs => ?_","premises":[{"full_name":"Set.Finite.subset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[633,8],"def_end_pos":[633,21]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\nS : Set (Set α)\nhSf : S.Finite\nhSc : ∀ t ∈ S, IsClosed t\nhSi : ∀ t ∈ S, IsIrreducible t\nhSU : Set.univ = ⋃₀ S\ns : Set α\nhs : s ∈ irreducibleComponents α\n⊢ s ∈ S","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\ns : Set α\nhs : s ∈ irreducibleComponents α\nS : Finset (Set α)\nhSc : ∀ t ∈ ↑S, IsClosed t\nhSi : ∀ t ∈ ↑S, IsIrreducible t\nhSU : Set.univ = ⋃₀ ↑S\n⊢ s ∈ ↑S","tactic":"lift S to Finset (Set α) using hSf","premises":[{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]}]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\ns : Set α\nhs : s ∈ irreducibleComponents α\nS : Finset (Set α)\nhSc : ∀ t ∈ ↑S, IsClosed t\nhSi : ∀ t ∈ ↑S, IsIrreducible t\nhSU : Set.univ = ⋃₀ ↑S\n⊢ s ∈ ↑S","state_after":"case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\ns : Set α\nhs : s ∈ irreducibleComponents α\nS : Finset (Set α)\nhSc : ∀ t ∈ ↑S, IsClosed t\nhSi : ∀ t ∈ ↑S, IsIrreducible t\nhSU : Set.univ = ⋃₀ ↑S\nt : Set α\nhtS : t ∈ S\nht : s ⊆ t\n⊢ s ∈ ↑S","tactic":"rcases isIrreducible_iff_sUnion_closed.1 hs.1 S hSc (hSU ▸ Set.subset_univ _) with ⟨t, htS, ht⟩","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.subset_univ","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[550,8],"def_end_pos":[550,19]},{"full_name":"isIrreducible_iff_sUnion_closed","def_path":"Mathlib/Topology/Irreducible.lean","def_pos":[247,8],"def_end_pos":[247,39]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NoetherianSpace α\ns : Set α\nhs : s ∈ irreducibleComponents α\nS : Finset (Set α)\nhSc : ∀ t ∈ ↑S, IsClosed t\nhSi : ∀ t ∈ ↑S, IsIrreducible t\nhSU : Set.univ = ⋃₀ ↑S\nt : Set α\nhtS : t ∈ S\nht : s ⊆ t\n⊢ s ∈ ↑S","state_after":"no goals","tactic":"rwa [ht.antisymm (hs.2 (hSi _ htS) ht)]","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
{"url":"Mathlib/RingTheory/OreLocalization/Basic.lean","commit":"","full_name":"OreLocalization.oreDiv_smul_char","start":[287,0],"end":[293,59],"file_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\nr₂ : X\ns₁ s₂ : ↥S\nr' : R\ns' : ↥S\nhuv : ↑s' * r₁ = r' * ↑s₂\n⊢ (r₁ /ₒ s₁) • (r₂ /ₒ s₂) = r' • r₂ /ₒ (s' * s₁)","state_after":"no goals","tactic":"with_unfolding_all exact smul'_char r₁ r₂ s₁ s₂ s' r' huv","premises":[{"full_name":"_private.Mathlib.RingTheory.OreLocalization.Basic.0.OreLocalization.smul'_char","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[200,16],"def_end_pos":[200,26]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Real.sqrtTwoAddSeries_one","start":[621,0],"end":[621,67],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"x : ℝ\n⊢ sqrtTwoAddSeries 0 1 = √2","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Topology/FiberBundle/Constructions.lean","commit":"","full_name":"Trivialization.Prod.left_inv","start":[167,0],"end":[172,68],"file_path":"Mathlib/Topology/FiberBundle/Constructions.lean","tactics":[{"state_before":"B : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_4\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_5\ninst✝² : TopologicalSpace (TotalSpace F₂ E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x\nh : x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ x) = x","state_after":"case mk.mk\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_4\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_5\ninst✝² : TopologicalSpace (TotalSpace F₂ E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh : { proj := x, snd := (v₁, v₂) } ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { proj := x, snd := (v₁, v₂) }) = { proj := x, snd := (v₁, v₂) }","tactic":"obtain ⟨x, v₁, v₂⟩ := x","premises":[]},{"state_before":"case mk.mk\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_4\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_5\ninst✝² : TopologicalSpace (TotalSpace F₂ E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh : { proj := x, snd := (v₁, v₂) } ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { proj := x, snd := (v₁, v₂) }) = { proj := x, snd := (v₁, v₂) }","state_after":"case mk.mk.intro\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_4\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_5\ninst✝² : TopologicalSpace (TotalSpace F₂ E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh₁ : x ∈ e₁.baseSet\nh₂ : x ∈ e₂.baseSet\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { proj := x, snd := (v₁, v₂) }) = { proj := x, snd := (v₁, v₂) }","tactic":"obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Trivialization.baseSet","def_path":"Mathlib/Topology/FiberBundle/Trivialization.lean","def_pos":[266,2],"def_end_pos":[266,9]}]},{"state_before":"case mk.mk.intro\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_4\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_5\ninst✝² : TopologicalSpace (TotalSpace F₂ E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh₁ : x ∈ e₁.baseSet\nh₂ : x ∈ e₂.baseSet\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { proj := x, snd := (v₁, v₂) }) = { proj := x, snd := (v₁, v₂) }","state_after":"no goals","tactic":"simp only [Prod.toFun', Prod.invFun', symm_apply_apply_mk, h₁, h₂]","premises":[{"full_name":"Trivialization.Prod.invFun'","def_path":"Mathlib/Topology/FiberBundle/Constructions.lean","def_pos":[162,18],"def_end_pos":[162,30]},{"full_name":"Trivialization.Prod.toFun'","def_path":"Mathlib/Topology/FiberBundle/Constructions.lean","def_pos":[132,4],"def_end_pos":[132,15]},{"full_name":"Trivialization.symm_apply_apply_mk","def_path":"Mathlib/Topology/FiberBundle/Trivialization.lean","def_pos":[554,8],"def_end_pos":[554,27]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Maps.lean","commit":"","full_name":"SimpleGraph.Iso.mapNeighborSet_apply_coe","start":[500,0],"end":[508,24],"file_path":"Mathlib/Combinatorics/SimpleGraph/Maps.lean","tactics":[{"state_before":"V : Type u_1\nW : Type u_2\nX : Type u_3\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nf : G ≃g G'\nv : V\nw : ↑(G'.neighborSet (f v))\n⊢ f.symm ↑w ∈ G.neighborSet v","state_after":"no goals","tactic":"simpa [RelIso.symm_apply_apply] using f.symm.apply_mem_neighborSet_iff.mpr w.2","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"RelIso.symm_apply_apply","def_path":"Mathlib/Order/RelIso/Basic.lean","def_pos":[651,8],"def_end_pos":[651,24]},{"full_name":"SimpleGraph.Iso.apply_mem_neighborSet_iff","def_path":"Mathlib/Combinatorics/SimpleGraph/Maps.lean","def_pos":[477,8],"def_end_pos":[477,33]},{"full_name":"SimpleGraph.Iso.symm","def_path":"Mathlib/Combinatorics/SimpleGraph/Maps.lean","def_pos":[468,7],"def_end_pos":[468,11]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"V : Type u_1\nW : Type u_2\nX : Type u_3\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nf : G ≃g G'\nv : V\nw : ↑(G.neighborSet v)\n⊢ (fun w => ⟨f.symm ↑w, ⋯⟩) ((fun w => ⟨f ↑w, ⋯⟩) w) = w","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"V : Type u_1\nW : Type u_2\nX : Type u_3\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nf : G ≃g G'\nv : V\nw : ↑(G'.neighborSet (f v))\n⊢ (fun w => ⟨f ↑w, ⋯⟩) ((fun w => ⟨f.symm ↑w, ⋯⟩) w) = w","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/GroupTheory/Complement.lean","commit":"","full_name":"Subgroup.exists_left_transversal_of_le","start":[278,0],"end":[292,76],"file_path":"Mathlib/GroupTheory/Complement.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\n⊢ ∃ S, S * ↑H' = ↑H ∧ Nat.card ↑S * Nat.card ↥H' = Nat.card ↥H","state_after":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\n⊢ ∃ S, S * ↑H' = ↑H ∧ Nat.card ↑S * Nat.card ↥H' = Nat.card ↥H","tactic":"let H'' : Subgroup H := H'.comap H.subtype","premises":[{"full_name":"Subgroup","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[293,10],"def_end_pos":[293,18]},{"full_name":"Subgroup.comap","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1051,4],"def_end_pos":[1051,9]},{"full_name":"Subgroup.subtype","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[601,14],"def_end_pos":[601,21]}]},{"state_before":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\n⊢ ∃ S, S * ↑H' = ↑H ∧ Nat.card ↑S * Nat.card ↥H' = Nat.card ↥H","state_after":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\n⊢ ∃ S, S * ↑H' = ↑H ∧ Nat.card ↑S * Nat.card ↥H' = Nat.card ↥H","tactic":"have : H' = H''.map H.subtype := by simp [H'', h]","premises":[{"full_name":"Subgroup.map","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1091,4],"def_end_pos":[1091,7]},{"full_name":"Subgroup.subtype","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[601,14],"def_end_pos":[601,21]}]},{"state_before":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\n⊢ ∃ S, S * ↑H' = ↑H ∧ Nat.card ↑S * Nat.card ↥H' = Nat.card ↥H","state_after":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\n⊢ ∃ S, S * ↑(map H.subtype H'') = ↑H ∧ Nat.card ↑S * Nat.card ↥(map H.subtype H'') = Nat.card ↥H","tactic":"rw [this]","premises":[]},{"state_before":"G : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\n⊢ ∃ S, S * ↑(map H.subtype H'') = ↑H ∧ Nat.card ↑S * Nat.card ↥(map H.subtype H'') = Nat.card ↥H","state_after":"case intro.intro\nG : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS✝ T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\nS : Set ↥H\ncmem : S ∈ leftTransversals ↑H''\n⊢ ∃ S, S * ↑(map H.subtype H'') = ↑H ∧ Nat.card ↑S * Nat.card ↥(map H.subtype H'') = Nat.card ↥H","tactic":"obtain ⟨S, cmem, -⟩ := H''.exists_left_transversal 1","premises":[{"full_name":"Subgroup.exists_left_transversal","def_path":"Mathlib/GroupTheory/Complement.lean","def_pos":[256,6],"def_end_pos":[256,29]}]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS✝ T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\nS : Set ↥H\ncmem : S ∈ leftTransversals ↑H''\n⊢ ∃ S, S * ↑(map H.subtype H'') = ↑H ∧ Nat.card ↑S * Nat.card ↥(map H.subtype H'') = Nat.card ↥H","state_after":"case intro.intro.refine_1\nG : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS✝ T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\nS : Set ↥H\ncmem : S ∈ leftTransversals ↑H''\n⊢ ⇑H.subtype '' S * ↑(map H.subtype H'') = ↑H\n\ncase intro.intro.refine_2\nG : Type u_1\ninst✝ : Group G\nH✝ K : Subgroup G\nS✝ T : Set G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\nS : Set ↥H\ncmem : S ∈ leftTransversals ↑H''\n⊢ Nat.card ↑(⇑H.subtype '' S) * Nat.card ↥(map H.subtype H'') = Nat.card ↥H","tactic":"refine ⟨H.subtype '' S, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Subgroup.subtype","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[601,14],"def_end_pos":[601,21]}]}]}
{"url":"Mathlib/Data/Nat/Cast/Defs.lean","commit":"","full_name":"three_add_one_eq_four","start":[198,0],"end":[202,8],"file_path":"Mathlib/Data/Nat/Cast/Defs.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ 3 + 1 = 4","state_after":"R : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ ↑(1 + 1 + 1 + 1) = 4","tactic":"rw [← two_add_one_eq_three, ← one_add_one_eq_two, ← Nat.cast_one,\n    ← Nat.cast_add, ← Nat.cast_add, ← Nat.cast_add]","premises":[{"full_name":"Nat.cast_add","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"Nat.cast_one","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"one_add_one_eq_two","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[188,8],"def_end_pos":[188,26]},{"full_name":"two_add_one_eq_three","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[193,8],"def_end_pos":[193,28]}]},{"state_before":"R : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ ↑(1 + 1 + 1 + 1) = 4","state_after":"case h\nR : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ 1 + 1 + 1 + 1 = 4","tactic":"apply congrArg","premises":[{"full_name":"congrArg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[362,8],"def_end_pos":[362,16]}]},{"state_before":"case h\nR : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ 1 + 1 + 1 + 1 = 4","state_after":"no goals","tactic":"decide","premises":[]}]}
{"url":"Mathlib/RingTheory/Kaehler/CotangentComplex.lean","commit":"","full_name":"Algebra.Generators.cotangentSpaceBasis_repr_tmul","start":[50,0],"end":[56,5],"file_path":"Mathlib/RingTheory/Kaehler/CotangentComplex.lean","tactics":[{"state_before":"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Generators R S\nr : S\nx : P.Ring\ni : P.vars\n⊢ (P.cotangentSpaceBasis.repr (r ⊗ₜ[P.Ring] (D R P.Ring) x)) i = r * (aeval P.val) ((pderiv i) x)","state_after":"no goals","tactic":"classical\n  simp only [cotangentSpaceBasis, Basis.baseChange_repr_tmul, mvPolynomialBasis_repr_apply,\n    Algebra.smul_def, mul_comm r]\n  rfl","premises":[{"full_name":"Algebra.Generators.cotangentSpaceBasis","def_path":"Mathlib/RingTheory/Kaehler/CotangentComplex.lean","def_pos":[47,4],"def_end_pos":[47,23]},{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"Basis.baseChange_repr_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basis.lean","def_pos":[60,6],"def_end_pos":[60,32]},{"full_name":"KaehlerDifferential.mvPolynomialBasis_repr_apply","def_path":"Mathlib/RingTheory/Kaehler/Polynomial.lean","def_pos":[73,6],"def_end_pos":[73,54]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","commit":"","full_name":"Asymptotics.IsBigO.sum","start":[1534,0],"end":[1537,30],"file_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_17\nA : ι → α → E'\nC : ι → ℝ\ns : Finset ι\nh : ∀ i ∈ s, A i =O[l] g\n⊢ (fun x => ∑ i ∈ s, A i x) =O[l] g","state_after":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_17\nA : ι → α → E'\nC : ι → ℝ\ns : Finset ι\nh : ∀ i ∈ s, ∃ c, IsBigOWith c l (A i) g\n⊢ ∃ c, IsBigOWith c l (fun x => ∑ i ∈ s, A i x) g","tactic":"simp only [IsBigO_def] at *","premises":[{"full_name":"Asymptotics.IsBigO_def","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[93,16],"def_end_pos":[93,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_17\nA : ι → α → E'\nC : ι → ℝ\ns : Finset ι\nh : ∀ i ∈ s, ∃ c, IsBigOWith c l (A i) g\n⊢ ∃ c, IsBigOWith c l (fun x => ∑ i ∈ s, A i x) g","state_after":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_17\nA : ι → α → E'\nC✝ : ι → ℝ\ns : Finset ι\nC : ι → ℝ\nhC : ∀ i ∈ s, IsBigOWith (C i) l (A i) g\n⊢ ∃ c, IsBigOWith c l (fun x => ∑ i ∈ s, A i x) g","tactic":"choose! C hC using h","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_17\nA : ι → α → E'\nC✝ : ι → ℝ\ns : Finset ι\nC : ι → ℝ\nhC : ∀ i ∈ s, IsBigOWith (C i) l (A i) g\n⊢ ∃ c, IsBigOWith c l (fun x => ∑ i ∈ s, A i x) g","state_after":"no goals","tactic":"exact ⟨_, IsBigOWith.sum hC⟩","premises":[{"full_name":"Asymptotics.IsBigOWith.sum","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1527,8],"def_end_pos":[1527,22]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Monic.lean","commit":"","full_name":"Polynomial.Monic.map","start":[58,0],"end":[66,80],"file_path":"Mathlib/Algebra/Polynomial/Monic.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\n⊢ (Polynomial.map f p).Monic","state_after":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\n⊢ (Polynomial.map f p).leadingCoeff = 1","tactic":"unfold Monic","premises":[{"full_name":"Polynomial.Monic","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[69,4],"def_end_pos":[69,9]}]},{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\n⊢ (Polynomial.map f p).leadingCoeff = 1","state_after":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\n⊢ (Polynomial.map f p).leadingCoeff = 1","tactic":"nontriviality","premises":[]},{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\n⊢ (Polynomial.map f p).leadingCoeff = 1","state_after":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\nthis : f p.leadingCoeff ≠ 0\n⊢ (Polynomial.map f p).leadingCoeff = 1","tactic":"have : f p.leadingCoeff ≠ 0 := by\n    rw [show _ = _ from hp, f.map_one]\n    exact one_ne_zero","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.leadingCoeff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[65,4],"def_end_pos":[65,16]},{"full_name":"RingHom.map_one","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[476,18],"def_end_pos":[476,25]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]}]},{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\nthis : f p.leadingCoeff ≠ 0\n⊢ (Polynomial.map f p).leadingCoeff = 1","state_after":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\nthis : f p.leadingCoeff ≠ 0\n⊢ f (p.coeff (Polynomial.map f p).natDegree) = 1","tactic":"rw [Polynomial.leadingCoeff, coeff_map]","premises":[{"full_name":"Polynomial.coeff_map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[671,8],"def_end_pos":[671,17]},{"full_name":"Polynomial.leadingCoeff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[65,4],"def_end_pos":[65,16]}]},{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\nthis : f p.leadingCoeff ≠ 0\n⊢ f (p.coeff (Polynomial.map f p).natDegree) = 1","state_after":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\nthis : f p.leadingCoeff ≠ 0\n⊢ p.coeff (Polynomial.map f p).natDegree = 1","tactic":"suffices p.coeff (p.map f).natDegree = 1 by simp [this]","premises":[{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[597,4],"def_end_pos":[597,7]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]}]},{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : p.Monic\na✝ : Nontrivial S\nthis : f p.leadingCoeff ≠ 0\n⊢ p.coeff (Polynomial.map f p).natDegree = 1","state_after":"no goals","tactic":"rwa [natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f this)]","premises":[{"full_name":"Polynomial.degree_map_eq_of_leadingCoeff_ne_zero","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[749,8],"def_end_pos":[749,45]},{"full_name":"Polynomial.natDegree_eq_of_degree_eq","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[147,8],"def_end_pos":[147,33]}]}]}
{"url":"Mathlib/Algebra/Group/Action/Pi.lean","commit":"","full_name":"Pi.faithfulVAdd_at","start":[71,0],"end":[81,98],"file_path":"Mathlib/Algebra/Group/Action/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nM : Type u_2\nN : Type u_3\nα : ι → Type u_4\nβ : ι → Type u_5\nγ : ι → Type u_6\nx y : (i : ι) → α i\ni✝ : ι\ninst✝² : (i : ι) → SMul M (α i)\ninst✝¹ : ∀ (i : ι), Nonempty (α i)\ni : ι\ninst✝ : FaithfulSMul M (α i)\nm₁✝ m₂✝ : M\nh : ∀ (a : (i : ι) → α i), m₁✝ • a = m₂✝ • a\na : α i\n⊢ m₁✝ • a = m₂✝ • a","state_after":"no goals","tactic":"classical\n    simpa using\n      congr_fun (h <| Function.update (fun j => Classical.choice (‹∀ i, Nonempty (α i)› j)) i a) i","premises":[{"full_name":"Classical.choice","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[735,6],"def_end_pos":[735,22]},{"full_name":"Function.update","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[462,4],"def_end_pos":[462,10]}]}]}
{"url":"Mathlib/NumberTheory/Zsqrtd/Basic.lean","commit":"","full_name":"Zsqrtd.le_of_add_le_add_left","start":[680,0],"end":[681,47],"file_path":"Mathlib/NumberTheory/Zsqrtd/Basic.lean","tactics":[{"state_before":"d : ℕ\na b c : ℤ√↑d\nh : c + a ≤ c + b\n⊢ a ≤ b","state_after":"no goals","tactic":"simpa using Zsqrtd.add_le_add_left _ _ h (-c)","premises":[{"full_name":"Zsqrtd.add_le_add_left","def_path":"Mathlib/NumberTheory/Zsqrtd/Basic.lean","def_pos":[677,18],"def_end_pos":[677,33]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean","commit":"","full_name":"szemeredi_regularity","start":[70,0],"end":[149,32],"file_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","state_after":"case inl\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : Fintype.card α ≤ bound ε l\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε\n\ncase inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","tactic":"obtain hα | hα := le_total (card α) (bound ε l)","premises":[{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"SzemerediRegularity.bound","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[187,18],"def_end_pos":[187,23]},{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","state_after":"case inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","tactic":"let t := initialBound ε l","premises":[{"full_name":"SzemerediRegularity.initialBound","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[164,18],"def_end_pos":[164,30]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","state_after":"case inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","tactic":"have htα : t ≤ (univ : Finset α).card :=\n    (initialBound_le_bound _ _).trans (by rwa [Finset.card_univ])","premises":[{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Finset.card_univ","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[228,8],"def_end_pos":[228,24]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"SzemerediRegularity.initialBound_le_bound","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[191,8],"def_end_pos":[191,29]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","state_after":"case inr.intro.intro\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","tactic":"obtain ⟨dum, hdum₁, hdum₂⟩ :=\n    exists_equipartition_card_eq (univ : Finset α) (initialBound_pos _ _).ne' htα","premises":[{"full_name":"Finpartition.exists_equipartition_card_eq","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean","def_pos":[195,8],"def_end_pos":[195,36]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"SzemerediRegularity.initialBound_pos","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[173,8],"def_end_pos":[173,24]}]},{"state_before":"case inr.intro.intro\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","state_after":"case inr.intro.intro.inl\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : 1 ≤ ε\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε\n\ncase inr.intro.intro.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","tactic":"obtain hε₁ | hε₁ := le_total 1 ε","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]},{"state_before":"case inr.intro.intro.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","state_after":"case inr.intro.intro.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\n⊢ ∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε","tactic":"have : Nonempty α := by\n    rw [← Fintype.card_pos_iff]\n    exact (bound_pos _ _).trans_le hα","premises":[{"full_name":"Fintype.card_pos_iff","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[488,8],"def_end_pos":[488,20]},{"full_name":"Nonempty","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[709,16],"def_end_pos":[709,24]},{"full_name":"SzemerediRegularity.bound_pos","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[197,8],"def_end_pos":[197,17]}]},{"state_before":"case inr.intro.intro.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\n⊢ ∀ (i : ℕ),\n    ∃ P,\n      P.IsEquipartition ∧\n        t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))","state_after":"case inr.intro.intro.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))","tactic":"intro i","premises":[]},{"state_before":"case inr.intro.intro.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))","state_after":"case inr.intro.intro.inr.zero\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[0] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑0 ≤ ↑(P.energy G))\n\ncase inr.intro.intro.inr.succ\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nih :\n  ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"induction' i with i ih","premises":[]},{"state_before":"case inr.intro.intro.inr.succ\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nih :\n  ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case inr.intro.intro.inr.succ.intro.intro.intro.intro\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"obtain ⟨P, hP₁, hP₂, hP₃, hP₄⟩ := ih","premises":[]},{"state_before":"case inr.intro.intro.inr.succ.intro.intro.intro.intro\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case pos\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhuniform : P.IsUniform G ε\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))\n\ncase neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhuniform : ¬P.IsUniform G ε\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"by_cases huniform : P.IsUniform G ε","premises":[{"full_name":"Finpartition.IsUniform","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","def_pos":[226,4],"def_end_pos":[226,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhuniform : ¬P.IsUniform G ε\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"replace hP₄ := hP₄.resolve_left huniform","premises":[{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]}]},{"state_before":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"have hεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5 :=\n    (hundred_lt_pow_initialBound_mul hε l).le.trans\n      (mul_le_mul_of_nonneg_right (pow_le_pow_right (by norm_num) hP₂) <| by positivity)","premises":[{"full_name":"Finpartition.parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[65,2],"def_end_pos":[65,7]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"SzemerediRegularity.hundred_lt_pow_initialBound_mul","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[176,8],"def_end_pos":[176,39]},{"full_name":"mul_le_mul_of_nonneg_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[194,8],"def_end_pos":[194,34]},{"full_name":"pow_le_pow_right","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[84,8],"def_end_pos":[84,24]}]},{"state_before":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"have hi : (i : ℝ) ≤ 4 / ε ^ 5 := by\n    have hi : ε ^ 5 / 4 * ↑i ≤ 1 := hP₄.trans (mod_cast P.energy_le_one G)\n    rw [div_mul_eq_mul_div, div_le_iff (show (0 : ℝ) < 4 by norm_num)] at hi\n    norm_num at hi\n    rwa [le_div_iff' (pow_pos hε _)]","premises":[{"full_name":"Finpartition.energy_le_one","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean","def_pos":[42,8],"def_end_pos":[42,21]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"div_le_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[51,8],"def_end_pos":[51,18]},{"full_name":"div_mul_eq_mul_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[558,8],"def_end_pos":[558,26]},{"full_name":"le_div_iff'","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[49,8],"def_end_pos":[49,19]},{"full_name":"pow_pos","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[385,8],"def_end_pos":[385,15]}]},{"state_before":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"have hsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t :=\n    hP₃.trans (monotone_iterate_of_id_le le_stepBound (Nat.le_floor hi) _)","premises":[{"full_name":"Finpartition.parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[65,2],"def_end_pos":[65,7]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Function.monotone_iterate_of_id_le","def_path":"Mathlib/Order/Iterate.lean","def_pos":[129,8],"def_end_pos":[129,33]},{"full_name":"Nat.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[98,4],"def_end_pos":[98,9]},{"full_name":"Nat.iterate","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[36,4],"def_end_pos":[36,15]},{"full_name":"Nat.le_floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[128,8],"def_end_pos":[128,16]},{"full_name":"SzemerediRegularity.le_stepBound","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[40,8],"def_end_pos":[40,20]},{"full_name":"SzemerediRegularity.stepBound","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[37,4],"def_end_pos":[37,13]}]},{"state_before":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","tactic":"have hPα : P.parts.card * 16 ^ P.parts.card ≤ card α :=\n    (Nat.mul_le_mul hsize (Nat.pow_le_pow_of_le_right (by norm_num) hsize)).trans hα","premises":[{"full_name":"Finpartition.parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[65,2],"def_end_pos":[65,7]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Nat.mul_le_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[722,18],"def_end_pos":[722,28]},{"full_name":"Nat.pow_le_pow_of_le_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[761,8],"def_end_pos":[761,30]}]},{"state_before":"case neg\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\n⊢ ∃ P,\n    P.IsEquipartition ∧\n      t ≤ P.parts.card ∧ P.parts.card ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))","state_after":"case neg.refine_1\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\n⊢ t ≤ (increment hP₁ G ε).parts.card\n\ncase neg.refine_2\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\n⊢ (increment hP₁ G ε).parts.card ≤ stepBound^[i + 1] t\n\ncase neg.refine_3\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ univ.card\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : dum.parts.card = initialBound ε l\nhε₁ : ε ≤ 1\nthis : Nonempty α\ni : ℕ\nP : Finpartition univ\nhP₁ : P.IsEquipartition\nhP₂ : t ≤ P.parts.card\nhP₃ : P.parts.card ≤ stepBound^[i] t\nhuniform : ¬P.IsUniform G ε\nhP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)\nhεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nhi : ↑i ≤ 4 / ε ^ 5\nhsize : P.parts.card ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\n⊢ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G) + ε ^ 5 / 4","tactic":"refine ⟨increment hP₁ G ε, increment_isEquipartition hP₁ G ε, ?_, ?_, Or.inr <| le_trans ?_ <|\n    energy_increment hP₁ ((seven_le_initialBound ε l).trans hP₂) hεl' hPα huniform hε.le hε₁⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"SzemerediRegularity.energy_increment","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean","def_pos":[133,8],"def_end_pos":[133,24]},{"full_name":"SzemerediRegularity.increment","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean","def_pos":[54,18],"def_end_pos":[54,27]},{"full_name":"SzemerediRegularity.increment_isEquipartition","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean","def_pos":[78,8],"def_end_pos":[78,33]},{"full_name":"SzemerediRegularity.seven_le_initialBound","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean","def_pos":[170,8],"def_end_pos":[170,29]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]}]}
{"url":"Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean","commit":"","full_name":"HomologicalComplex.biprodXIso_hom_fst","start":[55,0],"end":[58,19],"file_path":"Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean","tactics":[{"state_before":"C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK L : HomologicalComplex C c\ninst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)\ni : ι\n⊢ (K.biprodXIso L i).hom ≫ biprod.fst = biprod.fst.f i","state_after":"no goals","tactic":"simp [biprodXIso]","premises":[{"full_name":"HomologicalComplex.biprodXIso","def_path":"Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean","def_pos":[42,18],"def_end_pos":[42,28]}]}]}
{"url":"Mathlib/CategoryTheory/Preadditive/EndoFunctor.lean","commit":"","full_name":"CategoryTheory.Endofunctor.algebraPreadditive_homGroup_zero_f","start":[27,0],"end":[103,18],"file_path":"Mathlib/CategoryTheory/Preadditive/EndoFunctor.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nα β : A₁ ⟶ A₂\n⊢ F.map (α.f + β.f) ≫ A₂.str = A₁.str ≫ (α.f + β.f)","state_after":"no goals","tactic":"simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add]","premises":[{"full_name":"CategoryTheory.Endofunctor.Algebra.Hom.h","def_path":"Mathlib/CategoryTheory/Endofunctor/Algebra.lean","def_pos":[65,2],"def_end_pos":[65,3]},{"full_name":"CategoryTheory.Functor.map_add","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[52,8],"def_end_pos":[52,15]},{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ F.map 0 ≫ A₂.str = A₁.str ≫ 0","state_after":"no goals","tactic":"simp only [Functor.map_zero, zero_comp, comp_zero]","premises":[{"full_name":"CategoryTheory.Functor.map_zero","def_path":"Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean","def_pos":[49,18],"def_end_pos":[49,26]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn : ℕ\nα : A₁ ⟶ A₂\n⊢ F.map (n • α.f) ≫ A₂.str = A₁.str ≫ (n • α.f)","state_after":"no goals","tactic":"rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h]","premises":[{"full_name":"CategoryTheory.Endofunctor.Algebra.Hom.h","def_path":"Mathlib/CategoryTheory/Endofunctor/Algebra.lean","def_pos":[65,2],"def_end_pos":[65,3]},{"full_name":"CategoryTheory.Functor.map_nsmul","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"CategoryTheory.Preadditive.comp_nsmul","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[152,8],"def_end_pos":[152,18]},{"full_name":"CategoryTheory.Preadditive.nsmul_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[149,8],"def_end_pos":[149,18]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nα : A₁ ⟶ A₂\n⊢ F.map (-α.f) ≫ A₂.str = A₁.str ≫ (-α.f)","state_after":"no goals","tactic":"simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg]","premises":[{"full_name":"CategoryTheory.Endofunctor.Algebra.Hom.h","def_path":"Mathlib/CategoryTheory/Endofunctor/Algebra.lean","def_pos":[65,2],"def_end_pos":[65,3]},{"full_name":"CategoryTheory.Functor.map_neg","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[73,8],"def_end_pos":[73,15]},{"full_name":"CategoryTheory.Preadditive.comp_neg","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[143,8],"def_end_pos":[143,16]},{"full_name":"CategoryTheory.Preadditive.neg_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[138,8],"def_end_pos":[138,16]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nα β : A₁ ⟶ A₂\n⊢ F.map (α.f - β.f) ≫ A₂.str = A₁.str ≫ (α.f - β.f)","state_after":"no goals","tactic":"simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub]","premises":[{"full_name":"CategoryTheory.Endofunctor.Algebra.Hom.h","def_path":"Mathlib/CategoryTheory/Endofunctor/Algebra.lean","def_pos":[65,2],"def_end_pos":[65,3]},{"full_name":"CategoryTheory.Functor.map_sub","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[77,8],"def_end_pos":[77,15]},{"full_name":"CategoryTheory.Preadditive.comp_sub","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"CategoryTheory.Preadditive.sub_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[128,8],"def_end_pos":[128,16]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nr : ℤ\nα : A₁ ⟶ A₂\n⊢ F.map (r • α.f) ≫ A₂.str = A₁.str ≫ (r • α.f)","state_after":"no goals","tactic":"rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h]","premises":[{"full_name":"CategoryTheory.Endofunctor.Algebra.Hom.h","def_path":"Mathlib/CategoryTheory/Endofunctor/Algebra.lean","def_pos":[65,2],"def_end_pos":[65,3]},{"full_name":"CategoryTheory.Functor.map_zsmul","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[84,8],"def_end_pos":[84,17]},{"full_name":"CategoryTheory.Preadditive.comp_zsmul","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[158,8],"def_end_pos":[158,18]},{"full_name":"CategoryTheory.Preadditive.zsmul_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[155,8],"def_end_pos":[155,18]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a b c : A₁ ⟶ A₂), a + b + c = a + (b + c)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ c✝ : A₁ ⟶ A₂\n⊢ a✝ + b✝ + c✝ = a✝ + (b✝ + c✝)","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ c✝ : A₁ ⟶ A₂\n⊢ a✝ + b✝ + c✝ = a✝ + (b✝ + c✝)","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ c✝ : A₁ ⟶ A₂\n⊢ (a✝ + b✝ + c✝).f = (a✝ + (b✝ + c✝)).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ c✝ : A₁ ⟶ A₂\n⊢ (a✝ + b✝ + c✝).f = (a✝ + (b✝ + c✝)).f","state_after":"no goals","tactic":"apply add_assoc","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a : A₁ ⟶ A₂), 0 + a = a","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ 0 + a✝ = a✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ 0 + a✝ = a✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (0 + a✝).f = a✝.f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (0 + a✝).f = a✝.f","state_after":"no goals","tactic":"apply zero_add","premises":[{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a : A₁ ⟶ A₂), a + 0 = a","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ a✝ + 0 = a✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ a✝ + 0 = a✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (a✝ + 0).f = a✝.f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (a✝ + 0).f = a✝.f","state_after":"no goals","tactic":"apply add_zero","premises":[{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (x : A₁ ⟶ A₂), (fun n α => { f := n • α.f, h := ⋯ }) 0 x = 0","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nx✝ : A₁ ⟶ A₂\n⊢ (fun n α => { f := n • α.f, h := ⋯ }) 0 x✝ = 0","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nx✝ : A₁ ⟶ A₂\n⊢ (fun n α => { f := n • α.f, h := ⋯ }) 0 x✝ = 0","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nx✝ : A₁ ⟶ A₂\n⊢ ((fun n α => { f := n • α.f, h := ⋯ }) 0 x✝).f = Algebra.Hom.f 0","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nx✝ : A₁ ⟶ A₂\n⊢ ((fun n α => { f := n • α.f, h := ⋯ }) 0 x✝).f = Algebra.Hom.f 0","state_after":"no goals","tactic":"apply zero_smul","premises":[{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (n : ℕ) (x : A₁ ⟶ A₂),\n    (fun n α => { f := n • α.f, h := ⋯ }) (n + 1) x = (fun n α => { f := n • α.f, h := ⋯ }) n x + x","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\nx✝ : A₁ ⟶ A₂\n⊢ (fun n α => { f := n • α.f, h := ⋯ }) (n✝ + 1) x✝ = (fun n α => { f := n • α.f, h := ⋯ }) n✝ x✝ + x✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\nx✝ : A₁ ⟶ A₂\n⊢ (fun n α => { f := n • α.f, h := ⋯ }) (n✝ + 1) x✝ = (fun n α => { f := n • α.f, h := ⋯ }) n✝ x✝ + x✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\nx✝ : A₁ ⟶ A₂\n⊢ ((fun n α => { f := n • α.f, h := ⋯ }) (n✝ + 1) x✝).f = ((fun n α => { f := n • α.f, h := ⋯ }) n✝ x✝ + x✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\nx✝ : A₁ ⟶ A₂\n⊢ ((fun n α => { f := n • α.f, h := ⋯ }) (n✝ + 1) x✝).f = ((fun n α => { f := n • α.f, h := ⋯ }) n✝ x✝ + x✝).f","state_after":"no goals","tactic":"apply succ_nsmul","premises":[{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a b : A₁ ⟶ A₂), a - b = a + -b","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ a✝ - b✝ = a✝ + -b✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ a✝ - b✝ = a✝ + -b✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ (a✝ - b✝).f = (a✝ + -b✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ (a✝ - b✝).f = (a✝ + -b✝).f","state_after":"no goals","tactic":"apply sub_eq_add_neg","premises":[{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a : A₁ ⟶ A₂), (fun r α => { f := r • α.f, h := ⋯ }) 0 a = 0","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (fun r α => { f := r • α.f, h := ⋯ }) 0 a✝ = 0","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (fun r α => { f := r • α.f, h := ⋯ }) 0 a✝ = 0","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ ((fun r α => { f := r • α.f, h := ⋯ }) 0 a✝).f = Algebra.Hom.f 0","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ ((fun r α => { f := r • α.f, h := ⋯ }) 0 a✝).f = Algebra.Hom.f 0","state_after":"no goals","tactic":"apply zero_smul","premises":[{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (n : ℕ) (a : A₁ ⟶ A₂),\n    (fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n.succ) a =\n      (fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n) a + a","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ (fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝.succ) a✝ =\n    (fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝) a✝ + a✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ (fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝.succ) a✝ =\n    (fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝) a✝ + a✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ ((fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝.succ) a✝).f =\n    ((fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝) a✝ + a✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ ((fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝.succ) a✝).f =\n    ((fun r α => { f := r • α.f, h := ⋯ }) (Int.ofNat n✝) a✝ + a✝).f","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ ↑(n✝ + 1) • a✝.f = ({ f := ↑n✝ • a✝.f, h := ⋯ } + a✝).f","tactic":"dsimp","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ ↑(n✝ + 1) • a✝.f = ({ f := ↑n✝ • a✝.f, h := ⋯ } + a✝).f","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ n✝ • a✝.f + a✝.f = ({ f := n✝ • a✝.f, h := ⋯ } + a✝).f","tactic":"simp only [natCast_zsmul, succ_nsmul]","premises":[{"full_name":"natCast_zsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[874,40],"def_end_pos":[874,53]},{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ n✝ • a✝.f + a✝.f = ({ f := n✝ • a✝.f, h := ⋯ } + a✝).f","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (n : ℕ) (a : A₁ ⟶ A₂),\n    (fun r α => { f := r • α.f, h := ⋯ }) (Int.negSucc n) a = -(fun r α => { f := r • α.f, h := ⋯ }) (↑n.succ) a","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ (fun r α => { f := r • α.f, h := ⋯ }) (Int.negSucc n✝) a✝ = -(fun r α => { f := r • α.f, h := ⋯ }) (↑n✝.succ) a✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ (fun r α => { f := r • α.f, h := ⋯ }) (Int.negSucc n✝) a✝ = -(fun r α => { f := r • α.f, h := ⋯ }) (↑n✝.succ) a✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ ((fun r α => { f := r • α.f, h := ⋯ }) (Int.negSucc n✝) a✝).f =\n    (-(fun r α => { f := r • α.f, h := ⋯ }) (↑n✝.succ) a✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\nn✝ : ℕ\na✝ : A₁ ⟶ A₂\n⊢ ((fun r α => { f := r • α.f, h := ⋯ }) (Int.negSucc n✝) a✝).f =\n    (-(fun r α => { f := r • α.f, h := ⋯ }) (↑n✝.succ) a✝).f","state_after":"no goals","tactic":"simp only [negSucc_zsmul, neg_inj, ← Nat.cast_smul_eq_nsmul ℤ]","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Nat.cast_smul_eq_nsmul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[329,6],"def_end_pos":[329,28]},{"full_name":"negSucc_zsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[894,8],"def_end_pos":[894,21]},{"full_name":"neg_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[294,2],"def_end_pos":[294,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a : A₁ ⟶ A₂), -a + a = 0","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ -a✝ + a✝ = 0","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ -a✝ + a✝ = 0","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (-a✝ + a✝).f = Algebra.Hom.f 0","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ : A₁ ⟶ A₂\n⊢ (-a✝ + a✝).f = Algebra.Hom.f 0","state_after":"no goals","tactic":"apply add_left_neg","premises":[{"full_name":"add_left_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1038,2],"def_end_pos":[1038,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a b : A₁ ⟶ A₂), a + b = b + a","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ a✝ + b✝ = b✝ + a✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ a✝ + b✝ = b✝ + a✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ (a✝ + b✝).f = (b✝ + a✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\na✝ b✝ : A₁ ⟶ A₂\n⊢ (a✝ + b✝).f = (b✝ + a✝).f","state_after":"no goals","tactic":"apply add_comm","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\n⊢ ∀ (P Q R : Algebra F) (f f' : P ⟶ Q) (g : Q ⟶ R), (f + f') ≫ g = f ≫ g + f' ≫ g","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ f'✝ : P✝ ⟶ Q✝\ng✝ : Q✝ ⟶ R✝\n⊢ (f✝ + f'✝) ≫ g✝ = f✝ ≫ g✝ + f'✝ ≫ g✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ f'✝ : P✝ ⟶ Q✝\ng✝ : Q✝ ⟶ R✝\n⊢ (f✝ + f'✝) ≫ g✝ = f✝ ≫ g✝ + f'✝ ≫ g✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ f'✝ : P✝ ⟶ Q✝\ng✝ : Q✝ ⟶ R✝\n⊢ ((f✝ + f'✝) ≫ g✝).f = (f✝ ≫ g✝ + f'✝ ≫ g✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ f'✝ : P✝ ⟶ Q✝\ng✝ : Q✝ ⟶ R✝\n⊢ ((f✝ + f'✝) ≫ g✝).f = (f✝ ≫ g✝ + f'✝ ≫ g✝).f","state_after":"no goals","tactic":"apply add_comp","premises":[{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\n⊢ ∀ (P Q R : Algebra F) (f : P ⟶ Q) (g g' : Q ⟶ R), f ≫ (g + g') = f ≫ g + f ≫ g'","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ : P✝ ⟶ Q✝\ng✝ g'✝ : Q✝ ⟶ R✝\n⊢ f✝ ≫ (g✝ + g'✝) = f✝ ≫ g✝ + f✝ ≫ g'✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ : P✝ ⟶ Q✝\ng✝ g'✝ : Q✝ ⟶ R✝\n⊢ f✝ ≫ (g✝ + g'✝) = f✝ ≫ g✝ + f✝ ≫ g'✝","state_after":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ : P✝ ⟶ Q✝\ng✝ g'✝ : Q✝ ⟶ R✝\n⊢ (f✝ ≫ (g✝ + g'✝)).f = (f✝ ≫ g✝ + f✝ ≫ g'✝).f","tactic":"apply Algebra.Hom.ext","premises":[]},{"state_before":"case f\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nP✝ Q✝ R✝ : Algebra F\nf✝ : P✝ ⟶ Q✝\ng✝ g'✝ : Q✝ ⟶ R✝\n⊢ (f✝ ≫ (g✝ + g'✝)).f = (f✝ ≫ g✝ + f✝ ≫ g'✝).f","state_after":"no goals","tactic":"apply comp_add","premises":[{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]}]}]}
{"url":"Mathlib/Algebra/Algebra/Spectrum.lean","commit":"","full_name":"AlgEquiv.spectrum_eq","start":[396,0],"end":[402,61],"file_path":"Mathlib/Algebra/Algebra/Spectrum.lean","tactics":[{"state_before":"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Ring A\ninst✝⁴ : Ring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : EquivLike F A B\ninst✝ : AlgEquivClass F R A B\nf : F\na : A\n⊢ spectrum R a ⊆ spectrum R (f a)","state_after":"no goals","tactic":"simpa only [AlgEquiv.coe_algHom, AlgEquiv.coe_coe_symm_apply_coe_apply] using\n      AlgHom.spectrum_apply_subset (f : A ≃ₐ[R] B).symm (f a)","premises":[{"full_name":"AlgEquiv","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[26,10],"def_end_pos":[26,18]},{"full_name":"AlgEquiv.coe_algHom","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[230,8],"def_end_pos":[230,18]},{"full_name":"AlgEquiv.coe_coe_symm_apply_coe_apply","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[296,8],"def_end_pos":[296,36]},{"full_name":"AlgEquiv.symm","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[275,4],"def_end_pos":[275,8]},{"full_name":"AlgHom.spectrum_apply_subset","def_path":"Mathlib/Algebra/Algebra/Spectrum.lean","def_pos":[371,8],"def_end_pos":[371,29]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Circumcenter.lean","commit":"","full_name":"Affine.Simplex.circumradius_reindex","start":[380,0],"end":[383,98],"file_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ (s.reindex e).circumradius = s.circumradius","state_after":"no goals","tactic":"simp_rw [circumradius, circumsphere_reindex]","premises":[{"full_name":"Affine.Simplex.circumradius","def_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","def_pos":[258,4],"def_end_pos":[258,16]},{"full_name":"Affine.Simplex.circumsphere_reindex","def_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","def_pos":[369,8],"def_end_pos":[369,28]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","commit":"","full_name":"SimpleGraph.Walk.map_injective_of_injective","start":[1115,0],"end":[1131,21],"file_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝ v✝ u' v' : V\np : G.Walk u✝ v✝\nf : G →g G'\nhinj : Injective ⇑f\nu v : V\n⊢ Injective (Walk.map f)","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝ v✝ u' v' : V\np✝ : G.Walk u✝ v✝\nf : G →g G'\nhinj : Injective ⇑f\nu v : V\np p' : G.Walk u v\nh : Walk.map f p = Walk.map f p'\n⊢ p = p'","tactic":"intro p p' h","premises":[]}]}
{"url":"Mathlib/Data/Set/Pairwise/Basic.lean","commit":"","full_name":"Set.pairwise_insert_of_not_mem","start":[134,0],"end":[137,93],"file_path":"Mathlib/Data/Set/Pairwise/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nr p q : α → α → Prop\nf g : ι → α\ns t u : Set α\na b✝ : α\nha : a ∉ s\nb : α\nhb : b ∈ s\n⊢ a ≠ b → r a b ∧ r b a ↔ r a b ∧ r b a","state_after":"no goals","tactic":"simp [(ne_of_mem_of_not_mem hb ha).symm]","premises":[{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"ne_of_mem_of_not_mem","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[116,8],"def_end_pos":[116,28]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Quotient.lean","commit":"","full_name":"MeasureTheory.leftInvariantIsQuotientMeasureEqMeasurePreimage","start":[180,0],"end":[201,26],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Quotient.lean","tactics":[{"state_before":"G : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\nh : covolume (↥Γ.op) G ν = μ univ\n⊢ QuotientMeasureEqMeasurePreimage ν μ","state_after":"case intro\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\nh : covolume (↥Γ.op) G ν = μ univ\ns : Set G\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\n⊢ QuotientMeasureEqMeasurePreimage ν μ","tactic":"obtain ⟨s, fund_dom_s⟩ := hasFun.ExistsIsFundamentalDomain","premises":[{"full_name":"MeasureTheory.HasFundamentalDomain.ExistsIsFundamentalDomain","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[688,2],"def_end_pos":[688,27]}]},{"state_before":"case intro\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\nh : covolume (↥Γ.op) G ν = μ univ\ns : Set G\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\n⊢ QuotientMeasureEqMeasurePreimage ν μ","state_after":"case intro\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\nh : covolume (↥Γ.op) G ν = μ univ\ns : Set G\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\n⊢ QuotientMeasureEqMeasurePreimage ν μ","tactic":"have finiteCovol : μ univ < ⊤ := measure_lt_top μ univ","premises":[{"full_name":"MeasureTheory.measure_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[48,8],"def_end_pos":[48,22]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\nh : covolume (↥Γ.op) G ν = μ univ\ns : Set G\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\n⊢ QuotientMeasureEqMeasurePreimage ν μ","state_after":"case intro\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\n⊢ QuotientMeasureEqMeasurePreimage ν μ","tactic":"rw [fund_dom_s.covolume_eq_volume] at h","premises":[{"full_name":"MeasureTheory.IsFundamentalDomain.covolume_eq_volume","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[711,6],"def_end_pos":[711,44]}]},{"state_before":"case intro\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\n⊢ QuotientMeasureEqMeasurePreimage ν μ","state_after":"case pos\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\nmeas_s_ne_zero : ν s = 0\n⊢ QuotientMeasureEqMeasurePreimage ν μ\n\ncase neg\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\nmeas_s_ne_zero : ¬ν s = 0\n⊢ QuotientMeasureEqMeasurePreimage ν μ","tactic":"by_cases meas_s_ne_zero : ν s = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\nmeas_s_ne_zero : ¬ν s = 0\n⊢ QuotientMeasureEqMeasurePreimage ν μ","state_after":"case neg.neZeroV\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\nmeas_s_ne_zero : ¬ν s = 0\n⊢ μ univ ≠ 0\n\ncase neg.hV\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\nmeas_s_ne_zero : ¬ν s = 0\n⊢ μ univ = ν (QuotientGroup.mk ⁻¹' univ ∩ s)\n\ncase neg.neTopV\nG : Type u_1\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MeasurableSpace G\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : TopologicalGroup G\ninst✝¹¹ : BorelSpace G\ninst✝¹⁰ : PolishSpace G\nΓ : Subgroup G\ninst✝⁹ : Countable ↥Γ\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst✝⁵ : ν.IsMulLeftInvariant\ninst✝⁴ : ν.IsMulRightInvariant\ninst✝³ : SigmaFinite ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : SigmaFinite μ\ninst✝ : IsFiniteMeasure μ\nhasFun : HasFundamentalDomain (↥Γ.op) G ν\ns : Set G\nh : ν s = μ univ\nfund_dom_s : IsFundamentalDomain (↥Γ.op) s ν\nfiniteCovol : μ univ < ⊤\nmeas_s_ne_zero : ¬ν s = 0\n⊢ μ univ ≠ ⊤","tactic":"apply IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set (fund_dom_s := fund_dom_s)\n    (meas_V := MeasurableSet.univ)","premises":[{"full_name":"MeasurableSet.univ","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[95,18],"def_end_pos":[95,36]},{"full_name":"MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set","def_path":"Mathlib/MeasureTheory/Measure/Haar/Quotient.lean","def_pos":[154,8],"def_end_pos":[154,88]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.tail_eq_tailD","start":[235,0],"end":[235,72],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nl : List α\n⊢ l.tail = l.tailD []","state_after":"no goals","tactic":"cases l <;> rfl","premises":[]}]}
{"url":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","commit":"","full_name":"LipschitzWith.edist_iterate_succ_le_geometric","start":[258,0],"end":[261,59],"file_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y : α\nr : ℝ≥0∞\nf : α → α\nhf : LipschitzWith K f\nx : α\nn : ℕ\n⊢ edist (f^[n] x) (f^[n + 1] x) ≤ edist x (f x) * ↑K ^ n","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y : α\nr : ℝ≥0∞\nf : α → α\nhf : LipschitzWith K f\nx : α\nn : ℕ\n⊢ edist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * edist x (f x)","tactic":"rw [iterate_succ, mul_comm]","premises":[{"full_name":"Function.iterate_succ","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[57,8],"def_end_pos":[57,20]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y : α\nr : ℝ≥0∞\nf : α → α\nhf : LipschitzWith K f\nx : α\nn : ℕ\n⊢ edist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * edist x (f x)","state_after":"no goals","tactic":"simpa only [ENNReal.coe_pow] using (hf.iterate n) x (f x)","premises":[{"full_name":"ENNReal.coe_pow","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[350,25],"def_end_pos":[350,32]},{"full_name":"LipschitzWith.iterate","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[254,18],"def_end_pos":[254,25]}]}]}
{"url":"Mathlib/GroupTheory/Perm/Support.lean","commit":"","full_name":"Equiv.Perm.pow_apply_eq_of_apply_apply_eq_self","start":[143,0],"end":[149,61],"file_path":"Mathlib/GroupTheory/Perm/Support.lean","tactics":[{"state_before":"α : Type u_1\nf g h✝ : Perm α\nx : α\nhffx : f (f x) = x\nn : ℕ\nh : (f ^ n) x = x\n⊢ (f ^ (n + 1)) x = f x","state_after":"no goals","tactic":"rw [pow_succ', mul_apply, h]","premises":[{"full_name":"Equiv.Perm.mul_apply","def_path":"Mathlib/GroupTheory/Perm/Basic.lean","def_pos":[65,8],"def_end_pos":[65,17]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"α : Type u_1\nf g h✝ : Perm α\nx : α\nhffx : f (f x) = x\nn : ℕ\nh : (f ^ n) x = f x\n⊢ (f ^ (n + 1)) x = x","state_after":"no goals","tactic":"rw [pow_succ', mul_apply, h, hffx]","premises":[{"full_name":"Equiv.Perm.mul_apply","def_path":"Mathlib/GroupTheory/Perm/Basic.lean","def_pos":[65,8],"def_end_pos":[65,17]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]}]}
{"url":"Mathlib/Data/Nat/Factorization/Root.lean","commit":"","full_name":"Nat.ceilRoot_ne_zero","start":[129,0],"end":[130,77],"file_path":"Mathlib/Data/Nat/Factorization/Root.lean","tactics":[{"state_before":"a b n : ℕ\n⊢ n.ceilRoot a ≠ 0 ↔ n ≠ 0 ∧ a ≠ 0","state_after":"no goals","tactic":"simp (config := { contextual := true }) [ceilRoot_def, not_imp_not, not_or]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Nat.ceilRoot_def","def_path":"Mathlib/Data/Nat/Factorization/Root.lean","def_pos":[114,6],"def_end_pos":[114,18]},{"full_name":"not_imp_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[290,8],"def_end_pos":[290,19]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]}]}
{"url":"Mathlib/Order/Irreducible.lean","commit":"","full_name":"supPrime_iff_not_isMin","start":[279,0],"end":[280,25],"file_path":"Mathlib/Order/Irreducible.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝ : LinearOrder α\na : α\n⊢ ∀ ⦃b c : α⦄, a ≤ b ⊔ c → a ≤ b ∨ a ≤ c","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Probability/Martingale/Upcrossing.lean","commit":"","full_name":"MeasureTheory.Adapted.integrable_upcrossingsBefore","start":[751,0],"end":[758,38],"file_path":"Mathlib/Probability/Martingale/Upcrossing.lean","tactics":[{"state_before":"Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Adapted ℱ f\nhab : a < b\n⊢ ∀ᵐ (ω : Ω) ∂μ, ‖↑(upcrossingsBefore a b f N ω)‖ ≤ ↑N","state_after":"case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω✝ : Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Adapted ℱ f\nhab : a < b\nω : Ω\n⊢ ‖↑(upcrossingsBefore a b f N ω)‖ ≤ ↑N","tactic":"filter_upwards with ω","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω✝ : Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Adapted ℱ f\nhab : a < b\nω : Ω\n⊢ ‖↑(upcrossingsBefore a b f N ω)‖ ≤ ↑N","state_after":"case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω✝ : Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Adapted ℱ f\nhab : a < b\nω : Ω\n⊢ upcrossingsBefore a b f N ω ≤ N","tactic":"rw [Real.norm_eq_abs, Nat.abs_cast, Nat.cast_le]","premises":[{"full_name":"Nat.abs_cast","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[84,8],"def_end_pos":[84,16]},{"full_name":"Nat.cast_le","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[78,8],"def_end_pos":[78,15]},{"full_name":"Real.norm_eq_abs","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1132,8],"def_end_pos":[1132,19]}]},{"state_before":"case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω✝ : Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Adapted ℱ f\nhab : a < b\nω : Ω\n⊢ upcrossingsBefore a b f N ω ≤ N","state_after":"no goals","tactic":"exact upcrossingsBefore_le _ _ hab","premises":[{"full_name":"MeasureTheory.upcrossingsBefore_le","def_path":"Mathlib/Probability/Martingale/Upcrossing.lean","def_pos":[435,8],"def_end_pos":[435,28]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Basic.lean","commit":"","full_name":"Finset.Ico_eq_empty_iff","start":[75,0],"end":[77,52],"file_path":"Mathlib/Order/Interval/Finset/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ Ico a b = ∅ ↔ ¬a < b","state_after":"no goals","tactic":"rw [← coe_eq_empty, coe_Ico, Set.Ico_eq_empty_iff]","premises":[{"full_name":"Finset.coe_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[315,8],"def_end_pos":[315,15]},{"full_name":"Finset.coe_eq_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[536,8],"def_end_pos":[536,20]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.Ico_eq_empty_iff","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[554,8],"def_end_pos":[554,24]}]}]}
{"url":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","commit":"","full_name":"NonUnitalSubring.mem_iSup_of_directed","start":[863,0],"end":[873,53],"file_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","tactics":[{"state_before":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i","state_after":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i","tactic":"refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"le_iSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[617,8],"def_end_pos":[617,15]}]},{"state_before":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i","state_after":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i","tactic":"let U : NonUnitalSubring R :=\n    NonUnitalSubring.mk' (⋃ i, (S i : Set R)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup)\n      (Subsemigroup.coe_iSup_of_directed hS) (AddSubgroup.coe_iSup_of_directed hS)","premises":[{"full_name":"AddSubgroup.coe_iSup_of_directed","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1033,2],"def_end_pos":[1033,13]},{"full_name":"NonUnitalSubring","def_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","def_pos":[132,10],"def_end_pos":[132,26]},{"full_name":"NonUnitalSubring.mk'","def_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","def_pos":[234,14],"def_end_pos":[234,17]},{"full_name":"NonUnitalSubring.toSubsemigroup","def_path":"Mathlib/RingTheory/NonUnitalSubring/Basic.lean","def_pos":[144,4],"def_end_pos":[144,18]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Subsemigroup.coe_iSup_of_directed","def_path":"Mathlib/Algebra/Group/Subsemigroup/Membership.lean","def_pos":[56,8],"def_end_pos":[56,28]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]},{"state_before":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i","state_after":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\n⊢ ⨆ i, S i ≤ U","tactic":"suffices ⨆ i, S i ≤ U by simpa [U] using @this x","premises":[{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]},{"state_before":"F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\n⊢ ⨆ i, S i ≤ U","state_after":"no goals","tactic":"exact iSup_le fun i x hx ↦ Set.mem_iUnion.2 ⟨i, hx⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"iSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[661,8],"def_end_pos":[661,15]}]}]}
{"url":"Mathlib/Data/List/Sort.lean","commit":"","full_name":"List.Sorted.le_head!","start":[80,0],"end":[85,59],"file_path":"Mathlib/Data/List/Sort.lean","tactics":[{"state_before":"α : Type u\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝¹ : Inhabited α\ninst✝ : Preorder α\na : α\nl : List α\nh : Sorted (fun x x_1 => x > x_1) l\nha : a ∈ l\n⊢ a ≤ l.head!","state_after":"α : Type u\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝¹ : Inhabited α\ninst✝ : Preorder α\na : α\nl : List α\nh : Sorted (fun x x_1 => x > x_1) (l.head! :: l.tail)\nha : a ∈ l.head! :: l.tail\n⊢ a ≤ l.head!","tactic":"rw [← List.cons_head!_tail (List.ne_nil_of_mem ha)] at h ha","premises":[{"full_name":"List.cons_head!_tail","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[556,8],"def_end_pos":[556,23]},{"full_name":"List.ne_nil_of_mem","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[330,8],"def_end_pos":[330,21]}]},{"state_before":"α : Type u\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝¹ : Inhabited α\ninst✝ : Preorder α\na : α\nl : List α\nh : Sorted (fun x x_1 => x > x_1) (l.head! :: l.tail)\nha : a ∈ l.head! :: l.tail\n⊢ a ≤ l.head!","state_after":"case head\nα : Type u\nr : α → α → Prop\na : α\nl✝ : List α\ninst✝¹ : Inhabited α\ninst✝ : Preorder α\nl : List α\nh : Sorted (fun x x_1 => x > x_1) (l.head! :: l.tail)\n⊢ l.head! ≤ l.head!\n\ncase tail\nα : Type u\nr : α → α → Prop\na✝¹ : α\nl✝ : List α\ninst✝¹ : Inhabited α\ninst✝ : Preorder α\na : α\nl : List α\nh : Sorted (fun x x_1 => x > x_1) (l.head! :: l.tail)\na✝ : Mem a l.tail\n⊢ a ≤ l.head!","tactic":"cases ha","premises":[]}]}
{"url":"Mathlib/Data/Real/Hyperreal.lean","commit":"","full_name":"Hyperreal.st_neg","start":[528,0],"end":[531,62],"file_path":"Mathlib/Data/Real/Hyperreal.lean","tactics":[{"state_before":"x : ℝ*\nh : x.Infinite\n⊢ (-x).st = -x.st","state_after":"no goals","tactic":"rw [h.st_eq, (infinite_neg.2 h).st_eq, neg_zero]","premises":[{"full_name":"Hyperreal.Infinite.st_eq","def_path":"Mathlib/Data/Real/Hyperreal.lean","def_pos":[244,8],"def_end_pos":[244,22]},{"full_name":"Hyperreal.infinite_neg","def_path":"Mathlib/Data/Real/Hyperreal.lean","def_pos":[389,16],"def_end_pos":[389,28]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"neg_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[950,2],"def_end_pos":[950,13]}]}]}
{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.get_tails","start":[568,0],"end":[574,53],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\n⊢ ∀ (n : ℕ) (s : Stream' α), s.tails.get n = drop n s.tail","state_after":"α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\n⊢ ∀ (s : Stream' α), s.tails.get n = drop n s.tail","tactic":"intro n","premises":[]},{"state_before":"α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\n⊢ ∀ (s : Stream' α), s.tails.get n = drop n s.tail","state_after":"case zero\nα : Type u\nβ : Type v\nδ : Type w\n⊢ ∀ (s : Stream' α), s.tails.get 0 = drop 0 s.tail\n\ncase succ\nα : Type u\nβ : Type v\nδ : Type w\nn' : ℕ\nih : ∀ (s : Stream' α), s.tails.get n' = drop n' s.tail\n⊢ ∀ (s : Stream' α), s.tails.get (n' + 1) = drop (n' + 1) s.tail","tactic":"induction' n with n' ih","premises":[]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.Sublist.eraseP","start":[546,0],"end":[553,24],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\n⊢ eraseP p l₁ <+ eraseP p (a :: l₂✝)","state_after":"case pos\nα✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\nh : p a = true\n⊢ eraseP p l₁ <+ l₂✝\n\ncase neg\nα✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\nh : ¬p a = true\n⊢ eraseP p l₁ <+ a :: eraseP p l₂✝","tactic":"by_cases h : p a <;> simp [h]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nα✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\nh : p a = true\n⊢ eraseP p l₁ <+ l₂✝\n\ncase neg\nα✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\nh : ¬p a = true\n⊢ eraseP p l₁ <+ a :: eraseP p l₂✝","state_after":"no goals","tactic":"exacts [s.eraseP.trans (eraseP_sublist _), s.eraseP.cons _]","premises":[{"full_name":"List.Sublist.cons","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[147,4],"def_end_pos":[147,8]},{"full_name":"List.Sublist.trans","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[91,8],"def_end_pos":[91,21]},{"full_name":"List.eraseP_sublist","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[539,8],"def_end_pos":[539,22]}]},{"state_before":"α✝ : Type u_1\nl₁ l₂ : List α✝\np : α✝ → Bool\nl₁✝ l₂✝ : List α✝\na : α✝\ns : l₁✝ <+ l₂✝\n⊢ eraseP p (a :: l₁✝) <+ eraseP p (a :: l₂✝)","state_after":"case pos\nα✝ : Type u_1\nl₁ l₂ : List α✝\np : α✝ → Bool\nl₁✝ l₂✝ : List α✝\na : α✝\ns : l₁✝ <+ l₂✝\nh : p a = true\n⊢ l₁✝ <+ l₂✝\n\ncase neg\nα✝ : Type u_1\nl₁ l₂ : List α✝\np : α✝ → Bool\nl₁✝ l₂✝ : List α✝\na : α✝\ns : l₁✝ <+ l₂✝\nh : ¬p a = true\n⊢ eraseP p l₁✝ <+ eraseP p l₂✝","tactic":"by_cases h : p a <;> simp [h]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nα✝ : Type u_1\nl₁ l₂ : List α✝\np : α✝ → Bool\nl₁✝ l₂✝ : List α✝\na : α✝\ns : l₁✝ <+ l₂✝\nh : p a = true\n⊢ l₁✝ <+ l₂✝\n\ncase neg\nα✝ : Type u_1\nl₁ l₂ : List α✝\np : α✝ → Bool\nl₁✝ l₂✝ : List α✝\na : α✝\ns : l₁✝ <+ l₂✝\nh : ¬p a = true\n⊢ eraseP p l₁✝ <+ eraseP p l₂✝","state_after":"no goals","tactic":"exacts [s, s.eraseP]","premises":[]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"IsOpen.vadd_left","start":[1051,0],"end":[1054,43],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (s • t)","state_after":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (⋃ a ∈ s, a • t)","tactic":"rw [← iUnion_smul_set]","premises":[{"full_name":"Set.iUnion_smul_set","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[222,8],"def_end_pos":[222,23]}]},{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (⋃ a ∈ s, a • t)","state_after":"no goals","tactic":"exact isOpen_biUnion fun a _ => ht.smul _","premises":[{"full_name":"IsOpen.smul","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[220,8],"def_end_pos":[220,19]},{"full_name":"isOpen_biUnion","def_path":"Mathlib/Topology/Basic.lean","def_pos":[102,8],"def_end_pos":[102,22]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","commit":"","full_name":"WeierstrassCurve.leadingCoeff_Φ","start":[440,0],"end":[442,41],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : Nontrivial R\nn : ℤ\n⊢ (W.Φ n).leadingCoeff = 1","state_after":"no goals","tactic":"rw [leadingCoeff, natDegree_Φ, coeff_Φ]","premises":[{"full_name":"Polynomial.leadingCoeff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[65,4],"def_end_pos":[65,16]},{"full_name":"WeierstrassCurve.coeff_Φ","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","def_pos":[425,6],"def_end_pos":[425,13]},{"full_name":"WeierstrassCurve.natDegree_Φ","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean","def_pos":[434,6],"def_end_pos":[434,17]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Content.lean","commit":"","full_name":"Polynomial.natDegree_primPart","start":[236,0],"end":[241,95],"file_path":"Mathlib/RingTheory/Polynomial/Content.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ p.primPart.natDegree = p.natDegree","state_after":"case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : C p.content = 0\n⊢ p.primPart.natDegree = p.natDegree\n\ncase neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬C p.content = 0\n⊢ p.primPart.natDegree = p.natDegree","tactic":"by_cases h : C p.content = 0","premises":[{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.content","def_path":"Mathlib/RingTheory/Polynomial/Content.lean","def_pos":[70,4],"def_end_pos":[70,11]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬C p.content = 0\n⊢ p.primPart.natDegree = p.natDegree","state_after":"no goals","tactic":"conv_rhs =>\n    rw [p.eq_C_content_mul_primPart, natDegree_mul h p.primPart_ne_zero, natDegree_C, zero_add]","premises":[{"full_name":"Polynomial.eq_C_content_mul_primPart","def_path":"Mathlib/RingTheory/Polynomial/Content.lean","def_pos":[215,8],"def_end_pos":[215,33]},{"full_name":"Polynomial.natDegree_C","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[222,8],"def_end_pos":[222,19]},{"full_name":"Polynomial.natDegree_mul","def_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","def_pos":[115,8],"def_end_pos":[115,21]},{"full_name":"Polynomial.primPart_ne_zero","def_path":"Mathlib/RingTheory/Polynomial/Content.lean","def_pos":[233,8],"def_end_pos":[233,24]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
{"url":"Mathlib/Logic/Basic.lean","commit":"","full_name":"xor_not_not","start":[252,0],"end":[252,84],"file_path":"Mathlib/Logic/Basic.lean","tactics":[{"state_before":"a b : Prop\n⊢ Xor' (¬a) ¬b ↔ Xor' a b","state_after":"no goals","tactic":"simp [Xor', or_comm, and_comm]","premises":[{"full_name":"Xor'","def_path":"Mathlib/Init/Logic.lean","def_pos":[73,4],"def_end_pos":[73,8]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"or_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[823,8],"def_end_pos":[823,15]}]}]}
{"url":"Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean","commit":"","full_name":"AlgebraicGeometry.PresheafedSpace.colimitCocone_ι_app_c","start":[145,0],"end":[165,16],"file_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean","tactics":[{"state_before":"J : Type u'\ninst✝³ : Category.{v', u'} J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasColimitsOfShape J TopCat\ninst✝ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\nF : J ⥤ PresheafedSpace C\nj j' : J\nf : j ⟶ j'\n⊢ F.map f ≫\n      (fun j => { base := colimit.ι (F ⋙ forget C) j, c := limit.π (pushforwardDiagramToColimit F).leftOp (op j) }) j' =\n    (fun j => { base := colimit.ι (F ⋙ forget C) j, c := limit.π (pushforwardDiagramToColimit F).leftOp (op j) }) j ≫\n      ((Functor.const J).obj (colimit F)).map f","state_after":"case w\nJ : Type u'\ninst✝³ : Category.{v', u'} J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasColimitsOfShape J TopCat\ninst✝ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\nF : J ⥤ PresheafedSpace C\nj j' : J\nf : j ⟶ j'\n⊢ (F.map f ≫\n        (fun j => { base := colimit.ι (F ⋙ forget C) j, c := limit.π (pushforwardDiagramToColimit F).leftOp (op j) })\n          j').base =\n    ((fun j => { base := colimit.ι (F ⋙ forget C) j, c := limit.π (pushforwardDiagramToColimit F).leftOp (op j) }) j ≫\n        ((Functor.const J).obj (colimit F)).map f).base\n\ncase h\nJ : Type u'\ninst✝³ : Category.{v', u'} J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasColimitsOfShape J TopCat\ninst✝ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\nF : J ⥤ PresheafedSpace C\nj j' : J\nf : j ⟶ j'\n⊢ (F.map f ≫\n          (fun j => { base := colimit.ι (F ⋙ forget C) j, c := limit.π (pushforwardDiagramToColimit F).leftOp (op j) })\n            j').c ≫\n      whiskerRight (eqToHom ⋯) (F.obj j).presheaf =\n    ((fun j => { base := colimit.ι (F ⋙ forget C) j, c := limit.π (pushforwardDiagramToColimit F).leftOp (op j) }) j ≫\n        ((Functor.const J).obj (colimit F)).map f).c","tactic":"ext1","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","commit":"","full_name":"MeasureTheory.SimpleFunc.lintegral_restrict_iUnion_of_directed","start":[901,0],"end":[908,80],"file_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_5\ninst✝ : Countable ι\nf : α →ₛ ℝ≥0∞\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nμ : Measure α\n⊢ f.lintegral (μ.restrict (⋃ i, s i)) = ⨆ i, f.lintegral (μ.restrict (s i))","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_5\ninst✝ : Countable ι\nf : α →ₛ ℝ≥0∞\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nμ : Measure α\n⊢ ∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (↑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (↑f ⁻¹' {x})","tactic":"simp only [lintegral, Measure.restrict_iUnion_apply_eq_iSup hd (measurableSet_preimage ..),\n    ENNReal.mul_iSup]","premises":[{"full_name":"ENNReal.mul_iSup","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[571,8],"def_end_pos":[571,16]},{"full_name":"MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[271,8],"def_end_pos":[271,37]},{"full_name":"MeasureTheory.SimpleFunc.lintegral","def_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","def_pos":[795,4],"def_end_pos":[795,13]},{"full_name":"MeasureTheory.SimpleFunc.measurableSet_preimage","def_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","def_pos":[162,8],"def_end_pos":[162,30]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_5\ninst✝ : Countable ι\nf : α →ₛ ℝ≥0∞\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nμ : Measure α\n⊢ ∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (↑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (↑f ⁻¹' {x})","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_5\ninst✝ : Countable ι\nf : α →ₛ ℝ≥0∞\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nμ : Measure α\ni j k : ι\nx✝ : (fun x x_1 => x ⊆ x_1) (s i) (s k) ∧ (fun x x_1 => x ⊆ x_1) (s j) (s k)\nhik : (fun x x_1 => x ⊆ x_1) (s i) (s k)\nhjk : (fun x x_1 => x ⊆ x_1) (s j) (s k)\na : ℝ≥0∞\n⊢ a * (μ.restrict (s i)) (↑f ⁻¹' {a}) ≤ a * (μ.restrict (s k)) (↑f ⁻¹' {a}) ∧\n    a * (μ.restrict (s j)) (↑f ⁻¹' {a}) ≤ a * (μ.restrict (s k)) (↑f ⁻¹' {a})","tactic":"refine finsetSum_iSup fun i j ↦ (hd i j).imp fun k ⟨hik, hjk⟩ ↦ fun a ↦ ?_","premises":[{"full_name":"ENNReal.finsetSum_iSup","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[552,8],"def_end_pos":[552,22]},{"full_name":"Exists.imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[193,8],"def_end_pos":[193,18]}]}]}
{"url":"Mathlib/Probability/Kernel/Composition.lean","commit":"","full_name":"ProbabilityTheory.Kernel.prod_const","start":[1063,0],"end":[1066,56],"file_path":"Mathlib/Probability/Kernel/Composition.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nμ : Measure β\ninst✝¹ : SFinite μ\nν : Measure γ\ninst✝ : SFinite ν\n⊢ const α μ ×ₖ const α ν = const α (μ.prod ν)","state_after":"case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nμ : Measure β\ninst✝¹ : SFinite μ\nν : Measure γ\ninst✝ : SFinite ν\nx : α\ns✝ : Set (β × γ)\na✝ : MeasurableSet s✝\n⊢ ((const α μ ×ₖ const α ν) x) s✝ = ((const α (μ.prod ν)) x) s✝","tactic":"ext x","premises":[]},{"state_before":"case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nμ : Measure β\ninst✝¹ : SFinite μ\nν : Measure γ\ninst✝ : SFinite ν\nx : α\ns✝ : Set (β × γ)\na✝ : MeasurableSet s✝\n⊢ ((const α μ ×ₖ const α ν) x) s✝ = ((const α (μ.prod ν)) x) s✝","state_after":"no goals","tactic":"rw [const_apply, prod_apply, const_apply, const_apply]","premises":[{"full_name":"ProbabilityTheory.Kernel.const_apply","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[439,8],"def_end_pos":[439,19]},{"full_name":"ProbabilityTheory.Kernel.prod_apply","def_path":"Mathlib/Probability/Kernel/Composition.lean","def_pos":[1056,6],"def_end_pos":[1056,16]}]}]}
{"url":"Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean","commit":"","full_name":"MeasurableSpace.exists_measurableSet_of_ne","start":[143,0],"end":[146,29],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : SeparatesPoints α\nx y : α\nh : x ≠ y\n⊢ ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : SeparatesPoints α\nx y : α\nh : ∀ (s : Set α), MeasurableSet s → x ∈ s → y ∈ s\n⊢ x = y","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : SeparatesPoints α\nx y : α\nh : ∀ (s : Set α), MeasurableSet s → x ∈ s → y ∈ s\n⊢ x = y","state_after":"no goals","tactic":"exact separatesPoints_def h","premises":[{"full_name":"MeasurableSpace.separatesPoints_def","def_path":"Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean","def_pos":[140,8],"def_end_pos":[140,27]}]}]}
{"url":"Mathlib/RingTheory/Multiplicity.lean","commit":"","full_name":"multiplicity.unique","start":[113,0],"end":[118,39],"file_path":"Mathlib/RingTheory/Multiplicity.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\n⊢ multiplicity a b ≤ ↑k","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\nthis : Finite a b\n⊢ multiplicity a b ≤ ↑k","tactic":"have : Finite a b := ⟨k, hsucc⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"multiplicity.Finite","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[44,7],"def_end_pos":[44,13]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\nthis : Finite a b\n⊢ multiplicity a b ≤ ↑k","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\nthis : Finite a b\n⊢ ∃ (h : (multiplicity a b).Dom), (multiplicity a b).get h ≤ k","tactic":"rw [PartENat.le_coe_iff]","premises":[{"full_name":"PartENat.le_coe_iff","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[289,8],"def_end_pos":[289,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\nthis : Finite a b\n⊢ ∃ (h : (multiplicity a b).Dom), (multiplicity a b).get h ≤ k","state_after":"no goals","tactic":"exact ⟨this, Nat.find_min' _ hsucc⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Nat.find_min'","def_path":"Mathlib/Data/Nat/Find.lean","def_pos":[71,18],"def_end_pos":[71,27]}]}]}
{"url":"Mathlib/Algebra/Ring/Idempotents.lean","commit":"","full_name":"IsIdempotentElem.one_sub","start":[58,0],"end":[59,85],"file_path":"Mathlib/Algebra/Ring/Idempotents.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nS : Type u_3\nM₀ : Type u_4\nM₁ : Type u_5\nR : Type u_6\nG : Type u_7\nG₀ : Type u_8\ninst✝⁷ : Mul M\ninst✝⁶ : Monoid N\ninst✝⁵ : Semigroup S\ninst✝⁴ : MulZeroClass M₀\ninst✝³ : MulOneClass M₁\ninst✝² : NonAssocRing R\ninst✝¹ : Group G\ninst✝ : CancelMonoidWithZero G₀\np : R\nh : IsIdempotentElem p\n⊢ IsIdempotentElem (1 - p)","state_after":"no goals","tactic":"rw [IsIdempotentElem, mul_sub, mul_one, sub_mul, one_mul, h.eq, sub_self, sub_zero]","premises":[{"full_name":"IsIdempotentElem","def_path":"Mathlib/Algebra/Ring/Idempotents.lean","def_pos":[37,4],"def_end_pos":[37,20]},{"full_name":"IsIdempotentElem.eq","def_path":"Mathlib/Algebra/Ring/Idempotents.lean","def_pos":[45,8],"def_end_pos":[45,10]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[695,29],"def_end_pos":[695,37]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]}]}
{"url":"Mathlib/RingTheory/Localization/LocalizationLocalization.lean","commit":"","full_name":"IsLocalization.localization_localization_map_units","start":[63,0],"end":[67,92],"file_path":"Mathlib/RingTheory/Localization/LocalizationLocalization.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\n⊢ IsUnit ((algebraMap R T) ↑y)","state_after":"case intro.intro\nR : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\ny' : ↥N\nz : ↥M\neq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z\n⊢ IsUnit ((algebraMap R T) ↑y)","tactic":"obtain ⟨y', z, eq⟩ := mem_localizationLocalizationSubmodule.mp y.prop","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsLocalization.mem_localizationLocalizationSubmodule","def_path":"Mathlib/RingTheory/Localization/LocalizationLocalization.lean","def_pos":[50,8],"def_end_pos":[50,45]},{"full_name":"Subtype.prop","def_path":"Mathlib/Data/Subtype.lean","def_pos":[37,8],"def_end_pos":[37,12]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\ny' : ↥N\nz : ↥M\neq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z\n⊢ IsUnit ((algebraMap R T) ↑y)","state_after":"case intro.intro\nR : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\ny' : ↥N\nz : ↥M\neq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z\n⊢ IsUnit ((algebraMap S T) ↑y') ∧ IsUnit ((algebraMap S T) ((algebraMap R S) ↑z))","tactic":"rw [IsScalarTower.algebraMap_apply R S T, eq, RingHom.map_mul, IsUnit.mul_iff]","premises":[{"full_name":"IsScalarTower.algebraMap_apply","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[122,8],"def_end_pos":[122,24]},{"full_name":"IsUnit.mul_iff","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[638,8],"def_end_pos":[638,15]},{"full_name":"RingHom.map_mul","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[484,18],"def_end_pos":[484,25]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\ny' : ↥N\nz : ↥M\neq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z\n⊢ IsUnit ((algebraMap S T) ↑y') ∧ IsUnit ((algebraMap S T) ((algebraMap R S) ↑z))","state_after":"no goals","tactic":"exact ⟨IsLocalization.map_units T y', (IsLocalization.map_units _ z).map (algebraMap S T)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"IsLocalization.map_units","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"IsUnit.map","def_path":"Mathlib/Algebra/Group/Units/Hom.lean","def_pos":[162,8],"def_end_pos":[162,11]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]}]}
{"url":"Mathlib/Algebra/Algebra/Basic.lean","commit":"","full_name":"Algebra.mul_sub_algebraMap_commutes","start":[121,0],"end":[122,99],"file_path":"Mathlib/Algebra/Algebra/Basic.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nA : Type w\nB : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ x * (x - (algebraMap R A) r) = (x - (algebraMap R A) r) * x","state_after":"no goals","tactic":"rw [mul_sub, ← commutes, sub_mul]","premises":[{"full_name":"Algebra.commutes","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[282,8],"def_end_pos":[282,16]}]}]}
{"url":"Mathlib/Algebra/Order/Module/Defs.lean","commit":"","full_name":"smul_le_smul_iff_of_neg_left","start":[848,0],"end":[851,56],"file_path":"Mathlib/Algebra/Order/Module/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁴ : OrderedRing α\ninst✝³ : OrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : a < 0\n⊢ a • b₁ ≤ a • b₂ ↔ b₂ ≤ b₁","state_after":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁴ : OrderedRing α\ninst✝³ : OrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : a < 0\n⊢ -a • b₂ ≤ -a • b₁ ↔ b₂ ≤ b₁","tactic":"rw [← neg_neg a, neg_smul, neg_smul (-a), neg_le_neg_iff]","premises":[{"full_name":"neg_le_neg_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[216,2],"def_end_pos":[216,13]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁴ : OrderedRing α\ninst✝³ : OrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : a < 0\n⊢ -a • b₂ ≤ -a • b₁ ↔ b₂ ≤ b₁","state_after":"no goals","tactic":"exact smul_le_smul_iff_of_pos_left (neg_pos_of_neg ha)","premises":[{"full_name":"smul_le_smul_iff_of_pos_left","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[288,6],"def_end_pos":[288,34]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","commit":"","full_name":"MeasureTheory.Measure.haarScalarFactor_pos_of_isHaarMeasure","start":[383,0],"end":[388,80],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","tactics":[{"state_before":"G : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nH : μ'.haarScalarFactor μ = 0\n⊢ False","state_after":"no goals","tactic":"simpa [H] using haarScalarFactor_eq_mul μ' μ μ'","premises":[{"full_name":"MeasureTheory.Measure.haarScalarFactor_eq_mul","def_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","def_pos":[362,6],"def_end_pos":[362,29]}]}]}
{"url":"Mathlib/CategoryTheory/Functor/Category.lean","commit":"","full_name":"CategoryTheory.NatTrans.hcomp_app","start":[101,0],"end":[107,22],"file_path":"Mathlib/CategoryTheory/Functor/Category.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF G H✝ I✝ : C ⥤ D\nH I : D ⥤ E\nα : F ⟶ G\nβ : H ⟶ I\nX Y : C\nf : X ⟶ Y\n⊢ (F ⋙ H).map f ≫ (fun X => β.app (F.obj X) ≫ I.map (α.app X)) Y =\n    (fun X => β.app (F.obj X) ≫ I.map (α.app X)) X ≫ (G ⋙ I).map f","state_after":"no goals","tactic":"rw [Functor.comp_map, Functor.comp_map, ← assoc, naturality, assoc, ← map_comp I, naturality,\n      map_comp, assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.comp_map","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[109,8],"def_end_pos":[109,16]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]}]}]}
{"url":"Mathlib/RingTheory/MvPolynomial/Ideal.lean","commit":"","full_name":"MvPolynomial.mem_ideal_span_monomial_image_iff_dvd","start":[36,0],"end":[40,94],"file_path":"Mathlib/RingTheory/MvPolynomial/Ideal.lean","tactics":[{"state_before":"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\n⊢ x ∈ Ideal.span ((fun s => (monomial s) 1) '' s) ↔\n    ∀ xi ∈ x.support, ∃ si ∈ s, (monomial si) 1 ∣ (monomial xi) (coeff xi x)","state_after":"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\nxi : σ →₀ ℕ\nhxi : xi ∈ x.support\n⊢ (∃ si ∈ s, si ≤ xi) ↔ ∃ si ∈ s, (monomial si) 1 ∣ (monomial xi) (coeff xi x)","tactic":"refine mem_ideal_span_monomial_image.trans (forall₂_congr fun xi hxi => ?_)","premises":[{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"MvPolynomial.mem_ideal_span_monomial_image","def_path":"Mathlib/RingTheory/MvPolynomial/Ideal.lean","def_pos":[30,8],"def_end_pos":[30,37]},{"full_name":"forall₂_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[215,8],"def_end_pos":[215,21]}]},{"state_before":"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\nxi : σ →₀ ℕ\nhxi : xi ∈ x.support\n⊢ (∃ si ∈ s, si ≤ xi) ↔ ∃ si ∈ s, (monomial si) 1 ∣ (monomial xi) (coeff xi x)","state_after":"no goals","tactic":"simp_rw [monomial_dvd_monomial, one_dvd, and_true_iff, mem_support_iff.mp hxi, false_or_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MvPolynomial.mem_support_iff","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[515,8],"def_end_pos":[515,23]},{"full_name":"MvPolynomial.monomial_dvd_monomial","def_path":"Mathlib/Algebra/MvPolynomial/Division.lean","def_pos":[184,8],"def_end_pos":[184,29]},{"full_name":"and_true_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[93,8],"def_end_pos":[93,20]},{"full_name":"false_or_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[100,8],"def_end_pos":[100,20]},{"full_name":"one_dvd","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[130,8],"def_end_pos":[130,15]}]}]}
{"url":"Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean","commit":"","full_name":"DedekindDomain.FiniteIntegralAdeles.Coe.ringHom_apply","start":[112,0],"end":[123,57],"file_path":"Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean","tactics":[{"state_before":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx y : R_hat R K\n⊢ { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun (x * y) =\n    { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun x * { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun y","state_after":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx y : R_hat R K\np : HeightOneSpectrum R\n⊢ { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun (x * y) p =\n    ({ toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun x * { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun y) p","tactic":"refine funext fun p => ?_","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx y : R_hat R K\np : HeightOneSpectrum R\n⊢ { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun (x * y) p =\n    ({ toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun x * { toFun := fun x v => ↑(x v), map_one' := ⋯ }.toFun y) p","state_after":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx y : R_hat R K\np : HeightOneSpectrum R\n⊢ ↑((x * y) p) = ((fun v => ↑(x v)) * fun v => ↑(y v)) p","tactic":"simp only [Pi.mul_apply, Subring.coe_mul]","premises":[{"full_name":"Pi.mul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[82,8],"def_end_pos":[82,17]},{"full_name":"Subring.coe_mul","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[342,8],"def_end_pos":[342,15]}]},{"state_before":"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx y : R_hat R K\np : HeightOneSpectrum R\n⊢ ↑((x * y) p) = ((fun v => ↑(x v)) * fun v => ↑(y v)) p","state_after":"no goals","tactic":"erw [Pi.mul_apply, Pi.mul_apply, Subring.coe_mul]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Pi.mul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[82,8],"def_end_pos":[82,17]},{"full_name":"Subring.coe_mul","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[342,8],"def_end_pos":[342,15]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.replaceF_cons_of_none","start":[805,0],"end":[806,73],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\na : α\nl : List α\np : α → Option α\nh : p a = none\n⊢ replaceF p (a :: l) = a :: replaceF p l","state_after":"no goals","tactic":"simp [replaceF_cons, h]","premises":[{"full_name":"List.replaceF_cons","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[796,8],"def_end_pos":[796,21]}]}]}
{"url":"Mathlib/Topology/Algebra/Polynomial.lean","commit":"","full_name":"Polynomial.coeff_le_of_roots_le","start":[151,0],"end":[176,32],"file_path":"Mathlib/Topology/Algebra/Polynomial.lean","tactics":[{"state_before":"α : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\n⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)","state_after":"case inl\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : B < 0\n⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)\n\ncase inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\n⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)","tactic":"obtain hB | hB := lt_or_le B 0","premises":[{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"case inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\n⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)","state_after":"case inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\n⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)","tactic":"rw [← h1.natDegree_map f]","premises":[{"full_name":"Polynomial.Monic.natDegree_map","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[275,8],"def_end_pos":[275,27]}]},{"state_before":"case inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\n⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)","state_after":"case inr.inl\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : (map f p).natDegree < i\n⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)\n\ncase inr.inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)","tactic":"obtain hi | hi := lt_or_le (map f p).natDegree i","premises":[{"full_name":"Polynomial.map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[597,4],"def_end_pos":[597,7]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"case inr.inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)","state_after":"case inr.inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ‖(map f p).roots.esymm ((map f p).natDegree - i)‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)","tactic":"rw [coeff_eq_esymm_roots_of_splits ((splits_id_iff_splits f).2 h2) hi, (h1.map _).leadingCoeff,\n    one_mul, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"NormOneClass.norm_one","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[155,2],"def_end_pos":[155,10]},{"full_name":"Polynomial.Monic.leadingCoeff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[82,8],"def_end_pos":[82,26]},{"full_name":"Polynomial.Monic.map","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[58,8],"def_end_pos":[58,17]},{"full_name":"Polynomial.coeff_eq_esymm_roots_of_splits","def_path":"Mathlib/RingTheory/Polynomial/Vieta.lean","def_pos":[133,8],"def_end_pos":[133,56]},{"full_name":"Polynomial.splits_id_iff_splits","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[138,8],"def_end_pos":[138,28]},{"full_name":"norm_mul","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[661,8],"def_end_pos":[661,16]},{"full_name":"norm_neg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[375,29],"def_end_pos":[375,37]},{"full_name":"norm_pow","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[694,8],"def_end_pos":[694,16]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]}]},{"state_before":"case inr.inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ‖(map f p).roots.esymm ((map f p).natDegree - i)‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)","state_after":"case inr.inr\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ card (Multiset.map (fun x => ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree - i) (map f p).roots))) •\n      ?m.46254 ≤\n    B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)\n\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ℝ\n\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ∀ r ∈ Multiset.map (fun x => ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree - i) (map f p).roots)),\n    r ≤ ?m.46254","tactic":"apply ((norm_multiset_sum_le _).trans <| sum_le_card_nsmul _ _ fun r hr => _).trans","premises":[{"full_name":"Multiset.sum_le_card_nsmul","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean","def_pos":[36,14],"def_end_pos":[36,31]},{"full_name":"norm_multiset_sum_le","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[950,8],"def_end_pos":[950,28]}]},{"state_before":"α : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ∀ r ∈ Multiset.map (fun x => ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree - i) (map f p).roots)),\n    r ≤ B ^ ((map f p).natDegree - i)","state_after":"α : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\nr : ℝ\nhr : r ∈ Multiset.map (fun x => ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree - i) (map f p).roots))\n⊢ r ≤ B ^ ((map f p).natDegree - i)","tactic":"intro r hr","premises":[]},{"state_before":"α : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\nr : ℝ\nhr : r ∈ Multiset.map (fun x => ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree - i) (map f p).roots))\n⊢ r ≤ B ^ ((map f p).natDegree - i)","state_after":"α : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\nr : ℝ\nhr : ∃ a, (∃ a_1 ∈ powersetCard ((map f p).natDegree - i) (map f p).roots, a_1.prod = a) ∧ ‖a‖ = r\n⊢ r ≤ B ^ ((map f p).natDegree - i)","tactic":"simp_rw [Multiset.mem_map] at hr","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]}]},{"state_before":"α : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\nr : ℝ\nhr : ∃ a, (∃ a_1 ∈ powersetCard ((map f p).natDegree - i) (map f p).roots, a_1.prod = a) ∧ ‖a‖ = r\n⊢ r ≤ B ^ ((map f p).natDegree - i)","state_after":"case intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ∈ powersetCard ((map f p).natDegree - i) (map f p).roots\n⊢ ‖s.prod‖ ≤ B ^ ((map f p).natDegree - i)","tactic":"obtain ⟨_, ⟨s, hs, rfl⟩, rfl⟩ := hr","premises":[]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ∈ powersetCard ((map f p).natDegree - i) (map f p).roots\n⊢ ‖s.prod‖ ≤ B ^ ((map f p).natDegree - i)","state_after":"case intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\n⊢ ‖s.prod‖ ≤ B ^ ((map f p).natDegree - i)","tactic":"rw [mem_powersetCard] at hs","premises":[{"full_name":"Multiset.mem_powersetCard","def_path":"Mathlib/Data/Multiset/Powerset.lean","def_pos":[224,8],"def_end_pos":[224,24]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\n⊢ ‖s.prod‖ ≤ B ^ ((map f p).natDegree - i)","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\n⊢ ‖s.prod‖ ≤ ↑B ^ ((map f p).natDegree - i)","tactic":"lift B to ℝ≥0 using hB","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\n⊢ ‖s.prod‖ ≤ ↑B ^ ((map f p).natDegree - i)","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\n⊢ (Multiset.map (⇑↑nnnormHom) s).prod ≤ B ^ ((map f p).natDegree - i)","tactic":"rw [← coe_nnnorm, ← NNReal.coe_pow, NNReal.coe_le_coe, ← nnnormHom_apply, ← MonoidHom.coe_coe,\n    MonoidHom.map_multiset_prod]","premises":[{"full_name":"MonoidHom.coe_coe","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[375,8],"def_end_pos":[375,25]},{"full_name":"MonoidHom.map_multiset_prod","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[198,16],"def_end_pos":[198,50]},{"full_name":"NNReal.coe_le_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[325,25],"def_end_pos":[325,35]},{"full_name":"NNReal.coe_pow","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[258,8],"def_end_pos":[258,15]},{"full_name":"coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[615,40],"def_end_pos":[615,50]},{"full_name":"nnnormHom_apply","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[686,2],"def_end_pos":[686,7]}]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\n⊢ (Multiset.map (⇑↑nnnormHom) s).prod ≤ B ^ ((map f p).natDegree - i)","state_after":"case intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\nx : ℝ≥0\nhx : x ∈ Multiset.map (⇑↑nnnormHom) s\n⊢ x ≤ B","tactic":"refine (prod_le_pow_card _ B fun x hx => ?_).trans_eq (by rw [card_map, hs.2])","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Multiset.card_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1085,8],"def_end_pos":[1085,16]},{"full_name":"Multiset.prod_le_pow_card","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean","def_pos":[37,6],"def_end_pos":[37,22]}]},{"state_before":"case intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\nx : ℝ≥0\nhx : x ∈ Multiset.map (⇑↑nnnormHom) s\n⊢ x ≤ B","state_after":"case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\nz : K\nhz : z ∈ s\nhx : ↑nnnormHom z ∈ Multiset.map (⇑↑nnnormHom) s\n⊢ ↑nnnormHom z ≤ B","tactic":"obtain ⟨z, hz, rfl⟩ := Multiset.mem_map.1 hx","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\nz : K\nhz : z ∈ s\nhx : ↑nnnormHom z ∈ Multiset.map (⇑↑nnnormHom) s\n⊢ ↑nnnormHom z ≤ B","state_after":"no goals","tactic":"exact h3 z (mem_of_le hs.1 hz)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Multiset.mem_of_le","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[468,8],"def_end_pos":[468,17]}]}]}
{"url":"Mathlib/Algebra/Order/ToIntervalMod.lean","commit":"","full_name":"toIcoMod_zsmul_add'","start":[361,0],"end":[364,46],"file_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b","state_after":"no goals","tactic":"rw [add_comm, toIcoMod_add_zsmul', add_comm]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"toIcoMod_add_zsmul'","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[343,8],"def_end_pos":[343,27]}]}]}
{"url":"Mathlib/Analysis/LocallyConvex/AbsConvex.lean","commit":"","full_name":"gaugeSeminormFamily_ball","start":[129,0],"end":[134,79],"file_path":"Mathlib/Analysis/LocallyConvex/AbsConvex.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ (gaugeSeminormFamily 𝕜 E s).ball 0 1 = ↑s","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ (gaugeSeminorm ⋯ ⋯ ⋯).ball 0 1 = ↑s","tactic":"dsimp only [gaugeSeminormFamily]","premises":[{"full_name":"gaugeSeminormFamily","def_path":"Mathlib/Analysis/LocallyConvex/AbsConvex.lean","def_pos":[124,18],"def_end_pos":[124,37]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ (gaugeSeminorm ⋯ ⋯ ⋯).ball 0 1 = ↑s","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | (gaugeSeminorm ⋯ ⋯ ⋯) y < 1} = ↑s","tactic":"rw [Seminorm.ball_zero_eq]","premises":[{"full_name":"Seminorm.ball_zero_eq","def_path":"Mathlib/Analysis/Seminorm.lean","def_pos":[635,8],"def_end_pos":[635,20]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | (gaugeSeminorm ⋯ ⋯ ⋯) y < 1} = ↑s","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | gauge (↑s) y < 1} = ↑s","tactic":"simp_rw [gaugeSeminorm_toFun]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"gaugeSeminorm_toFun","def_path":"Mathlib/Analysis/Convex/Gauge.lean","def_pos":[475,2],"def_end_pos":[475,8]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | gauge (↑s) y < 1} = ↑s","state_after":"no goals","tactic":"exact gauge_lt_one_eq_self_of_isOpen s.coe_convex s.coe_zero_mem s.coe_isOpen","premises":[{"full_name":"AbsConvexOpenSets.coe_convex","def_path":"Mathlib/Analysis/LocallyConvex/AbsConvex.lean","def_pos":[104,8],"def_end_pos":[104,18]},{"full_name":"AbsConvexOpenSets.coe_isOpen","def_path":"Mathlib/Analysis/LocallyConvex/AbsConvex.lean","def_pos":[95,8],"def_end_pos":[95,18]},{"full_name":"AbsConvexOpenSets.coe_zero_mem","def_path":"Mathlib/Analysis/LocallyConvex/AbsConvex.lean","def_pos":[92,8],"def_end_pos":[92,20]},{"full_name":"gauge_lt_one_eq_self_of_isOpen","def_path":"Mathlib/Analysis/Convex/Gauge.lean","def_pos":[356,8],"def_end_pos":[356,38]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"ball_eq","start":[499,0],"end":[501,45],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ny : E\nε : ℝ\na : E\n⊢ a ∈ ball y ε ↔ a ∈ {x | ‖x / y‖ < ε}","state_after":"no goals","tactic":"simp [dist_eq_norm_div]","premises":[{"full_name":"dist_eq_norm_div","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[343,8],"def_end_pos":[343,24]}]}]}
{"url":"Mathlib/Topology/Algebra/InfiniteSum/Group.lean","commit":"","full_name":"tprod_const","start":[353,0],"end":[363,45],"file_path":"Mathlib/Topology/Algebra/InfiniteSum/Group.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : TopologicalGroup G\nf : α → G\ninst✝ : T2Space G\na : G\n⊢ ∏' (x : β), a = a ^ Nat.card β","state_after":"case inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : TopologicalGroup G\nf : α → G\ninst✝ : T2Space G\na : G\nhβ : Finite β\n⊢ ∏' (x : β), a = a ^ Nat.card β\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : TopologicalGroup G\nf : α → G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\n⊢ ∏' (x : β), a = a ^ Nat.card β","tactic":"rcases finite_or_infinite β with hβ|hβ","premises":[{"full_name":"finite_or_infinite","def_path":"Mathlib/Data/Finite/Defs.lean","def_pos":[129,8],"def_end_pos":[129,26]}]}]}
{"url":"Mathlib/RingTheory/ChainOfDivisors.lean","commit":"","full_name":"multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors","start":[406,0],"end":[431,32],"file_path":"Mathlib/RingTheory/ChainOfDivisors.lean","tactics":[{"state_before":"M : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n = multiplicity p m","state_after":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n","tactic":"apply Eq.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]},{"state_before":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n","state_after":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩))\n      (Associates.mk n)","tactic":"suffices multiplicity (Associates.mk p) (Associates.mk m) = multiplicity (Associates.mk\n    ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by\n      simp [dvd_of_mem_normalizedFactors hp]⟩)) (Associates.mk n) by\n    simpa only [multiplicity_mk_eq_multiplicity, associatesEquivOfUniqueUnits_symm_apply,\n      associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq] using this","premises":[{"full_name":"Associates.mk","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[737,17],"def_end_pos":[737,19]},{"full_name":"Associates.out_mk","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[197,8],"def_end_pos":[197,14]},{"full_name":"MulEquiv.symm","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[252,4],"def_end_pos":[252,8]},{"full_name":"UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[724,8],"def_end_pos":[724,36]},{"full_name":"associatesEquivOfUniqueUnits","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[877,4],"def_end_pos":[877,32]},{"full_name":"associatesEquivOfUniqueUnits_apply","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[876,2],"def_end_pos":[876,7]},{"full_name":"associatesEquivOfUniqueUnits_symm_apply","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[876,2],"def_end_pos":[876,7]},{"full_name":"multiplicity","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[34,4],"def_end_pos":[34,16]},{"full_name":"multiplicity.multiplicity_mk_eq_multiplicity","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[332,8],"def_end_pos":[332,39]},{"full_name":"normalize_eq","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[872,8],"def_end_pos":[872,20]}]},{"state_before":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩))\n      (Associates.mk n)","state_after":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩))\n      (Associates.mk n)","tactic":"have : Associates.mk (d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by\n    simp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_symm_apply,\n      associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq]⟩ : N) =\n    ↑(mkFactorOrderIsoOfFactorDvdEquiv hd ⟨associatesEquivOfUniqueUnits.symm p, by\n      rw [associatesEquivOfUniqueUnits_symm_apply]\n      exact mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp)⟩) := by\n    rw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]","premises":[{"full_name":"Associates.mk","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[737,17],"def_end_pos":[737,19]},{"full_name":"Associates.mk_le_mk_of_dvd","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[908,8],"def_end_pos":[908,23]},{"full_name":"Associates.out_mk","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[197,8],"def_end_pos":[197,14]},{"full_name":"MulEquiv.symm","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[252,4],"def_end_pos":[252,8]},{"full_name":"UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[724,8],"def_end_pos":[724,36]},{"full_name":"associatesEquivOfUniqueUnits","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[877,4],"def_end_pos":[877,32]},{"full_name":"associatesEquivOfUniqueUnits_apply","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[876,2],"def_end_pos":[876,7]},{"full_name":"associatesEquivOfUniqueUnits_symm_apply","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[876,2],"def_end_pos":[876,7]},{"full_name":"mkFactorOrderIsoOfFactorDvdEquiv","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[340,4],"def_end_pos":[340,36]},{"full_name":"mkFactorOrderIsoOfFactorDvdEquiv_apply_coe","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[339,2],"def_end_pos":[339,7]},{"full_name":"normalize_eq","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[872,8],"def_end_pos":[872,20]}]},{"state_before":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩))\n      (Associates.mk n)","state_after":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)) (Associates.mk n)","tactic":"rw [this]","premises":[]},{"state_before":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)) (Associates.mk n)","state_after":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ Associates.mk p ∈ normalizedFactors (Associates.mk m)","tactic":"refine\n    multiplicity_prime_eq_multiplicity_image_by_factor_orderIso (mk_ne_zero.mpr hn) ?_\n      (mkFactorOrderIsoOfFactorDvdEquiv hd)","premises":[{"full_name":"Associates.mk_ne_zero","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[958,8],"def_end_pos":[958,18]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"mkFactorOrderIsoOfFactorDvdEquiv","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[340,4],"def_end_pos":[340,36]},{"full_name":"multiplicity_prime_eq_multiplicity_image_by_factor_orderIso","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[320,8],"def_end_pos":[320,67]}]},{"state_before":"case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ Associates.mk p ∈ normalizedFactors (Associates.mk m)","state_after":"case h.intro.intro\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\nq : Associates M\nhq : q ∈ normalizedFactors (Associates.mk m)\nhq' : Associated (Associates.mk p) q\n⊢ Associates.mk p ∈ normalizedFactors (Associates.mk m)","tactic":"obtain ⟨q, hq, hq'⟩ :=\n    exists_mem_normalizedFactors_of_dvd (mk_ne_zero.mpr hm)\n      (prime_mk.mpr (prime_of_normalized_factor p hp)).irreducible\n      (mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp))","premises":[{"full_name":"Associates.mk_le_mk_of_dvd","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[908,8],"def_end_pos":[908,23]},{"full_name":"Associates.mk_ne_zero","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[958,8],"def_end_pos":[958,18]},{"full_name":"Associates.prime_mk","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[994,8],"def_end_pos":[994,16]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Prime.irreducible","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[320,18],"def_end_pos":[320,35]},{"full_name":"UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[724,8],"def_end_pos":[724,36]},{"full_name":"UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[610,8],"def_end_pos":[610,43]},{"full_name":"UniqueFactorizationMonoid.prime_of_normalized_factor","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[573,8],"def_end_pos":[573,34]}]},{"state_before":"case h.intro.intro\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\nq : Associates M\nhq : q ∈ normalizedFactors (Associates.mk m)\nhq' : Associated (Associates.mk p) q\n⊢ Associates.mk p ∈ normalizedFactors (Associates.mk m)","state_after":"no goals","tactic":"rwa [associated_iff_eq.mp hq']","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"associated_iff_eq","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[693,8],"def_end_pos":[693,25]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Bochner.lean","commit":"","full_name":"MeasureTheory.integral_tendsto_of_tendsto_of_antitone","start":[1200,0],"end":[1211,53],"file_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF✝ : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace ℝ F✝\ninst✝⁴ : CompleteSpace F✝\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))\n⊢ Tendsto (fun n => ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))","state_after":"α : Type u_1\nE : Type u_2\nF✝ : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace ℝ F✝\ninst✝⁴ : CompleteSpace F✝\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))\n⊢ Tendsto (fun n => ∫ (x : α), -f n x ∂μ) atTop (𝓝 (∫ (x : α), -F x ∂μ))","tactic":"suffices Tendsto (fun n ↦ ∫ x, -f n x ∂μ) atTop (𝓝 (∫ x, -F x ∂μ)) by\n    suffices Tendsto (fun n ↦ ∫ x, - -f n x ∂μ) atTop (𝓝 (∫ x, - -F x ∂μ)) by\n      simpa [neg_neg] using this\n    convert this.neg <;> rw [integral_neg]","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.neg","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[214,2],"def_end_pos":[214,13]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"MeasureTheory.integral_neg","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[803,8],"def_end_pos":[803,20]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF✝ : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace ℝ F✝\ninst✝⁴ : CompleteSpace F✝\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))\n⊢ Tendsto (fun n => ∫ (x : α), -f n x ∂μ) atTop (𝓝 (∫ (x : α), -F x ∂μ))","state_after":"case refine_1\nα : Type u_1\nE : Type u_2\nF✝ : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace ℝ F✝\ninst✝⁴ : CompleteSpace F✝\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Monotone fun n => -f n x\n\ncase refine_2\nα : Type u_1\nE : Type u_2\nF✝ : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F✝\ninst✝⁵ : NormedSpace ℝ F✝\ninst✝⁴ : CompleteSpace F✝\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => -f n x) atTop (𝓝 (-F x))","tactic":"refine integral_tendsto_of_tendsto_of_monotone (fun n ↦ (hf n).neg) hF.neg ?_ ?_","premises":[{"full_name":"MeasureTheory.Integrable.neg","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[601,8],"def_end_pos":[601,22]},{"full_name":"MeasureTheory.integral_tendsto_of_tendsto_of_monotone","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1163,6],"def_end_pos":[1163,45]}]}]}
{"url":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","commit":"","full_name":"MeasureTheory.MeasurePreserving.aemeasurable_comp_iff","start":[80,0],"end":[82,43],"file_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nf : α → β\nhf : MeasurePreserving f μa μb\nh₂ : MeasurableEmbedding f\ng : β → γ\n⊢ AEMeasurable (g ∘ f) μa ↔ AEMeasurable g μb","state_after":"no goals","tactic":"rw [← hf.map_eq, h₂.aemeasurable_map_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"MeasurableEmbedding.aemeasurable_map_iff","def_path":"Mathlib/MeasureTheory/Measure/AEMeasurable.lean","def_pos":[232,8],"def_end_pos":[232,48]},{"full_name":"MeasureTheory.MeasurePreserving.map_eq","def_path":"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean","def_pos":[43,12],"def_end_pos":[43,18]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean","commit":"","full_name":"Convex.isLittleO_alternate_sum_square","start":[172,0],"end":[224,6],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean","tactics":[{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[62,4],"def_end_pos":[62,7]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Icc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[54,4],"def_end_pos":[54,7]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => by simp only [two_smul]","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"two_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[96,8],"def_end_pos":[96,16]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have h2v2w : x + (2 : ℝ) • v + (2 : ℝ) • w ∈ interior s := by\n    convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1\n    simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]\n    norm_num\n    simp only [show (4 : ℝ) = (2 : ℝ) + (2 : ℝ) by norm_num, _root_.add_smul]\n    abel","premises":[{"full_name":"Convex.add_smul","def_path":"Mathlib/Analysis/Convex/Basic.lean","def_pos":[518,8],"def_end_pos":[518,23]},{"full_name":"Convex.add_smul_sub_mem","def_path":"Mathlib/Analysis/Convex/Basic.lean","def_pos":[444,8],"def_end_pos":[444,31]},{"full_name":"Convex.interior","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[212,18],"def_end_pos":[212,33]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"add_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[87,8],"def_end_pos":[87,16]},{"full_name":"add_sub_add_left_eq_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[889,2],"def_end_pos":[889,13]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have h2vww : x + (2 • v + w) + w ∈ interior s := by\n    convert h2v2w using 1\n    simp only [two_smul]\n    abel","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"two_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[96,8],"def_end_pos":[96,16]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have h2v : x + (2 : ℝ) • v ∈ interior s := by\n    convert s_conv.add_smul_sub_mem_interior xs h4v A using 1\n    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]\n    norm_num","premises":[{"full_name":"Convex.add_smul_sub_mem_interior","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[197,8],"def_end_pos":[197,40]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"add_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[63,2],"def_end_pos":[63,13]},{"full_name":"add_sub_cancel_left","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[923,2],"def_end_pos":[923,13]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have h2w : x + (2 : ℝ) • w ∈ interior s := by\n    convert s_conv.add_smul_sub_mem_interior xs h4w A using 1\n    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]\n    norm_num","premises":[{"full_name":"Convex.add_smul_sub_mem_interior","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[197,8],"def_end_pos":[197,40]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"add_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[63,2],"def_end_pos":[63,13]},{"full_name":"add_sub_cancel_left","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[923,2],"def_end_pos":[923,13]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have hvw : x + (v + w) ∈ interior s := by\n    convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1\n    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj, smul_add, smul_sub]\n    norm_num\n    abel","premises":[{"full_name":"Convex.add_smul_sub_mem_interior","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[197,8],"def_end_pos":[197,40]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"add_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[63,2],"def_end_pos":[63,13]},{"full_name":"add_sub_cancel_left","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[923,2],"def_end_pos":[923,13]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have h2vw : x + (2 • v + w) ∈ interior s := by\n    convert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1\n    simp only [smul_add, smul_sub, smul_smul, ← C]\n    norm_num\n    abel","premises":[{"full_name":"Convex.add_smul_sub_mem","def_path":"Mathlib/Analysis/Convex/Basic.lean","def_pos":[444,8],"def_end_pos":[444,31]},{"full_name":"Convex.interior","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[212,18],"def_end_pos":[212,33]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have hvww : x + (v + w) + w ∈ interior s := by\n    convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1\n    rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel_left, inv_smul_smul₀ two_ne_zero,\n      two_smul]\n    abel","premises":[{"full_name":"Convex.add_smul_sub_mem","def_path":"Mathlib/Analysis/Convex/Basic.lean","def_pos":[444,8],"def_end_pos":[444,31]},{"full_name":"Convex.interior","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[212,18],"def_end_pos":[212,33]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"add_sub_add_right_eq_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[705,2],"def_end_pos":[705,13]},{"full_name":"add_sub_cancel_left","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[923,2],"def_end_pos":[923,13]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"inv_smul_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[35,8],"def_end_pos":[35,22]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"two_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[54,6],"def_end_pos":[54,17]},{"full_name":"two_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[96,8],"def_end_pos":[96,16]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have TA1 := s_conv.taylor_approx_two_segment hf xs hx h2vw h2vww","premises":[{"full_name":"Convex.taylor_approx_two_segment","def_path":"Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean","def_pos":[66,8],"def_end_pos":[66,40]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","tactic":"have TA2 := s_conv.taylor_approx_two_segment hf xs hx hvw hvww","premises":[{"full_name":"Convex.taylor_approx_two_segment","def_path":"Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean","def_pos":[66,8],"def_end_pos":[66,40]}]},{"state_before":"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2","state_after":"case h.e'_7\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =\n    fun x_1 =>\n    f (x + x_1 • (2 • v + w) + x_1 • w) - f (x + x_1 • (2 • v + w)) - x_1 • (f' x) w - x_1 ^ 2 • (f'' (2 • v + w)) w -\n        (x_1 ^ 2 / 2) • (f'' w) w -\n      (f (x + x_1 • (v + w) + x_1 • w) - f (x + x_1 • (v + w)) - x_1 • (f' x) w - x_1 ^ 2 • (f'' (v + w)) w -\n        (x_1 ^ 2 / 2) • (f'' w) w)","tactic":"convert TA1.sub TA2 using 1","premises":[{"full_name":"Asymptotics.IsLittleO.sub","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[937,8],"def_end_pos":[937,21]}]},{"state_before":"case h.e'_7\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =\n    fun x_1 =>\n    f (x + x_1 • (2 • v + w) + x_1 • w) - f (x + x_1 • (2 • v + w)) - x_1 • (f' x) w - x_1 ^ 2 • (f'' (2 • v + w)) w -\n        (x_1 ^ 2 / 2) • (f'' w) w -\n      (f (x + x_1 • (v + w) + x_1 • w) - f (x + x_1 • (v + w)) - x_1 • (f' x) w - x_1 ^ 2 • (f'' (v + w)) w -\n        (x_1 ^ 2 / 2) • (f'' w) w)","state_after":"case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n      h ^ 2 • (f'' v) w =\n    f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w -\n      (f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w)","tactic":"ext h","premises":[]},{"state_before":"case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n      h ^ 2 • (f'' v) w =\n    f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w -\n      (f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w)","state_after":"case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • v + h • v + h • w + h • w) + f (x + h • v + h • w) - f (x + h • v + h • v + h • w) -\n        f (x + h • v + h • w + h • w) -\n      h ^ 2 • (f'' v) w =\n    f (x + h • v + h • v + h • w + h • w) - f (x + h • v + h • v + h • w) - h • (f' x) w -\n          (h ^ 2 • (f'' v) w + h ^ 2 • (f'' v) w + h ^ 2 • (f'' w) w) -\n        (h ^ 2 / 2) • (f'' w) w -\n      (f (x + h • v + h • w + h • w) - f (x + h • v + h • w) - h • (f' x) w - (h ^ 2 • (f'' v) w + h ^ 2 • (f'' w) w) -\n        (h ^ 2 / 2) • (f'' w) w)","tactic":"simp only [two_smul, smul_add, ← add_assoc, ContinuousLinearMap.map_add,\n    ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul',\n    ContinuousLinearMap.map_smul]","premises":[{"full_name":"ContinuousLinearMap.add_apply","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[632,8],"def_end_pos":[632,17]},{"full_name":"ContinuousLinearMap.coe_smul'","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[536,8],"def_end_pos":[536,17]},{"full_name":"ContinuousLinearMap.map_add","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[448,18],"def_end_pos":[448,25]},{"full_name":"ContinuousLinearMap.map_smul","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[456,18],"def_end_pos":[456,26]},{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]},{"full_name":"two_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[96,8],"def_end_pos":[96,16]}]},{"state_before":"case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • v + h • v + h • w + h • w) + f (x + h • v + h • w) - f (x + h • v + h • v + h • w) -\n        f (x + h • v + h • w + h • w) -\n      h ^ 2 • (f'' v) w =\n    f (x + h • v + h • v + h • w + h • w) - f (x + h • v + h • v + h • w) - h • (f' x) w -\n          (h ^ 2 • (f'' v) w + h ^ 2 • (f'' v) w + h ^ 2 • (f'' w) w) -\n        (h ^ 2 / 2) • (f'' w) w -\n      (f (x + h • v + h • w + h • w) - f (x + h • v + h • w) - h • (f' x) w - (h ^ 2 • (f'' v) w + h ^ 2 • (f'' w) w) -\n        (h ^ 2 / 2) • (f'' w) w)","state_after":"no goals","tactic":"abel","premises":[]}]}
{"url":"Mathlib/SetTheory/Ordinal/Exponential.lean","commit":"","full_name":"Ordinal.sup_opow_nat","start":[406,0],"end":[412,33],"file_path":"Mathlib/SetTheory/Ordinal/Exponential.lean","tactics":[{"state_before":"o : Ordinal.{u_1}\nho : 0 < o\n⊢ (sup fun n => o ^ ↑n) = o ^ ω","state_after":"case inl\no : Ordinal.{u_1}\nho : 0 < o\nho₁ : 1 < o\n⊢ (sup fun n => o ^ ↑n) = o ^ ω\n\ncase inr\nho : 0 < 1\n⊢ (sup fun n => 1 ^ ↑n) = 1 ^ ω","tactic":"rcases lt_or_eq_of_le (one_le_iff_pos.2 ho) with (ho₁ | rfl)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ordinal.one_le_iff_pos","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[937,8],"def_end_pos":[937,22]},{"full_name":"lt_or_eq_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[191,8],"def_end_pos":[191,22]}]}]}
{"url":"Mathlib/Data/PFun.lean","commit":"","full_name":"PFun.Part.bind_comp","start":[515,0],"end":[521,21],"file_path":"Mathlib/Data/PFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\n⊢ a.bind (f.comp g) = (a.bind g).bind f","state_after":"case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ c ∈ a.bind (f.comp g) ↔ c ∈ (a.bind g).bind f","tactic":"ext c","premises":[]},{"state_before":"case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ c ∈ a.bind (f.comp g) ↔ c ∈ (a.bind g).bind f","state_after":"case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ (∃ a_1 x, a_1 ∈ a ∧ x ∈ g a_1 ∧ c ∈ f x) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1","tactic":"simp_rw [Part.mem_bind_iff, comp_apply, Part.mem_bind_iff, ← exists_and_right, ← exists_and_left]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"PFun.comp_apply","def_path":"Mathlib/Data/PFun.lean","def_pos":[489,8],"def_end_pos":[489,18]},{"full_name":"Part.mem_bind_iff","def_path":"Mathlib/Data/Part.lean","def_pos":[413,8],"def_end_pos":[413,20]},{"full_name":"exists_and_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[288,16],"def_end_pos":[288,31]},{"full_name":"exists_and_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[291,16],"def_end_pos":[291,32]}]},{"state_before":"case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ (∃ a_1 x, a_1 ∈ a ∧ x ∈ g a_1 ∧ c ∈ f x) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1","state_after":"case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ (∃ b, ∃ a_1 ∈ a, b ∈ g a_1 ∧ c ∈ f b) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1","tactic":"rw [exists_comm]","premises":[{"full_name":"exists_comm","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[318,8],"def_end_pos":[318,19]}]},{"state_before":"case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ (∃ b, ∃ a_1 ∈ a, b ∈ g a_1 ∧ c ∈ f b) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1","state_after":"no goals","tactic":"simp_rw [and_assoc]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[148,8],"def_end_pos":[148,17]}]}]}
{"url":"Mathlib/Data/TypeVec.lean","commit":"","full_name":"TypeVec.lastFun_toSubtype","start":[594,0],"end":[598,37],"file_path":"Mathlib/Data/TypeVec.lean","tactics":[{"state_before":"n : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\n⊢ lastFun (toSubtype p) = _root_.id","state_after":"case h\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\ni : last fun i => { x // ofRepeat (p i x) }\n⊢ lastFun (toSubtype p) i = _root_.id i","tactic":"ext i : 2","premises":[]},{"state_before":"case h\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\ni : last fun i => { x // ofRepeat (p i x) }\n⊢ lastFun (toSubtype p) i = _root_.id i","state_after":"case h.mk\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\nval✝ : α Fin2.fz\nproperty✝ : ofRepeat (p Fin2.fz val✝)\n⊢ lastFun (toSubtype p) ⟨val✝, property✝⟩ = _root_.id ⟨val✝, property✝⟩","tactic":"induction i","premises":[]},{"state_before":"case h.mk\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\nval✝ : α Fin2.fz\nproperty✝ : ofRepeat (p Fin2.fz val✝)\n⊢ lastFun (toSubtype p) ⟨val✝, property✝⟩ = _root_.id ⟨val✝, property✝⟩","state_after":"case h.mk\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\nval✝ : α Fin2.fz\nproperty✝ : ofRepeat (p Fin2.fz val✝)\n⊢ lastFun (toSubtype p) ⟨val✝, property✝⟩ = ⟨val✝, property✝⟩","tactic":"simp [dropFun, *]","premises":[{"full_name":"TypeVec.dropFun","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[136,4],"def_end_pos":[136,11]}]},{"state_before":"case h.mk\nn : ℕ\nα : TypeVec.{u_1} (n + 1)\np : α ⟹ repeat (n + 1) Prop\nval✝ : α Fin2.fz\nproperty✝ : ofRepeat (p Fin2.fz val✝)\n⊢ lastFun (toSubtype p) ⟨val✝, property✝⟩ = ⟨val✝, property✝⟩","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Probability/Moments.lean","commit":"","full_name":"ProbabilityTheory.cgf_neg","start":[184,0],"end":[184,74],"file_path":"Mathlib/Probability/Moments.lean","tactics":[{"state_before":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\n⊢ cgf (-X) μ t = cgf X μ (-t)","state_after":"no goals","tactic":"simp_rw [cgf, mgf_neg]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"ProbabilityTheory.cgf","def_path":"Mathlib/Probability/Moments.lean","def_pos":[98,4],"def_end_pos":[98,7]},{"full_name":"ProbabilityTheory.mgf_neg","def_path":"Mathlib/Probability/Moments.lean","def_pos":[182,8],"def_end_pos":[182,15]}]}]}
{"url":"Mathlib/Topology/Instances/AddCircle.lean","commit":"","full_name":"continuous_left_toIocMod","start":[79,0],"end":[86,98],"file_path":"Mathlib/Topology/Instances/AddCircle.lean","tactics":[{"state_before":"𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt (toIocMod hp a) (Iic x) x","state_after":"𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x","tactic":"rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :\n      toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)]","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Neg.neg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1329,2],"def_end_pos":[1329,5]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"toIcoMod","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[66,4],"def_end_pos":[66,12]},{"full_name":"toIocMod","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[70,4],"def_end_pos":[70,12]},{"full_name":"toIocMod_neg","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[462,8],"def_end_pos":[462,20]}]},{"state_before":"𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x","state_after":"𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nthis : ContinuousNeg 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x","tactic":"have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg","premises":[{"full_name":"ContinuousNeg","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[140,6],"def_end_pos":[140,19]}]},{"state_before":"𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nthis : ContinuousNeg 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x","state_after":"no goals","tactic":"exact\n    (continuous_sub_left _).continuousAt.comp_continuousWithinAt <|\n      (continuous_right_toIcoMod _ _ _).comp continuous_neg.continuousWithinAt fun y => neg_le_neg","premises":[{"full_name":"Continuous.continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1424,8],"def_end_pos":[1424,31]},{"full_name":"Continuous.continuousWithinAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[814,8],"def_end_pos":[814,37]},{"full_name":"ContinuousAt.comp_continuousWithinAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[785,8],"def_end_pos":[785,44]},{"full_name":"ContinuousNeg.continuous_neg","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[141,2],"def_end_pos":[141,16]},{"full_name":"ContinuousWithinAt.comp","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[775,8],"def_end_pos":[775,31]},{"full_name":"continuous_right_toIcoMod","def_path":"Mathlib/Topology/Instances/AddCircle.lean","def_pos":[62,8],"def_end_pos":[62,33]},{"full_name":"continuous_sub_left","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[969,35],"def_end_pos":[969,54]},{"full_name":"neg_le_neg","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[172,31],"def_end_pos":[172,41]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.count_filter","start":[2193,0],"end":[2197,31],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ns✝ : Multiset α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\n⊢ count a (filter p s) = if p a then count a s else 0","state_after":"case pos\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ns✝ : Multiset α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : p a\n⊢ count a (filter p s) = count a s\n\ncase neg\nα : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α\ns✝ : Multiset α\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nh : ¬p a\n⊢ count a (filter p s) = 0","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","commit":"","full_name":"CategoryTheory.Limits.biproduct.map_eq_map'","start":[550,0],"end":[561,8],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","tactics":[{"state_before":"J : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\n⊢ map p = map' p","state_after":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ ι f j✝ ≫ map p ≫ π g j✝¹ = ι f j✝ ≫ map' p ≫ π g j✝¹","tactic":"ext","premises":[]},{"state_before":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ ι f j✝ ≫ map p ≫ π g j✝¹ = ι f j✝ ≫ map' p ≫ π g j✝¹","state_after":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ (bicone f).toCocone.ι.app { as := j✝ } ≫ (bicone f).toCone.π.app { as := j✝¹ } ≫ p j✝¹ =\n    p j✝ ≫ (bicone g).toCocone.ι.app { as := j✝ } ≫ (bicone g).toCone.π.app { as := j✝¹ }","tactic":"simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,\n    Category.assoc, ← Bicone.toCone_π_app_mk, ← biproduct.bicone_π, ← Bicone.toCocone_ι_app_mk,\n    ← biproduct.bicone_ι]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Discrete.natTrans_app","def_path":"Mathlib/CategoryTheory/DiscreteCategory.lean","def_pos":[197,2],"def_end_pos":[197,7]},{"full_name":"CategoryTheory.Limits.Bicone.toCocone_ι_app_mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[214,8],"def_end_pos":[214,25]},{"full_name":"CategoryTheory.Limits.Bicone.toCone_π_app_mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[192,8],"def_end_pos":[192,23]},{"full_name":"CategoryTheory.Limits.IsLimit.map_π","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[79,8],"def_end_pos":[79,13]},{"full_name":"CategoryTheory.Limits.biproduct.bicone_ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[444,8],"def_end_pos":[444,26]},{"full_name":"CategoryTheory.Limits.biproduct.bicone_π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[436,8],"def_end_pos":[436,26]}]},{"state_before":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ (bicone f).toCocone.ι.app { as := j✝ } ≫ (bicone f).toCone.π.app { as := j✝¹ } ≫ p j✝¹ =\n    p j✝ ≫ (bicone g).toCocone.ι.app { as := j✝ } ≫ (bicone g).toCone.π.app { as := j✝¹ }","state_after":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ ι f j✝ ≫ π f j✝¹ ≫ p j✝¹ = p j✝ ≫ ι g j✝ ≫ π g j✝¹","tactic":"dsimp","premises":[]},{"state_before":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ ι f j✝ ≫ π f j✝¹ ≫ p j✝¹ = p j✝ ≫ ι g j✝ ≫ π g j✝¹","state_after":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ (if h : j✝ = j✝¹ then eqToHom ⋯ else 0) ≫ p j✝¹ = p j✝ ≫ if h : j✝ = j✝¹ then eqToHom ⋯ else 0","tactic":"rw [biproduct.ι_π_assoc, biproduct.ι_π]","premises":[{"full_name":"CategoryTheory.Limits.biproduct.ι_π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[450,8],"def_end_pos":[450,21]}]},{"state_before":"case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ (if h : j✝ = j✝¹ then eqToHom ⋯ else 0) ≫ p j✝¹ = p j✝ ≫ if h : j✝ = j✝¹ then eqToHom ⋯ else 0","state_after":"case pos\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\nh : j✝ = j✝¹\n⊢ eqToHom ⋯ ≫ p j✝¹ = p j✝ ≫ eqToHom ⋯\n\ncase neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\nh : ¬j✝ = j✝¹\n⊢ 0 ≫ p j✝¹ = p j✝ ≫ 0","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/NumberTheory/SumFourSquares.lean","commit":"","full_name":"euler_four_squares","start":[25,0],"end":[29,82],"file_path":"Mathlib/NumberTheory/SumFourSquares.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\na b c d x y z w : R\n⊢ (a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 + (a * z - b * w + c * x + d * y) ^ 2 +\n      (a * w + b * z - c * y + d * x) ^ 2 =\n    (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)","state_after":"no goals","tactic":"ring","premises":[]}]}
{"url":"Mathlib/RingTheory/FreeCommRing.lean","commit":"","full_name":"FreeCommRing.isSupported_of","start":[210,0],"end":[240,96],"file_path":"Mathlib/RingTheory/FreeCommRing.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\n⊢ p ∈ s","state_after":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred s\n⊢ p ∈ s","tactic":"haveI := Classical.decPred s","premises":[{"full_name":"Classical.decPred","def_path":"Mathlib/Logic/Basic.lean","def_pos":[731,18],"def_end_pos":[731,25]}]},{"state_before":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nthis : ∀ (x : FreeCommRing α), x.IsSupported s → ∃ n, (lift fun a => if a ∈ s then 0 else X) x = ↑n\n⊢ p ∈ s","state_after":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nthis : ∃ n, (lift fun a => if a ∈ s then 0 else X) (of p) = ↑n\n⊢ p ∈ s","tactic":"specialize this (of p) hps","premises":[{"full_name":"FreeCommRing.of","def_path":"Mathlib/RingTheory/FreeCommRing.lean","def_pos":[76,4],"def_end_pos":[76,6]}]},{"state_before":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nthis : ∃ n, (lift fun a => if a ∈ s then 0 else X) (of p) = ↑n\n⊢ p ∈ s","state_after":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nthis : ∃ n, (if p ∈ s then 0 else X) = ↑n\n⊢ p ∈ s","tactic":"rw [lift_of] at this","premises":[{"full_name":"FreeCommRing.lift_of","def_path":"Mathlib/RingTheory/FreeCommRing.lean","def_pos":[133,8],"def_end_pos":[133,15]}]},{"state_before":"α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nthis : ∃ n, (if p ∈ s then 0 else X) = ↑n\n⊢ p ∈ s","state_after":"case pos\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∈ s\nthis : ∃ n, 0 = ↑n\n⊢ p ∈ s\n\ncase neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ p ∈ s","tactic":"split_ifs at this with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ p ∈ s","state_after":"case neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ False","tactic":"exfalso","premises":[{"full_name":"False.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[236,20],"def_end_pos":[236,30]}]},{"state_before":"case neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ False","state_after":"case neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ 1 = 0","tactic":"apply Ne.symm Int.zero_ne_one","premises":[{"full_name":"Int.zero_ne_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[67,18],"def_end_pos":[67,29]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]}]},{"state_before":"case neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ 1 = 0","state_after":"case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = ↑w\n⊢ 1 = 0","tactic":"rcases this with ⟨w, H⟩","premises":[]},{"state_before":"case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = ↑w\n⊢ 1 = 0","state_after":"case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = C ↑w\n⊢ 1 = 0","tactic":"rw [← Polynomial.C_eq_intCast] at H","premises":[{"full_name":"Polynomial.C_eq_intCast","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[1033,8],"def_end_pos":[1033,20]}]},{"state_before":"case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = C ↑w\n⊢ 1 = 0","state_after":"case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = C ↑w\nthis : X.coeff 1 = (C w).coeff 1\n⊢ 1 = 0","tactic":"have : Polynomial.X.coeff 1 = (Polynomial.C ↑w).coeff 1 := by rw [H]; rfl","premises":[{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]}]},{"state_before":"case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = C ↑w\nthis : X.coeff 1 = (C w).coeff 1\n⊢ 1 = 0","state_after":"no goals","tactic":"rwa [Polynomial.coeff_C, if_neg (one_ne_zero : 1 ≠ 0), Polynomial.coeff_X, if_pos rfl] at this","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.coeff_C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[613,8],"def_end_pos":[613,15]},{"full_name":"Polynomial.coeff_X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[600,8],"def_end_pos":[600,15]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","commit":"","full_name":"intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae","start":[377,0],"end":[416,63],"file_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis : la'.IsMeasurablyGenerated\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖","tactic":"haveI := FTCFilter.meas_gen la","premises":[{"full_name":"intervalIntegral.FTCFilter.meas_gen","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[196,3],"def_end_pos":[196,11]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis : la'.IsMeasurablyGenerated\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖","tactic":"haveI := FTCFilter.meas_gen lb","premises":[{"full_name":"intervalIntegral.FTCFilter.meas_gen","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[196,3],"def_end_pos":[196,11]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","tactic":"refine\n    ((measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add\n          (measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr'\n      ?_ EventuallyEq.rfl","premises":[{"full_name":"Asymptotics.IsLittleO.add_add","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[908,8],"def_end_pos":[908,25]},{"full_name":"Asymptotics.IsLittleO.congr'","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[332,8],"def_end_pos":[332,24]},{"full_name":"Filter.EventuallyEq.rfl","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1303,18],"def_end_pos":[1303,34]},{"full_name":"intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[331,8],"def_end_pos":[331,59]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","tactic":"have A : ∀ᶠ t in lt, IntervalIntegrable f μ (ua t) (va t) :=\n    ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la) hua hva","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Tendsto.eventually_intervalIntegrable_ae","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[379,8],"def_end_pos":[379,55]},{"full_name":"IntervalIntegrable","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[67,4],"def_end_pos":[67,22]},{"full_name":"intervalIntegral.FTCFilter.finiteAt_inner","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[208,8],"def_end_pos":[208,22]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","tactic":"have A' : ∀ᶠ t in lt, IntervalIntegrable f μ a (ua t) :=\n    ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la)\n      (tendsto_const_pure.mono_right FTCFilter.pure_le) hua","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Tendsto.eventually_intervalIntegrable_ae","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[379,8],"def_end_pos":[379,55]},{"full_name":"Filter.Tendsto.mono_right","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2664,8],"def_end_pos":[2664,26]},{"full_name":"Filter.tendsto_const_pure","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2777,8],"def_end_pos":[2777,26]},{"full_name":"IntervalIntegrable","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[67,4],"def_end_pos":[67,22]},{"full_name":"intervalIntegral.FTCFilter.finiteAt_inner","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[208,8],"def_end_pos":[208,22]},{"full_name":"intervalIntegral.FTCFilter.pure_le","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[194,2],"def_end_pos":[194,9]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","tactic":"have B : ∀ᶠ t in lt, IntervalIntegrable f μ (ub t) (vb t) :=\n    hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb) hub hvb","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Tendsto.eventually_intervalIntegrable_ae","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[379,8],"def_end_pos":[379,55]},{"full_name":"IntervalIntegrable","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[67,4],"def_end_pos":[67,22]},{"full_name":"intervalIntegral.FTCFilter.finiteAt_inner","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[208,8],"def_end_pos":[208,22]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","tactic":"have B' : ∀ᶠ t in lt, IntervalIntegrable f μ b (ub t) :=\n    hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb)\n      (tendsto_const_pure.mono_right FTCFilter.pure_le) hub","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Tendsto.eventually_intervalIntegrable_ae","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[379,8],"def_end_pos":[379,55]},{"full_name":"Filter.Tendsto.mono_right","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2664,8],"def_end_pos":[2664,26]},{"full_name":"Filter.tendsto_const_pure","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2777,8],"def_end_pos":[2777,26]},{"full_name":"IntervalIntegrable","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[67,4],"def_end_pos":[67,22]},{"full_name":"intervalIntegral.FTCFilter.finiteAt_inner","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[208,8],"def_end_pos":[208,22]},{"full_name":"intervalIntegral.FTCFilter.pure_le","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[194,2],"def_end_pos":[194,9]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)","state_after":"case h\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ -(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) +\n      (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) =\n    ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ -\n      (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)","tactic":"filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ -(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) +\n      (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) =\n    ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ -\n      (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)","state_after":"case h\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ -(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) +\n      (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) =\n    ∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ -\n      (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)\n\ncase h.hab\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ub a✝) (vb a✝)\n\ncase h.hcd\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ua a✝) (va a✝)\n\ncase h.hac\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ub a✝) (ua a✝)","tactic":"rw [← integral_interval_sub_interval_comm']","premises":[{"full_name":"intervalIntegral.integral_interval_sub_interval_comm'","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[853,8],"def_end_pos":[853,44]}]},{"state_before":"case h.hab\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ub a✝) (vb a✝)\n\ncase h.hcd\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ua a✝) (va a✝)\n\ncase h.hac\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ub a✝) (ua a✝)","state_after":"no goals","tactic":"exacts [ub_vb, ua_va, b_ub.symm.trans <| hab.symm.trans a_ua]","premises":[{"full_name":"IntervalIntegrable.symm","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[144,15],"def_end_pos":[144,19]},{"full_name":"IntervalIntegrable.trans","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[151,8],"def_end_pos":[151,13]}]}]}
{"url":"Mathlib/RingTheory/Localization/Integral.lean","commit":"","full_name":"isAlgebraic_of_isLocalization","start":[436,0],"end":[451,85],"file_path":"Mathlib/RingTheory/Localization/Integral.lean","tactics":[{"state_before":"R✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\n⊢ Algebra.IsAlgebraic R S","state_after":"case isAlgebraic\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\n⊢ ∀ (x : S), IsAlgebraic R x","tactic":"constructor","premises":[]},{"state_before":"case isAlgebraic\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\n⊢ ∀ (x : S), IsAlgebraic R x","state_after":"case isAlgebraic\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : S\n⊢ IsAlgebraic R x","tactic":"intro x","premises":[]},{"state_before":"case isAlgebraic\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : S\n⊢ IsAlgebraic R x","state_after":"case isAlgebraic.intro.intro\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\n⊢ IsAlgebraic R (mk' S x s)","tactic":"obtain ⟨x, s, rfl⟩ := IsLocalization.mk'_surjective M x","premises":[{"full_name":"IsLocalization.mk'_surjective","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[279,8],"def_end_pos":[279,22]}]},{"state_before":"case isAlgebraic.intro.intro\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\n⊢ IsAlgebraic R (mk' S x s)","state_after":"case pos\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\nhs : ↑s = 0\n⊢ IsAlgebraic R (mk' S x s)\n\ncase neg\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\nhs : ¬↑s = 0\n⊢ IsAlgebraic R (mk' S x s)","tactic":"by_cases hs : (s : R) = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\nhs : ¬↑s = 0\n⊢ IsAlgebraic R (mk' S x s)","state_after":"case neg.refine_1\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\nhs : ¬↑s = 0\n⊢ s • X - C x ≠ 0\n\ncase neg.refine_2\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ns : ↥M\nhs : ¬↑s = 0\n⊢ (aeval (mk' S x s)) (s • X - C x) = 0","tactic":"refine ⟨s • X - C x, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]}]}]}
{"url":"Mathlib/GroupTheory/Congruence/Basic.lean","commit":"","full_name":"Con.smul","start":[959,0],"end":[962,69],"file_path":"Mathlib/GroupTheory/Congruence/Basic.lean","tactics":[{"state_before":"M✝ : Type u_1\nN : Type u_2\nP : Type u_3\nα : Type u_4\nM : Type u_5\ninst✝² : MulOneClass M\ninst✝¹ : SMul α M\ninst✝ : IsScalarTower α M M\nc : Con M\na : α\nw x : M\nh : c w x\n⊢ c (a • w) (a • x)","state_after":"no goals","tactic":"simpa only [smul_one_mul] using c.mul (c.refl' (a • (1 : M) : M)) h","premises":[{"full_name":"Con.mul","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[143,18],"def_end_pos":[143,21]},{"full_name":"Setoid.refl'","def_path":"Mathlib/Data/Setoid/Basic.lean","def_pos":[74,8],"def_end_pos":[74,13]},{"full_name":"smul_one_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[585,6],"def_end_pos":[585,18]}]}]}
{"url":"Mathlib/Data/Setoid/Partition.lean","commit":"","full_name":"Setoid.IsPartition.sUnion_eq_univ","start":[210,0],"end":[216,14],"file_path":"Mathlib/Data/Setoid/Partition.lean","tactics":[{"state_before":"α : Type u_1\nc : Set (Set α)\nhc : IsPartition c\nx : α\nt : Set α\nht : (fun b => b ∈ c ∧ x ∈ b) t ∧ ∀ (y : Set α), (fun b => b ∈ c ∧ x ∈ b) y → y = t\n⊢ t ∈ c ∧ x ∈ t","state_after":"α : Type u_1\nc : Set (Set α)\nhc : IsPartition c\nx : α\nt : Set α\nht : (t ∈ c ∧ x ∈ t) ∧ ∀ (y : Set α), y ∈ c ∧ x ∈ y → y = t\n⊢ t ∈ c ∧ x ∈ t","tactic":"simp only [exists_unique_iff_exists] at ht","premises":[{"full_name":"exists_unique_iff_exists","def_path":"Mathlib/Logic/Basic.lean","def_pos":[511,16],"def_end_pos":[511,40]}]},{"state_before":"α : Type u_1\nc : Set (Set α)\nhc : IsPartition c\nx : α\nt : Set α\nht : (t ∈ c ∧ x ∈ t) ∧ ∀ (y : Set α), y ∈ c ∧ x ∈ y → y = t\n⊢ t ∈ c ∧ x ∈ t","state_after":"no goals","tactic":"tauto","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Holder.lean","commit":"","full_name":"HolderOnWith.edist_le_of_le","start":[100,0],"end":[102,39],"file_path":"Mathlib/Topology/MetricSpace/Holder.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\nC r : ℝ≥0\nf : X → Y\ns t : Set X\nh : HolderOnWith C r f s\nx y : X\nhx : x ∈ s\nhy : y ∈ s\nd : ℝ≥0∞\nhd : edist x y ≤ d\n⊢ ↑C * edist x y ^ ↑r ≤ ↑C * d ^ ↑r","state_after":"no goals","tactic":"gcongr","premises":[]}]}
{"url":"Mathlib/Data/Fintype/BigOperators.lean","commit":"","full_name":"Fintype.prod_extend_by_one","start":[48,0],"end":[51,53],"file_path":"Mathlib/Data/Fintype/BigOperators.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : CommMonoid α\ns : Finset ι\nf : ι → α\n⊢ (∏ i : ι, if i ∈ s then f i else 1) = ∏ i ∈ s, f i","state_after":"no goals","tactic":"rw [← prod_filter, filter_mem_eq_inter, univ_inter]","premises":[{"full_name":"Finset.filter_mem_eq_inter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2303,8],"def_end_pos":[2303,27]},{"full_name":"Finset.prod_filter","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[844,8],"def_end_pos":[844,19]},{"full_name":"Finset.univ_inter","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[246,14],"def_end_pos":[246,24]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.toFinsupp_X_pow","start":[778,0],"end":[780,44],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (X ^ n).toFinsupp = Finsupp.single n 1","state_after":"no goals","tactic":"rw [X_pow_eq_monomial, toFinsupp_monomial]","premises":[{"full_name":"Polynomial.X_pow_eq_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[773,8],"def_end_pos":[773,25]},{"full_name":"Polynomial.toFinsupp_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[379,8],"def_end_pos":[379,26]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.countP_map","start":[2004,0],"end":[2009,15],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\np✝ : α → Prop\ninst✝¹ : DecidablePred p✝\nf : α → β\ns : Multiset α\np : β → Prop\ninst✝ : DecidablePred p\n⊢ countP p (map f s) = card (filter (fun a => p (f a)) s)","state_after":"case refine_1\nα : Type u_1\nβ : Type v\nγ : Type u_2\np✝ : α → Prop\ninst✝¹ : DecidablePred p✝\nf : α → β\ns : Multiset α\np : β → Prop\ninst✝ : DecidablePred p\n⊢ countP p (map f 0) = card (filter (fun a => p (f a)) 0)\n\ncase refine_2\nα : Type u_1\nβ : Type v\nγ : Type u_2\np✝ : α → Prop\ninst✝¹ : DecidablePred p✝\nf : α → β\ns : Multiset α\np : β → Prop\ninst✝ : DecidablePred p\na : α\nt : Multiset α\nIH : countP p (map f t) = card (filter (fun a => p (f a)) t)\n⊢ countP p (map f (a ::ₘ t)) = card (filter (fun a => p (f a)) (a ::ₘ t))","tactic":"refine Multiset.induction_on s ?_ fun a t IH => ?_","premises":[{"full_name":"Multiset.induction_on","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[152,18],"def_end_pos":[152,30]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","commit":"","full_name":"Real.rpow_lt_one_iff_of_pos","start":[652,0],"end":[653,86],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","tactics":[{"state_before":"x y z : ℝ\nn : ℕ\nhx : 0 < x\n⊢ x ^ y < 1 ↔ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y","state_after":"no goals","tactic":"rw [rpow_def_of_pos hx, exp_lt_one_iff, mul_neg_iff, log_pos_iff hx, log_neg_iff hx]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Real.exp_lt_one_iff","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1035,8],"def_end_pos":[1035,22]},{"full_name":"Real.log_neg_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[153,8],"def_end_pos":[153,19]},{"full_name":"Real.log_pos_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[141,8],"def_end_pos":[141,19]},{"full_name":"Real.rpow_def_of_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[49,8],"def_end_pos":[49,23]},{"full_name":"mul_neg_iff","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[960,6],"def_end_pos":[960,17]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.contained_proj","start":[860,0],"end":[862,23],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\n⊢ contained (π C fun x => ord I x < o) o","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nx w✝ : I → Bool\nleft✝ : w✝ ∈ C\nh : Proj (fun x => ord I x < o) w✝ = x\nj : I\nhj : x j = true\n⊢ ord I j < o","tactic":"intro x ⟨_, _, h⟩ j hj","premises":[]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nx w✝ : I → Bool\nleft✝ : w✝ ∈ C\nh : Proj (fun x => ord I x < o) w✝ = x\nj : I\nhj : x j = true\n⊢ ord I j < o","state_after":"no goals","tactic":"aesop (add simp Proj)","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Stirling.lean","commit":"","full_name":"Stirling.log_stirlingSeq_bounded_by_constant","start":[157,0],"end":[160,64],"file_path":"Mathlib/Analysis/SpecialFunctions/Stirling.lean","tactics":[{"state_before":"⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))","state_after":"case intro\nd : ℝ\nh : ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ d\n⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))","tactic":"obtain ⟨d, h⟩ := log_stirlingSeq_bounded_aux","premises":[{"full_name":"Stirling.log_stirlingSeq_bounded_aux","def_path":"Mathlib/Analysis/SpecialFunctions/Stirling.lean","def_pos":[137,8],"def_end_pos":[137,35]}]},{"state_before":"case intro\nd : ℝ\nh : ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ d\n⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))","state_after":"no goals","tactic":"exact ⟨log (stirlingSeq 1) - d, fun n => sub_le_comm.mp (h n)⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Real.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[39,18],"def_end_pos":[39,21]},{"full_name":"Stirling.stirlingSeq","def_path":"Mathlib/Analysis/SpecialFunctions/Stirling.lean","def_pos":[49,18],"def_end_pos":[49,29]},{"full_name":"sub_le_comm","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[562,2],"def_end_pos":[562,13]}]}]}
{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.Finite.preimage'","start":[771,0],"end":[774,31],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : s.Finite\nhf : ∀ b ∈ s, (f ⁻¹' {b}).Finite\n⊢ (f ⁻¹' s).Finite","state_after":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : s.Finite\nhf : ∀ b ∈ s, (f ⁻¹' {b}).Finite\n⊢ (⋃ y ∈ s, f ⁻¹' {y}).Finite","tactic":"rw [← Set.biUnion_preimage_singleton]","premises":[{"full_name":"Set.biUnion_preimage_singleton","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1469,8],"def_end_pos":[1469,34]}]},{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : s.Finite\nhf : ∀ b ∈ s, (f ⁻¹' {b}).Finite\n⊢ (⋃ y ∈ s, f ⁻¹' {y}).Finite","state_after":"no goals","tactic":"exact Set.Finite.biUnion h hf","premises":[{"full_name":"Set.Finite.biUnion","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[683,8],"def_end_pos":[683,22]}]}]}
{"url":"Mathlib/CategoryTheory/Opposites.lean","commit":"","full_name":"CategoryTheory.Functor.opUnopEquiv_unitIso","start":[550,0],"end":[564,55],"file_path":"Mathlib/CategoryTheory/Opposites.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\n⊢ ∀ {X Y : (C ⥤ D)ᵒᵖ} (f : X ⟶ Y),\n    (𝟭 (C ⥤ D)ᵒᵖ).map f ≫ ((fun F => (unop F).opUnopIso.op) Y).hom =\n      ((fun F => (unop F).opUnopIso.op) X).hom ≫ (opHom C D ⋙ opInv C D).map f","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ (𝟭 (C ⥤ D)ᵒᵖ).map f ≫ ((fun F => (unop F).opUnopIso.op) G).hom =\n    ((fun F => (unop F).opUnopIso.op) F).hom ≫ (opHom C D ⋙ opInv C D).map f","tactic":"intro F G f","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ (𝟭 (C ⥤ D)ᵒᵖ).map f ≫ ((fun F => (unop F).opUnopIso.op) G).hom =\n    ((fun F => (unop F).opUnopIso.op) F).hom ≫ (opHom C D ⋙ opInv C D).map f","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ f ≫ (NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom.op =\n    (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom.op ≫\n      Quiver.Hom.op { app := fun X => f.unop.app X, naturality := ⋯ }","tactic":"dsimp [opUnopIso]","premises":[{"full_name":"CategoryTheory.Functor.opUnopIso","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[173,4],"def_end_pos":[173,13]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ f ≫ (NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom.op =\n    (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom.op ≫\n      Quiver.Hom.op { app := fun X => f.unop.app X, naturality := ⋯ }","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ ((NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom ≫ f.unop).op =\n    ({ app := fun X => f.unop.op.unop.app X, naturality := ⋯ } ≫\n        (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom).op","tactic":"rw [show f = f.unop.op by simp, ← op_comp, ← op_comp]","premises":[{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"Quiver.Hom.unop","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[139,4],"def_end_pos":[139,12]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ ((NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom ≫ f.unop).op =\n    ({ app := fun X => f.unop.op.unop.app X, naturality := ⋯ } ≫\n        (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom).op","state_after":"case e_f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ (NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom ≫ f.unop =\n    { app := fun X => f.unop.op.unop.app X, naturality := ⋯ } ≫\n      (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ (NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom ≫ f.unop =\n    { app := fun X => f.unop.op.unop.app X, naturality := ⋯ } ≫\n      (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Order/Hom/Basic.lean","commit":"","full_name":"Disjoint.map_orderIso","start":[1088,0],"end":[1092,28],"file_path":"Mathlib/Order/Hom/Basic.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ Disjoint (f a) (f b)","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ f (a ⊓ b) ≤ f ⊥","tactic":"rw [disjoint_iff_inf_le, ← f.map_inf, ← f.map_bot]","premises":[{"full_name":"OrderIso.map_bot","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[1060,8],"def_end_pos":[1060,24]},{"full_name":"OrderIso.map_inf","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[1078,8],"def_end_pos":[1078,24]},{"full_name":"disjoint_iff_inf_le","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[113,8],"def_end_pos":[113,27]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ f (a ⊓ b) ≤ f ⊥","state_after":"no goals","tactic":"exact f.monotone ha.le_bot","premises":[{"full_name":"Disjoint.le_bot","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[119,8],"def_end_pos":[119,23]},{"full_name":"OrderIso.monotone","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[913,18],"def_end_pos":[913,26]}]}]}
{"url":"Mathlib/Algebra/CubicDiscriminant.lean","commit":"","full_name":"Cubic.degree_of_b_ne_zero","start":[265,0],"end":[267,43],"file_path":"Mathlib/Algebra/CubicDiscriminant.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b ≠ 0\n⊢ P.toPoly.degree = 2","state_after":"no goals","tactic":"rw [of_a_eq_zero ha, degree_quadratic hb]","premises":[{"full_name":"Cubic.of_a_eq_zero","def_path":"Mathlib/Algebra/CubicDiscriminant.lean","def_pos":[119,8],"def_end_pos":[119,20]},{"full_name":"Polynomial.degree_quadratic","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[1099,8],"def_end_pos":[1099,24]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/SetToL1.lean","commit":"","full_name":"MeasureTheory.SimpleFunc.setToSimpleFunc_zero","start":[259,0],"end":[261,79],"file_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\n⊢ setToSimpleFunc 0 f = 0","state_after":"no goals","tactic":"simp [setToSimpleFunc]","premises":[{"full_name":"MeasureTheory.SimpleFunc.setToSimpleFunc","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[256,4],"def_end_pos":[256,19]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean","commit":"","full_name":"EulerSine.tendsto_integral_cos_pow_mul_div","start":[266,0],"end":[283,59],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean","tactics":[{"state_before":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\n⊢ Tendsto (fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * f x) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) atTop (𝓝 (f 0))","state_after":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","tactic":"simp_rw [div_eq_inv_mul (α := ℂ), ← Complex.ofReal_inv, integral_of_le pi_div_two_pos.le,\n    ← MeasureTheory.integral_Icc_eq_integral_Ioc, ← Complex.ofReal_pow, ← Complex.real_smul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.ofReal_inv","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[677,8],"def_end_pos":[677,18]},{"full_name":"Complex.ofReal_pow","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[650,8],"def_end_pos":[650,18]},{"full_name":"Complex.real_smul","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[313,8],"def_end_pos":[313,17]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.integral_Icc_eq_integral_Ioc","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[704,8],"def_end_pos":[704,36]},{"full_name":"Real.pi_div_two_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[157,8],"def_end_pos":[157,22]},{"full_name":"div_eq_inv_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[522,8],"def_end_pos":[522,22]},{"full_name":"intervalIntegral.integral_of_le","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[430,8],"def_end_pos":[430,22]}]},{"state_before":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","state_after":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","tactic":"have c_lt : ∀ y : ℝ, y ∈ Icc 0 (π / 2) → y ≠ 0 → cos y < cos 0 := fun y hy hy' =>\n    cos_lt_cos_of_nonneg_of_le_pi_div_two (le_refl 0) hy.2 (lt_of_le_of_ne hy.1 hy'.symm)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[112,11],"def_end_pos":[112,14]},{"full_name":"Real.cos_lt_cos_of_nonneg_of_le_pi_div_two","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[547,8],"def_end_pos":[547,45]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Set.Icc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[54,4],"def_end_pos":[54,7]},{"full_name":"le_refl","def_path":"Mathlib/Order/Defs.lean","def_pos":[39,8],"def_end_pos":[39,15]},{"full_name":"lt_of_le_of_ne","def_path":"Mathlib/Order/Defs.lean","def_pos":[164,8],"def_end_pos":[164,22]}]},{"state_before":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","state_after":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","tactic":"have c_nonneg : ∀ x : ℝ, x ∈ Icc 0 (π / 2) → 0 ≤ cos x := fun x hx =>\n    cos_nonneg_of_mem_Icc ((Icc_subset_Icc_left (neg_nonpos_of_nonneg pi_div_two_pos.le)) hx)","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[112,11],"def_end_pos":[112,14]},{"full_name":"Real.cos_nonneg_of_mem_Icc","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[442,8],"def_end_pos":[442,29]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Real.pi_div_two_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[157,8],"def_end_pos":[157,22]},{"full_name":"Set.Icc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[54,4],"def_end_pos":[54,7]},{"full_name":"Set.Icc_subset_Icc_left","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[381,8],"def_end_pos":[381,27]},{"full_name":"neg_nonpos_of_nonneg","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[188,2],"def_end_pos":[188,13]}]},{"state_before":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","state_after":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\nc_zero_pos : 0 < cos 0\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","tactic":"have c_zero_pos : 0 < cos 0 := by rw [cos_zero]; exact zero_lt_one","premises":[{"full_name":"Real.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[112,11],"def_end_pos":[112,14]},{"full_name":"Real.cos_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[732,8],"def_end_pos":[732,16]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\nc_zero_pos : 0 < cos 0\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","state_after":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\nc_zero_pos : 0 < cos 0\nzero_mem : 0 ∈ closure (interior (Icc 0 (π / 2)))\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","tactic":"have zero_mem : (0 : ℝ) ∈ closure (interior (Icc 0 (π / 2))) := by\n    rw [interior_Icc, closure_Ioo pi_div_two_pos.ne, left_mem_Icc]\n    exact pi_div_two_pos.le","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Real.pi_div_two_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[157,8],"def_end_pos":[157,22]},{"full_name":"Set.Icc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[54,4],"def_end_pos":[54,7]},{"full_name":"Set.left_mem_Icc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[155,8],"def_end_pos":[155,20]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]},{"full_name":"closure_Ioo","def_path":"Mathlib/Topology/Order/DenselyOrdered.lean","def_pos":[48,8],"def_end_pos":[48,19]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"interior_Icc","def_path":"Mathlib/Topology/Order/DenselyOrdered.lean","def_pos":[90,8],"def_end_pos":[90,20]}]},{"state_before":"f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\nc_zero_pos : 0 < cos 0\nzero_mem : 0 ∈ closure (interior (Icc 0 (π / 2)))\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))","state_after":"no goals","tactic":"exact\n    tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn isCompact_Icc\n      continuousOn_cos c_lt c_nonneg c_zero_pos zero_mem hf","premises":[{"full_name":"CompactIccSpace.isCompact_Icc","def_path":"Mathlib/Topology/Algebra/Order/Compact.lean","def_pos":[50,2],"def_end_pos":[50,15]},{"full_name":"Real.continuousOn_cos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[98,8],"def_end_pos":[98,24]},{"full_name":"tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn","def_path":"Mathlib/MeasureTheory/Integral/PeakFunction.lean","def_pos":[381,8],"def_end_pos":[381,83]}]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieHom.isIdealMorphism_of_surjective","start":[1086,0],"end":[1088,35],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : Function.Surjective ⇑f\n⊢ f.IsIdealMorphism","state_after":"no goals","tactic":"rw [isIdealMorphism_def, f.idealRange_eq_top_of_surjective h, f.range_eq_top.mpr h,\n    LieIdeal.top_coe_lieSubalgebra]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LieHom.idealRange_eq_top_of_surjective","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1078,8],"def_end_pos":[1078,39]},{"full_name":"LieHom.isIdealMorphism_def","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1015,8],"def_end_pos":[1015,27]},{"full_name":"LieHom.range_eq_top","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1073,8],"def_end_pos":[1073,20]},{"full_name":"LieIdeal.top_coe_lieSubalgebra","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[891,8],"def_end_pos":[891,29]}]}]}
{"url":"Mathlib/Data/Set/Pointwise/Interval.lean","commit":"","full_name":"Set.image_mul_left_Ioo","start":[713,0],"end":[715,72],"file_path":"Mathlib/Data/Set/Pointwise/Interval.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a : α\nh : 0 < a\nb c : α\n⊢ (fun x => a * x) '' Ioo b c = Ioo (a * b) (a * c)","state_after":"no goals","tactic":"convert image_mul_right_Ioo b c h using 1 <;> simp only [mul_comm _ a]","premises":[{"full_name":"Set.image_mul_right_Ioo","def_path":"Mathlib/Data/Set/Pointwise/Interval.lean","def_pos":[709,8],"def_end_pos":[709,27]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.map₂Left'_nil_right","start":[2419,0],"end":[2421,77],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → Option β → γ\nas : List α\n⊢ map₂Left' f as [] = (map (fun a => f a none) as, [])","state_after":"no goals","tactic":"cases as <;> rfl","premises":[]}]}
{"url":"Mathlib/RingTheory/Polynomial/Quotient.lean","commit":"","full_name":"Polynomial.modByMonic_eq_zero_iff_quotient_eq_zero","start":[66,0],"end":[68,68],"file_path":"Mathlib/RingTheory/Polynomial/Quotient.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhq : q.Monic\n⊢ p %ₘ q = 0 ↔ (Ideal.Quotient.mk (Ideal.span {q})) p = 0","state_after":"no goals","tactic":"rw [modByMonic_eq_zero_iff_dvd hq, Ideal.Quotient.eq_zero_iff_dvd]","premises":[{"full_name":"Ideal.Quotient.eq_zero_iff_dvd","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[115,8],"def_end_pos":[115,23]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Polynomial.modByMonic_eq_zero_iff_dvd","def_path":"Mathlib/Algebra/Polynomial/Div.lean","def_pos":[367,8],"def_end_pos":[367,34]}]}]}
{"url":"Mathlib/LinearAlgebra/BilinearForm/Properties.lean","commit":"","full_name":"LinearMap.BilinForm.IsAdjointPair.mul","start":[221,0],"end":[223,54],"file_path":"Mathlib/LinearAlgebra/BilinearForm/Properties.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝¹⁸ : CommSemiring R\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹⁵ : CommRing R₁\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nM'✝ : Type u_7\nM''✝ : Type u_8\ninst✝⁹ : AddCommMonoid M'✝\ninst✝⁸ : AddCommMonoid M''✝\ninst✝⁷ : Module R M'✝\ninst✝⁶ : Module R M''✝\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nF : BilinForm R M\nM' : Type u_9\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f'✝ : M →ₗ[R] M'\ng✝ g'✝ : M' →ₗ[R] M\nM₁' : Type u_10\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R M'\nf₂ f₂' : M →ₗ[R] M'\ng₂ g₂' : M' →ₗ[R] M\nM'' : Type u_11\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nf g f' g' : Module.End R M\nh : B.IsAdjointPair B f g\nh' : B.IsAdjointPair B f' g'\nx y : M\n⊢ (B ((f * f') x)) y = (B x) ((g' * g) y)","state_after":"no goals","tactic":"rw [LinearMap.mul_apply, LinearMap.mul_apply, h, h']","premises":[{"full_name":"LinearMap.mul_apply","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[57,8],"def_end_pos":[57,17]}]}]}
{"url":"Mathlib/Data/Matrix/Invertible.lean","commit":"","full_name":"Matrix.mul_mul_invOf_self_cancel","start":[46,0],"end":[48,79],"file_path":"Mathlib/Data/Matrix/Invertible.lean","tactics":[{"state_before":"m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\nA : Matrix m n α\nB : Matrix n n α\ninst✝ : Invertible B\n⊢ A * B * ⅟B = A","state_after":"no goals","tactic":"rw [Matrix.mul_assoc, mul_invOf_self, Matrix.mul_one]","premises":[{"full_name":"Matrix.mul_assoc","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1039,18],"def_end_pos":[1039,27]},{"full_name":"Matrix.mul_one","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[998,18],"def_end_pos":[998,25]},{"full_name":"mul_invOf_self","def_path":"Mathlib/Algebra/Group/Invertible/Defs.lean","def_pos":[108,8],"def_end_pos":[108,22]}]}]}
{"url":"Mathlib/Data/Fintype/Card.lean","commit":"","full_name":"Finset.card_add_card_compl","start":[264,0],"end":[267,87],"file_path":"Mathlib/Data/Fintype/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\n⊢ s.card + sᶜ.card = Fintype.card α","state_after":"no goals","tactic":"rw [Finset.card_compl, ← Nat.add_sub_assoc (card_le_univ s), Nat.add_sub_cancel_left]","premises":[{"full_name":"Finset.card_compl","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[260,8],"def_end_pos":[260,25]},{"full_name":"Finset.card_le_univ","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[241,8],"def_end_pos":[241,27]},{"full_name":"Nat.add_sub_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[912,18],"def_end_pos":[912,31]},{"full_name":"Nat.add_sub_cancel_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[908,18],"def_end_pos":[908,37]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean","commit":"","full_name":"CongruenceSubgroup.conj_cong_is_cong","start":[224,0],"end":[229,10],"file_path":"Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean","tactics":[{"state_before":"N : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nh : IsCongruenceSubgroup Γ\n⊢ IsCongruenceSubgroup (g • Γ)","state_after":"case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ IsCongruenceSubgroup (g • Γ)","tactic":"obtain ⟨N, HN⟩ := h","premises":[]},{"state_before":"case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ IsCongruenceSubgroup (g • Γ)","state_after":"case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ Gamma ↑N ≤ g • Γ","tactic":"refine ⟨N, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ Gamma ↑N ≤ g • Γ","state_after":"case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ Gamma ↑N ≤ Γ","tactic":"rw [← Gamma_cong_eq_self N g, Subgroup.pointwise_smul_le_pointwise_smul_iff]","premises":[{"full_name":"CongruenceSubgroup.Gamma_cong_eq_self","def_path":"Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean","def_pos":[221,8],"def_end_pos":[221,26]},{"full_name":"Subgroup.pointwise_smul_le_pointwise_smul_iff","def_path":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","def_pos":[345,8],"def_end_pos":[345,44]}]},{"state_before":"case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ Gamma ↑N ≤ Γ","state_after":"no goals","tactic":"exact HN","premises":[]}]}
{"url":"Mathlib/NumberTheory/RamificationInertia.lean","commit":"","full_name":"Ideal.inertiaDeg_algebraMap","start":[187,0],"end":[198,69],"file_path":"Mathlib/NumberTheory/RamificationInertia.lean","tactics":[{"state_before":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)","state_after":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)","tactic":"nontriviality S ⧸ P using inertiaDeg_of_subsingleton, finrank_zero_of_subsingleton","premises":[{"full_name":"FiniteDimensional.finrank_zero_of_subsingleton","def_path":"Mathlib/LinearAlgebra/Dimension/Finite.lean","def_pos":[361,8],"def_end_pos":[361,54]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Ideal.inertiaDeg_of_subsingleton","def_path":"Mathlib/NumberTheory/RamificationInertia.lean","def_pos":[181,8],"def_end_pos":[181,34]}]},{"state_before":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)","state_after":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)","tactic":"have := comap_eq_of_scalar_tower_quotient (algebraMap (R ⧸ p) (S ⧸ P)).injective","premises":[{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Ideal.comap_eq_of_scalar_tower_quotient","def_path":"Mathlib/RingTheory/Ideal/Over.lean","def_pos":[131,8],"def_end_pos":[131,41]},{"full_name":"RingHom.injective","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[214,18],"def_end_pos":[214,27]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)","state_after":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ finrank (R ⧸ p) (S ⧸ P) = finrank (R ⧸ p) (S ⧸ P)","tactic":"rw [inertiaDeg, dif_pos this]","premises":[{"full_name":"Ideal.inertiaDeg","def_path":"Mathlib/NumberTheory/RamificationInertia.lean","def_pos":[170,18],"def_end_pos":[170,28]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]},{"state_before":"R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ finrank (R ⧸ p) (S ⧸ P) = finrank (R ⧸ p) (S ⧸ P)","state_after":"case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ (Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯).toAlgebra = inst✝¹","tactic":"congr","premises":[]},{"state_before":"case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ (Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯).toAlgebra = inst✝¹","state_after":"case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\nx' : R ⧸ p\nx : R\n⊢ (algebraMap (R ⧸ p) (S ⧸ P)) (Quotient.mk'' x) = (algebraMap (R ⧸ p) (S ⧸ P)) (Quotient.mk'' x)","tactic":"refine Algebra.algebra_ext _ _ fun x' => Quotient.inductionOn' x' fun x => ?_","premises":[{"full_name":"Algebra.algebra_ext","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[247,8],"def_end_pos":[247,19]},{"full_name":"Quotient.inductionOn'","def_path":"Mathlib/Data/Quot.lean","def_pos":[597,18],"def_end_pos":[597,30]}]},{"state_before":"case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\nx' : R ⧸ p\nx : R\n⊢ (algebraMap (R ⧸ p) (S ⧸ P)) (Quotient.mk'' x) = (algebraMap (R ⧸ p) (S ⧸ P)) (Quotient.mk'' x)","state_after":"case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\nx' : R ⧸ p\nx : R\n⊢ (Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯) ((Quotient.mk p) x) =\n    (algebraMap (R ⧸ p) (S ⧸ P)) ((Quotient.mk p) x)","tactic":"change Ideal.Quotient.lift p _ _ (Ideal.Quotient.mk p x) = algebraMap _ _ (Ideal.Quotient.mk p x)","premises":[{"full_name":"Ideal.Quotient.lift","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[232,4],"def_end_pos":[232,8]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\nx' : R ⧸ p\nx : R\n⊢ (Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯) ((Quotient.mk p) x) =\n    (algebraMap (R ⧸ p) (S ⧸ P)) ((Quotient.mk p) x)","state_after":"no goals","tactic":"rw [Ideal.Quotient.lift_mk, ← Ideal.Quotient.algebraMap_eq P, ← IsScalarTower.algebraMap_eq,\n    ← Ideal.Quotient.algebraMap_eq, ← IsScalarTower.algebraMap_apply]","premises":[{"full_name":"Ideal.Quotient.algebraMap_eq","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[362,8],"def_end_pos":[362,30]},{"full_name":"Ideal.Quotient.lift_mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[238,8],"def_end_pos":[238,15]},{"full_name":"IsScalarTower.algebraMap_apply","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[122,8],"def_end_pos":[122,24]},{"full_name":"IsScalarTower.algebraMap_eq","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[118,8],"def_end_pos":[118,21]}]}]}
{"url":"Mathlib/Algebra/Module/Basic.lean","commit":"","full_name":"map_inv_natCast_smul","start":[28,0],"end":[43,79],"file_path":"Mathlib/Algebra/Module/Basic.lean","tactics":[{"state_before":"α : Type u_1\nR✝ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\n⊢ f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x","state_after":"case pos\nα : Type u_1\nR✝ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\nhR : ↑n = 0\nhS : ↑n = 0\n⊢ f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x\n\ncase neg\nα : Type u_1\nR✝ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\nhR : ↑n = 0\nhS : ¬↑n = 0\n⊢ f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x\n\ncase pos\nα : Type u_1\nR✝ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\nhR : ¬↑n = 0\nhS : ↑n = 0\n⊢ f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x\n\ncase neg\nα : Type u_1\nR✝ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\nhR : ¬↑n = 0\nhS : ¬↑n = 0\n⊢ f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x","tactic":"by_cases hR : (n : R) = 0 <;> by_cases hS : (n : S) = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Analysis/PSeries.lean","commit":"","full_name":"sum_Ioc_inv_sq_le_sub","start":[378,0],"end":[395,14],"file_path":"Mathlib/Analysis/PSeries.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedField α\nk n : ℕ\nhk : k ≠ 0\nh : k ≤ n\n⊢ ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹","state_after":"case refine_1\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n : ℕ\nhk : k ≠ 0\nh : k ≤ n\n⊢ ∑ i ∈ Ioc k k, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑k)⁻¹\n\ncase refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n : ℕ\nhk : k ≠ 0\nh : k ≤ n\n⊢ ∀ (n : ℕ),\n    k ≤ n → ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹ → ∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","tactic":"refine Nat.le_induction ?_ ?_ n h","premises":[{"full_name":"Nat.le_induction","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[829,6],"def_end_pos":[829,18]}]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n : ℕ\nhk : k ≠ 0\nh : k ≤ n\n⊢ ∀ (n : ℕ),\n    k ≤ n → ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹ → ∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","tactic":"intro n hn IH","premises":[]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ∑ k ∈ Ioc k n, (↑k ^ 2)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","tactic":"rw [sum_Ioc_succ_top hn]","premises":[{"full_name":"Finset.sum_Ioc_succ_top","def_path":"Mathlib/Algebra/BigOperators/Intervals.lean","def_pos":[126,2],"def_end_pos":[126,13]}]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ∑ k ∈ Ioc k n, (↑k ^ 2)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ (↑k)⁻¹ - (↑n)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","tactic":"apply (add_le_add IH le_rfl).trans","premises":[{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ (↑k)⁻¹ - (↑n)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹","tactic":"simp only [sub_eq_add_neg, add_assoc, Nat.cast_add, Nat.cast_one, le_add_neg_iff_add_le,\n    add_le_iff_nonpos_right, neg_add_le_iff_le_add, add_zero]","premises":[{"full_name":"Nat.cast_add","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"Nat.cast_one","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_le_iff_nonpos_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[393,29],"def_end_pos":[393,52]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"le_add_neg_iff_add_le","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[147,2],"def_end_pos":[147,13]},{"full_name":"neg_add_le_iff_le_add","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[55,2],"def_end_pos":[55,13]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹","tactic":"have A : 0 < (n : α) := by simpa using hk.bot_lt.trans_le hn","premises":[{"full_name":"Ne.bot_lt","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[301,8],"def_end_pos":[301,17]}]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹","tactic":"have B : 0 < (n : α) + 1 := by linarith","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]}]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ (↑n + 1 + (↑n + 1) ^ 2) / ((↑n + 1) ^ 2 * (↑n + 1)) ≤ 1 / ↑n","tactic":"field_simp","premises":[]},{"state_before":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ (↑n + 1 + (↑n + 1) ^ 2) / ((↑n + 1) ^ 2 * (↑n + 1)) ≤ 1 / ↑n","state_after":"case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ 0 ≤ 1 * ((↑n + 1) ^ 2 * (↑n + 1)) - (↑n + 1 + (↑n + 1) ^ 2) * ↑n\n\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ 0 < (↑n + 1) ^ 2 * (↑n + 1)","tactic":"rw [div_le_div_iff _ A, ← sub_nonneg]","premises":[{"full_name":"div_le_div_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[268,8],"def_end_pos":[268,22]},{"full_name":"sub_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[480,29],"def_end_pos":[480,39]}]}]}
{"url":"Mathlib/Topology/Constructions.lean","commit":"","full_name":"mem_nhds_of_pi_mem_nhds","start":[1308,0],"end":[1310,50],"file_path":"Mathlib/Topology/Constructions.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\nι : Type u_5\nπ : ι → Type u_6\nκ : Type u_7\ninst✝ : TopologicalSpace X\nT : (i : ι) → TopologicalSpace (π i)\nf : X → (i : ι) → π i\nI : Set ι\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\nhs : I.pi s ∈ 𝓝 a\ni : ι\nhi : i ∈ I\n⊢ s i ∈ 𝓝 (a i)","state_after":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\nι : Type u_5\nπ : ι → Type u_6\nκ : Type u_7\ninst✝ : TopologicalSpace X\nT : (i : ι) → TopologicalSpace (π i)\nf : X → (i : ι) → π i\nI : Set ι\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\nhs : I.pi s ∈ Filter.pi fun i => 𝓝 (a i)\ni : ι\nhi : i ∈ I\n⊢ s i ∈ 𝓝 (a i)","tactic":"rw [nhds_pi] at hs","premises":[{"full_name":"nhds_pi","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1156,8],"def_end_pos":[1156,15]}]},{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\nι : Type u_5\nπ : ι → Type u_6\nκ : Type u_7\ninst✝ : TopologicalSpace X\nT : (i : ι) → TopologicalSpace (π i)\nf : X → (i : ι) → π i\nI : Set ι\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\nhs : I.pi s ∈ Filter.pi fun i => 𝓝 (a i)\ni : ι\nhi : i ∈ I\n⊢ s i ∈ 𝓝 (a i)","state_after":"no goals","tactic":"exact mem_of_pi_mem_pi hs hi","premises":[{"full_name":"Filter.mem_of_pi_mem_pi","def_path":"Mathlib/Order/Filter/Pi.lean","def_pos":[84,8],"def_end_pos":[84,24]}]}]}
{"url":"Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean","commit":"","full_name":"NumberField.Units.logEmbeddingQuot_injective","start":[390,0],"end":[396,95],"file_path":"Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean","tactics":[{"state_before":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\n⊢ Function.Injective ⇑(logEmbeddingQuot K)","state_after":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\n⊢ Function.Injective\n    ⇑(MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))","tactic":"unfold logEmbeddingQuot","premises":[{"full_name":"NumberField.Units.logEmbeddingQuot","def_path":"Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean","def_pos":[376,4],"def_end_pos":[376,20]}]},{"state_before":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\n⊢ Function.Injective\n    ⇑(MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))","state_after":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n  (MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))\n      a₁✝ =\n    (MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))\n      a₂✝\n⊢ a₁✝ = a₂✝","tactic":"intro _ _ h","premises":[]},{"state_before":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n  (MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))\n      a₁✝ =\n    (MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))\n      a₂✝\n⊢ a₁✝ = a₂✝","state_after":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n  (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))\n      ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₁✝)) =\n    (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))\n      ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₂✝))\n⊢ a₁✝ = a₂✝","tactic":"simp_rw [MonoidHom.toAdditive'_apply_apply, MonoidHom.coe_comp, MulEquiv.coe_toMonoidHom,\n    Function.comp_apply, EmbeddingLike.apply_eq_iff_eq] at h","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"EmbeddingLike.apply_eq_iff_eq","def_path":"Mathlib/Data/FunLike/Embedding.lean","def_pos":[143,8],"def_end_pos":[143,23]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MonoidHom.coe_comp","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[706,8],"def_end_pos":[706,26]},{"full_name":"MonoidHom.toAdditive'_apply_apply","def_path":"Mathlib/Algebra/Group/TypeTags.lean","def_pos":[498,2],"def_end_pos":[498,8]},{"full_name":"MulEquiv.coe_toMonoidHom","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[492,8],"def_end_pos":[492,23]}]},{"state_before":"K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n  (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))\n      ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₁✝)) =\n    (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))\n      ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₂✝))\n⊢ a₁✝ = a₂✝","state_after":"no goals","tactic":"exact (EmbeddingLike.apply_eq_iff_eq _).mp <| (QuotientGroup.kerLift_injective _).eq_iff.mp h","premises":[{"full_name":"EmbeddingLike.apply_eq_iff_eq","def_path":"Mathlib/Data/FunLike/Embedding.lean","def_pos":[143,8],"def_end_pos":[143,23]},{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"QuotientGroup.kerLift_injective","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[335,8],"def_end_pos":[335,25]}]}]}
{"url":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean","commit":"","full_name":"PresheafOfModules.sheafificationAdjunction_homEquiv_apply","start":[104,0],"end":[123,52],"file_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean","tactics":[{"state_before":"C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ ((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g) =\n    (sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)","tactic":"apply (SheafOfModules.toSheaf _).map_injective","premises":[{"full_name":"CategoryTheory.Functor.map_injective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[57,8],"def_end_pos":[57,21]},{"full_name":"SheafOfModules.toSheaf","def_path":"Mathlib/Algebra/Category/ModuleCat/Sheaf.lean","def_pos":[92,4],"def_end_pos":[92,11]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫\n      (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)","tactic":"rw [Functor.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫\n      (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N)).symm\n      (f ≫ g).hom =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫\n      ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv Q₀.presheaf\n            ((SheafOfModules.toSheaf R).obj N)).symm\n        g.hom","tactic":"erw [toSheaf_map_sheafificationHomEquiv_symm,\n          toSheaf_map_sheafificationHomEquiv_symm]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"PresheafOfModules.toSheaf_map_sheafificationHomEquiv_symm","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean","def_pos":[93,6],"def_end_pos":[93,45]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N)).symm\n      (f ≫ g).hom =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫\n      ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv Q₀.presheaf\n            ((SheafOfModules.toSheaf R).obj N)).symm\n        g.hom","state_after":"no goals","tactic":"apply Adjunction.homEquiv_naturality_left_symm","premises":[{"full_name":"CategoryTheory.Adjunction.homEquiv_naturality_left_symm","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[141,8],"def_end_pos":[141,37]}]},{"state_before":"C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((fun x x_1 => sheafificationHomEquiv α) P₀ N) (f ≫ g) =\n    ((fun x x_1 => sheafificationHomEquiv α) P₀ M) f ≫ (SheafOfModules.forget R ⋙ restrictScalars α).map g","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ (toPresheaf R₀).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N) (f ≫ g)) =\n    (toPresheaf R₀).map\n      (((fun x x_1 => sheafificationHomEquiv α) P₀ M) f ≫ (SheafOfModules.forget R ⋙ restrictScalars α).map g)","tactic":"apply (toPresheaf _).map_injective","premises":[{"full_name":"CategoryTheory.Functor.map_injective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[57,8],"def_end_pos":[57,21]},{"full_name":"PresheafOfModules.toPresheaf","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf.lean","def_pos":[204,4],"def_end_pos":[204,14]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ (toPresheaf R₀).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N) (f ≫ g)) =\n    (toPresheaf R₀).map\n      (((fun x x_1 => sheafificationHomEquiv α) P₀ M) f ≫ (SheafOfModules.forget R ⋙ restrictScalars α).map g)","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((sheafificationHomEquiv α) (f ≫ g)).hom = ((sheafificationHomEquiv α) f).hom ≫ g.val.hom","tactic":"dsimp [toPresheaf]","premises":[{"full_name":"PresheafOfModules.toPresheaf","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf.lean","def_pos":[204,4],"def_end_pos":[204,14]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((sheafificationHomEquiv α) (f ≫ g)).hom = ((sheafificationHomEquiv α) f).hom ≫ g.val.hom","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N))\n      ((SheafOfModules.toSheaf R).map (f ≫ g)) =\n    ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj M))\n        ((SheafOfModules.toSheaf R).map f) ≫\n      g.val.hom","tactic":"erw [sheafificationHomEquiv_hom, sheafificationHomEquiv_hom]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"PresheafOfModules.sheafificationHomEquiv_hom","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean","def_pos":[84,6],"def_end_pos":[84,32]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N))\n      ((SheafOfModules.toSheaf R).map (f ≫ g)) =\n    ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj M))\n        ((SheafOfModules.toSheaf R).map f) ≫\n      g.val.hom","state_after":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N))\n      ((SheafOfModules.toSheaf R).map f ≫ (SheafOfModules.toSheaf R).map g) =\n    ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj M))\n        ((SheafOfModules.toSheaf R).map f) ≫\n      g.val.hom","tactic":"rw [Functor.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]},{"state_before":"case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N))\n      ((SheafOfModules.toSheaf R).map f ≫ (SheafOfModules.toSheaf R).map g) =\n    ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj M))\n        ((SheafOfModules.toSheaf R).map f) ≫\n      g.val.hom","state_after":"no goals","tactic":"apply Adjunction.homEquiv_naturality_right","premises":[{"full_name":"CategoryTheory.Adjunction.homEquiv_naturality_right","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[154,8],"def_end_pos":[154,33]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","commit":"","full_name":"Substring.Valid.nextn_stop","start":[983,0],"end":[984,78],"file_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","tactics":[{"state_before":"x✝ : Substring\nh✝ : x✝.Valid\nn : Nat\nl m r : List Char\nh : ValidFor l m r x✝\n⊢ x✝.nextn n { byteIdx := x✝.bsize } = { byteIdx := x✝.bsize }","state_after":"no goals","tactic":"simp [h.bsize, h.nextn_stop]","premises":[{"full_name":"Substring.ValidFor.bsize","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[788,8],"def_end_pos":[788,13]},{"full_name":"Substring.ValidFor.nextn_stop","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[830,8],"def_end_pos":[830,18]}]}]}
{"url":"Mathlib/Data/Seq/Computation.lean","commit":"","full_name":"Computation.destruct_eq_pure","start":[102,0],"end":[111,18],"file_path":"Mathlib/Data/Seq/Computation.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\n⊢ s.destruct = Sum.inl a → s = pure a","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\n⊢ (match ↑s 0 with\n      | none => Sum.inr s.tail\n      | some a => Sum.inl a) =\n      Sum.inl a →\n    s = pure a","tactic":"dsimp [destruct]","premises":[{"full_name":"Computation.destruct","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[89,4],"def_end_pos":[89,12]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\n⊢ (match ↑s 0 with\n      | none => Sum.inr s.tail\n      | some a => Sum.inl a) =\n      Sum.inl a →\n    s = pure a","state_after":"case none\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nf0 : ↑s 0 = none\nh :\n  (match none with\n    | none => Sum.inr s.tail\n    | some a => Sum.inl a) =\n    Sum.inl a\n⊢ s = pure a\n\ncase some\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na val✝ : α\nf0 : ↑s 0 = some val✝\nh :\n  (match some val✝ with\n    | none => Sum.inr s.tail\n    | some a => Sum.inl a) =\n    Sum.inl a\n⊢ s = pure a","tactic":"induction' f0 : s.1 0 with _ <;> intro h","premises":[{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]}]}
{"url":"Mathlib/LinearAlgebra/SesquilinearForm.lean","commit":"","full_name":"LinearMap.IsAlt.eq_of_add_add_eq_zero","start":[246,0],"end":[251,28],"file_path":"Mathlib/LinearAlgebra/SesquilinearForm.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : CommSemiring R₁\ninst✝² : AddCommMonoid M₁\ninst✝¹ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\ninst✝ : IsCancelAdd M\na b c : M₁\nhAdd : a + b + c = 0\n⊢ (B a) b = (B b) c","state_after":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : CommSemiring R₁\ninst✝² : AddCommMonoid M₁\ninst✝¹ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\ninst✝ : IsCancelAdd M\na b c : M₁\nhAdd : a + b + c = 0\nthis : (B a) a + (B a) b + (B a) c = (B a) c + (B b) c + (B c) c\n⊢ (B a) b = (B b) c","tactic":"have : B a a + B a b + B a c = B a c + B b c + B c c := by\n    simp_rw [← map_add, ← map_add₂, hAdd, map_zero, LinearMap.zero_apply]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearMap.map_add₂","def_path":"Mathlib/LinearAlgebra/BilinearMap.lean","def_pos":[136,8],"def_end_pos":[136,16]},{"full_name":"LinearMap.zero_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[786,8],"def_end_pos":[786,18]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]}]},{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : CommSemiring R₁\ninst✝² : AddCommMonoid M₁\ninst✝¹ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\ninst✝ : IsCancelAdd M\na b c : M₁\nhAdd : a + b + c = 0\nthis : (B a) a + (B a) b + (B a) c = (B a) c + (B b) c + (B c) c\n⊢ (B a) b = (B b) c","state_after":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : CommSemiring R₁\ninst✝² : AddCommMonoid M₁\ninst✝¹ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\ninst✝ : IsCancelAdd M\na b c : M₁\nhAdd : a + b + c = 0\nthis : (B a) c + (B a) b = (B a) c + (B b) c\n⊢ (B a) b = (B b) c","tactic":"rw [H, H, zero_add, add_zero, add_comm] at this","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : CommSemiring R₁\ninst✝² : AddCommMonoid M₁\ninst✝¹ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\ninst✝ : IsCancelAdd M\na b c : M₁\nhAdd : a + b + c = 0\nthis : (B a) c + (B a) b = (B a) c + (B b) c\n⊢ (B a) b = (B b) c","state_after":"no goals","tactic":"exact add_left_cancel this","premises":[{"full_name":"add_left_cancel","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[214,2],"def_end_pos":[214,13]}]}]}
{"url":"Mathlib/Data/Finset/Prod.lean","commit":"","full_name":"Finset.diag_singleton","start":[355,0],"end":[356,91],"file_path":"Mathlib/Data/Finset/Prod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\na : α\n⊢ {a}.diag = {(a, a)}","state_after":"no goals","tactic":"rw [← product_sdiff_offDiag, offDiag_singleton, sdiff_empty, singleton_product_singleton]","premises":[{"full_name":"Finset.offDiag_singleton","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[353,8],"def_end_pos":[353,25]},{"full_name":"Finset.product_sdiff_offDiag","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[327,8],"def_end_pos":[327,29]},{"full_name":"Finset.sdiff_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1806,8],"def_end_pos":[1806,19]},{"full_name":"Finset.singleton_product_singleton","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[204,8],"def_end_pos":[204,35]}]}]}
{"url":"Mathlib/Analysis/NormedSpace/QuaternionExponential.lean","commit":"","full_name":"Quaternion.exp_eq","start":[101,0],"end":[106,40],"file_path":"Mathlib/Analysis/NormedSpace/QuaternionExponential.lean","tactics":[{"state_before":"q : ℍ\n⊢ exp ℝ q = exp ℝ q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im)","state_after":"q : ℍ\n⊢ Commute (↑q.re) q.im","tactic":"rw [← exp_of_re_eq_zero q.im q.im_re, ← coe_mul_eq_smul, ← exp_coe, ← exp_add_of_commute,\n    re_add_im]","premises":[{"full_name":"NormedSpace.exp_add_of_commute","def_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","def_pos":[435,8],"def_end_pos":[435,26]},{"full_name":"Quaternion.coe_mul_eq_smul","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[967,8],"def_end_pos":[967,23]},{"full_name":"Quaternion.exp_coe","def_path":"Mathlib/Analysis/NormedSpace/QuaternionExponential.lean","def_pos":[32,8],"def_end_pos":[32,15]},{"full_name":"Quaternion.exp_of_re_eq_zero","def_path":"Mathlib/Analysis/NormedSpace/QuaternionExponential.lean","def_pos":[94,8],"def_end_pos":[94,25]},{"full_name":"Quaternion.im","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[740,11],"def_end_pos":[740,13]},{"full_name":"Quaternion.im_re","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[742,16],"def_end_pos":[742,21]},{"full_name":"Quaternion.re_add_im","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[752,23],"def_end_pos":[752,32]}]},{"state_before":"q : ℍ\n⊢ Commute (↑q.re) q.im","state_after":"no goals","tactic":"exact Algebra.commutes q.re (_ : ℍ[ℝ])","premises":[{"full_name":"Algebra.commutes","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[282,8],"def_end_pos":[282,16]},{"full_name":"Quaternion","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[675,4],"def_end_pos":[675,14]},{"full_name":"QuaternionAlgebra.re","def_path":"Mathlib/Algebra/Quaternion.lean","def_pos":[57,2],"def_end_pos":[57,4]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]}]}
{"url":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","commit":"","full_name":"FractionalIdeal.mul_inv_cancel_iff","start":[113,0],"end":[114,74],"file_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\nR₁ : Type u_4\ninst✝³ : CommRing R₁\ninst✝² : IsDomain R₁\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI✝ J✝ I : FractionalIdeal R₁⁰ K\nx✝ : ∃ J, I * J = 1\nJ : FractionalIdeal R₁⁰ K\nhJ : I * J = 1\n⊢ I * I⁻¹ = 1","state_after":"no goals","tactic":"rwa [← right_inverse_eq K I J hJ]","premises":[{"full_name":"FractionalIdeal.right_inverse_eq","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[97,8],"def_end_pos":[97,24]}]}]}
{"url":"Mathlib/RingTheory/Jacobson.lean","commit":"","full_name":"Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson","start":[649,0],"end":[654,16],"file_path":"Mathlib/RingTheory/Jacobson.lean","tactics":[{"state_before":"n : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : P.IsMaximal\n⊢ ((Quotient.mk P).comp MvPolynomial.C).IsIntegral","state_after":"n : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : P.IsMaximal\n⊢ (algebraMap R (MvPolynomial (Fin n) R ⧸ P)).IsIntegral","tactic":"change RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R ⧸ P))","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"RingHom.IsIntegral","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean","def_pos":[38,4],"def_end_pos":[38,22]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"n : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : P.IsMaximal\n⊢ (algebraMap R (MvPolynomial (Fin n) R ⧸ P)).IsIntegral","state_after":"case hP\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : P.IsMaximal\n⊢ P.IsMaximal","tactic":"apply quotient_mk_comp_C_isIntegral_of_jacobson'","premises":[{"full_name":"_private.Mathlib.RingTheory.Jacobson.0.Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson'","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[640,16],"def_end_pos":[640,58]}]},{"state_before":"case hP\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : P.IsMaximal\n⊢ P.IsMaximal","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/GroupTheory/Congruence/Basic.lean","commit":"","full_name":"Con.lift_range","start":[806,0],"end":[813,86],"file_path":"Mathlib/GroupTheory/Congruence/Basic.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf : M →* P\nH : c ≤ ker f\nx : P\n⊢ x ∈ MonoidHom.mrange (c.lift f H) → x ∈ MonoidHom.mrange f","state_after":"case intro.mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y✝ : M\nf : M →* P\nH : c ≤ ker f\nx : P\nw✝ : c.Quotient\ny : M\nhy : (c.lift f H) (Quot.mk r y) = x\n⊢ x ∈ MonoidHom.mrange f","tactic":"rintro ⟨⟨y⟩, hy⟩","premises":[]},{"state_before":"case intro.mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y✝ : M\nf : M →* P\nH : c ≤ ker f\nx : P\nw✝ : c.Quotient\ny : M\nhy : (c.lift f H) (Quot.mk r y) = x\n⊢ x ∈ MonoidHom.mrange f","state_after":"no goals","tactic":"exact ⟨y, hy⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/Analysis/Convex/Between.lean","commit":"","full_name":"mem_vsub_const_affineSegment","start":[111,0],"end":[114,78],"file_path":"Mathlib/Analysis/Convex/Between.lean","tactics":[{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z p : P\n⊢ z -ᵥ p ∈ affineSegment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affineSegment R x y","state_after":"no goals","tactic":"rw [← affineSegment_vsub_const_image, (vsub_left_injective p).mem_set_image]","premises":[{"full_name":"Function.Injective.mem_set_image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[187,8],"def_end_pos":[187,47]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"affineSegment_vsub_const_image","def_path":"Mathlib/Analysis/Convex/Between.lean","def_pos":[90,8],"def_end_pos":[90,38]},{"full_name":"vsub_left_injective","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[193,8],"def_end_pos":[193,27]}]}]}
{"url":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","commit":"","full_name":"PMF.toOuterMeasure_apply_singleton","start":[156,0],"end":[159,60],"file_path":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns t : Set α\na : α\n⊢ p.toOuterMeasure {a} = p a","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns t : Set α\na b : α\nhb : b ≠ a\n⊢ {a}.indicator (⇑p) b = 0\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns t : Set α\na : α\n⊢ {a}.indicator (⇑p) a = p a","tactic":"refine (p.toOuterMeasure_apply {a}).trans ((tsum_eq_single a fun b hb => ?_).trans ?_)","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"PMF.toOuterMeasure_apply","def_path":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","def_pos":[140,8],"def_end_pos":[140,28]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"tsum_eq_single","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]}]}]}
{"url":"Mathlib/Analysis/Normed/Lp/LpEquiv.lean","commit":"","full_name":"equiv_lpPiLp_norm","start":[84,0],"end":[88,70],"file_path":"Mathlib/Analysis/Normed/Lp/LpEquiv.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Fintype α\nf : ↥(lp E p)\n⊢ ‖Equiv.lpPiLp f‖ = ‖f‖","state_after":"case inl\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Fintype α\nf : ↥(lp E 0)\n⊢ ‖Equiv.lpPiLp f‖ = ‖f‖\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Fintype α\nf : ↥(lp E ⊤)\n⊢ ‖Equiv.lpPiLp f‖ = ‖f‖\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Fintype α\nf : ↥(lp E p)\nh : 0 < p.toReal\n⊢ ‖Equiv.lpPiLp f‖ = ‖f‖","tactic":"rcases p.trichotomy with (rfl | rfl | h)","premises":[{"full_name":"ENNReal.trichotomy","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[412,18],"def_end_pos":[412,28]}]}]}
{"url":"Mathlib/Topology/Order/LeftRightNhds.lean","commit":"","full_name":"LinearOrderedAddCommGroup.tendsto_nhds","start":[302,0],"end":[304,45],"file_path":"Mathlib/Topology/Order/LeftRightNhds.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : OrderTopology α\nl : Filter β\nf g : β → α\nx : Filter β\na : α\n⊢ Tendsto f x (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ (b : β) in x, |f b - a| < ε","state_after":"no goals","tactic":"simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]","premises":[{"full_name":"abs_sub_comm","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[72,2],"def_end_pos":[72,13]},{"full_name":"nhds_eq_iInf_abs_sub","def_path":"Mathlib/Topology/Order/LeftRightNhds.lean","def_pos":[289,8],"def_end_pos":[289,28]}]}]}
{"url":"Mathlib/GroupTheory/QuotientGroup.lean","commit":"","full_name":"QuotientGroup.quotientKerEquivOfRightInverse_symm_apply","start":[366,0],"end":[375,21],"file_path":"Mathlib/GroupTheory/QuotientGroup.lean","tactics":[{"state_before":"G : Type u\ninst✝² : Group G\nN : Subgroup G\nnN : N.Normal\nH : Type v\ninst✝¹ : Group H\nM : Type x\ninst✝ : Monoid M\nφ : G →* H\nψ : H → G\nhφ : Function.RightInverse ψ ⇑φ\nx : G ⧸ φ.ker\n⊢ (kerLift φ) ((mk ∘ ψ) ((kerLift φ) x)) = (kerLift φ) x","state_after":"no goals","tactic":"rw [Function.comp_apply, kerLift_mk', hφ]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"QuotientGroup.kerLift_mk'","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[331,8],"def_end_pos":[331,19]}]}]}
{"url":"Mathlib/Order/CompleteLattice.lean","commit":"","full_name":"iSup_iSup_eq_right","start":[961,0],"end":[966,10],"file_path":"Mathlib/Order/CompleteLattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nβ₂ : Type u_3\nγ : Type u_4\nι : Sort u_5\nι' : Sort u_6\nκ : ι → Sort u_7\nκ' : ι' → Sort u_8\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b✝ : α\nb : β\nf : (x : β) → b = x → α\nc : β\n⊢ ∀ (j : b = c), f c j ≤ f b ⋯","state_after":"α : Type u_1\nβ : Type u_2\nβ₂ : Type u_3\nγ : Type u_4\nι : Sort u_5\nι' : Sort u_6\nκ : ι → Sort u_7\nκ' : ι' → Sort u_8\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b✝ : α\nb : β\nf : (x : β) → b = x → α\n⊢ f b ⋯ ≤ f b ⋯","tactic":"rintro rfl","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nβ₂ : Type u_3\nγ : Type u_4\nι : Sort u_5\nι' : Sort u_6\nκ : ι → Sort u_7\nκ' : ι' → Sort u_8\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b✝ : α\nb : β\nf : (x : β) → b = x → α\n⊢ f b ⋯ ≤ f b ⋯","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Lie/Weights/RootSystem.lean","commit":"","full_name":"LieAlgebra.IsKilling.chainTopCoeff_zero_right","start":[247,0],"end":[276,55],"file_path":"Mathlib/Algebra/Lie/Weights/RootSystem.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ chainTopCoeff (⇑α) 0 = 1","state_after":"K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ 1 = chainTopCoeff (⇑α) 0","tactic":"symm","premises":[]},{"state_before":"K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ 1 = chainTopCoeff (⇑α) 0","state_after":"case hab\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ 1 ≤ chainTopCoeff (⇑α) 0\n\ncase hba\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"apply eq_of_le_of_not_lt","premises":[{"full_name":"eq_of_le_of_not_lt","def_path":"Mathlib/Order/Basic.lean","def_pos":[345,8],"def_end_pos":[345,26]}]},{"state_before":"case hba\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"obtain ⟨x, hx, x_ne0⟩ := (chainTop α (0 : Weight K H L)).exists_ne_zero","premises":[{"full_name":"LieModule.Weight","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[189,10],"def_end_pos":[189,16]},{"full_name":"LieModule.Weight.exists_ne_zero","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[213,6],"def_end_pos":[213,20]},{"full_name":"LieModule.chainTop","def_path":"Mathlib/Algebra/Lie/Weights/Chain.lean","def_pos":[300,4],"def_end_pos":[300,12]}]},{"state_before":"case hba.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\nh e f : L\nisSl2 : IsSl2Triple h e f\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"obtain ⟨h, e, f, isSl2, he, hf⟩ := exists_isSl2Triple_of_weight_isNonZero hα","premises":[{"full_name":"LieAlgebra.IsKilling.exists_isSl2Triple_of_weight_isNonZero","def_path":"Mathlib/Algebra/Lie/Weights/Killing.lean","def_pos":[472,6],"def_end_pos":[472,44]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\nh e f : L\nisSl2 : IsSl2Triple h e f\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"obtain rfl := isSl2.h_eq_coroot hα he hf","premises":[{"full_name":"IsSl2Triple.h_eq_coroot","def_path":"Mathlib/Algebra/Lie/Weights/Killing.lean","def_pos":[498,6],"def_end_pos":[498,36]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"have prim : isSl2.HasPrimitiveVectorWith x (chainLength α (0 : Weight K H L) : K) :=\n    have := lie_mem_weightSpace_of_mem_weightSpace he hx\n    ⟨x_ne0, (chainLength_smul _ _ hx).symm, by rwa [weightSpace_add_chainTop _ _ hα] at this⟩","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"IsSl2Triple.HasPrimitiveVectorWith","def_path":"Mathlib/Algebra/Lie/Sl2.lean","def_pos":[77,10],"def_end_pos":[77,32]},{"full_name":"LieAlgebra.IsKilling.chainLength","def_path":"Mathlib/Algebra/Lie/Weights/RootSystem.lean","def_pos":[60,4],"def_end_pos":[60,15]},{"full_name":"LieAlgebra.IsKilling.chainLength_smul","def_path":"Mathlib/Algebra/Lie/Weights/RootSystem.lean","def_pos":[79,6],"def_end_pos":[79,22]},{"full_name":"LieAlgebra.lie_mem_weightSpace_of_mem_weightSpace","def_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","def_pos":[58,8],"def_end_pos":[58,46]},{"full_name":"LieModule.Weight","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[189,10],"def_end_pos":[189,16]},{"full_name":"LieModule.weightSpace_add_chainTop","def_path":"Mathlib/Algebra/Lie/Weights/Chain.lean","def_pos":[320,6],"def_end_pos":[320,30]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : k • f = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1)) x\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"obtain ⟨k, hk⟩ : ∃ k : K, k • f =\n      (toEnd K L L f ^ (chainTopCoeff α (0 : Weight K H L) + 1)) x := by\n    have : (toEnd K L L f ^ (chainTopCoeff α (0 : Weight K H L) + 1)) x ∈ rootSpace H (-α) := by\n      convert toEnd_pow_apply_mem hf hx (chainTopCoeff α (0 : Weight K H L) + 1) using 2\n      rw [coe_chainTop', Weight.coe_zero, add_zero, succ_nsmul',\n        add_assoc, smul_neg, neg_add_self, add_zero]\n    simpa using (finrank_eq_one_iff_of_nonzero' ⟨f, hf⟩ (by simpa using isSl2.f_ne_zero)).mp\n      (finrank_rootSpace_eq_one _ hα.neg) ⟨_, this⟩","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsSl2Triple.f_ne_zero","def_path":"Mathlib/Algebra/Lie/Sl2.lean","def_pos":[67,6],"def_end_pos":[67,15]},{"full_name":"LieAlgebra.IsKilling.finrank_rootSpace_eq_one","def_path":"Mathlib/Algebra/Lie/Weights/Killing.lean","def_pos":[522,6],"def_end_pos":[522,30]},{"full_name":"LieAlgebra.rootSpace","def_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","def_pos":[44,7],"def_end_pos":[44,16]},{"full_name":"LieAlgebra.toEnd_pow_apply_mem","def_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","def_pos":[68,6],"def_end_pos":[68,25]},{"full_name":"LieModule.Weight","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[189,10],"def_end_pos":[189,16]},{"full_name":"LieModule.Weight.IsNonZero.neg","def_path":"Mathlib/Algebra/Lie/Weights/Killing.lean","def_pos":[583,6],"def_end_pos":[583,19]},{"full_name":"LieModule.Weight.coe_zero","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[224,6],"def_end_pos":[224,14]},{"full_name":"LieModule.chainTopCoeff","def_path":"Mathlib/Algebra/Lie/Weights/Chain.lean","def_pos":[229,4],"def_end_pos":[229,17]},{"full_name":"LieModule.coe_chainTop'","def_path":"Mathlib/Algebra/Lie/Weights/Chain.lean","def_pos":[308,6],"def_end_pos":[308,19]},{"full_name":"LieModule.toEnd","def_path":"Mathlib/Algebra/Lie/OfAssociative.lean","def_pos":[178,4],"def_end_pos":[178,19]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"finrank_eq_one_iff_of_nonzero'","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[761,8],"def_end_pos":[761,38]},{"full_name":"neg_add_self","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1042,2],"def_end_pos":[1042,13]},{"full_name":"smul_neg","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[276,8],"def_end_pos":[276,16]},{"full_name":"succ_nsmul'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,14],"def_end_pos":[573,25]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : ⁅f, k • f⁆ = ⁅f, ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1)) x⁆\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\n⊢ ¬1 < chainTopCoeff (⇑α) 0","tactic":"simp only [lie_smul, lie_self, smul_zero, prim.lie_f_pow_toEnd_f] at hk","premises":[{"full_name":"IsSl2Triple.HasPrimitiveVectorWith.lie_f_pow_toEnd_f","def_path":"Mathlib/Algebra/Lie/Sl2.lean","def_pos":[113,6],"def_end_pos":[113,23]},{"full_name":"lie_self","def_path":"Mathlib/Algebra/Lie/Basic.lean","def_pos":[131,8],"def_end_pos":[131,16]},{"full_name":"lie_smul","def_path":"Mathlib/Algebra/Lie/Basic.lean","def_pos":[116,8],"def_end_pos":[116,16]},{"full_name":"smul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[63,8],"def_end_pos":[63,17]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\n⊢ ¬1 < chainTopCoeff (⇑α) 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\n⊢ False","tactic":"intro e","premises":[]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\n⊢ False","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0","tactic":"refine prim.pow_toEnd_f_ne_zero_of_eq_nat rfl ?_ hk.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"IsSl2Triple.HasPrimitiveVectorWith.pow_toEnd_f_ne_zero_of_eq_nat","def_path":"Mathlib/Algebra/Lie/Sl2.lean","def_pos":[148,6],"def_end_pos":[148,35]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\nthis : ↑(↑(chainLength α 0) - 2 * ↑(chainTopCoeff (⇑α) 0)) = 0 (coroot α)\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0","tactic":"have := (apply_coroot_eq_cast' α 0).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"LieAlgebra.IsKilling.apply_coroot_eq_cast'","def_path":"Mathlib/Algebra/Lie/Weights/RootSystem.lean","def_pos":[83,6],"def_end_pos":[83,27]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\nthis : ↑(↑(chainLength α 0) - 2 * ↑(chainTopCoeff (⇑α) 0)) = 0 (coroot α)\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0","state_after":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\nthis : chainLength α 0 = 2 * chainTopCoeff (⇑α) 0\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0","tactic":"simp only [← @Nat.cast_two ℤ, ← Nat.cast_mul, Weight.zero_apply, Int.cast_eq_zero, sub_eq_zero,\n    Nat.cast_inj] at this","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.cast_eq_zero","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[57,14],"def_end_pos":[57,26]},{"full_name":"LieModule.Weight.zero_apply","def_path":"Mathlib/Algebra/Lie/Weights/Basic.lean","def_pos":[226,6],"def_end_pos":[226,16]},{"full_name":"Nat.cast_inj","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"Nat.cast_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[56,25],"def_end_pos":[56,33]},{"full_name":"Nat.cast_two","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[168,8],"def_end_pos":[168,16]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\nthis : chainLength α 0 = 2 * chainTopCoeff (⇑α) 0\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0","state_after":"no goals","tactic":"rwa [this, Nat.succ_le, two_mul, add_lt_add_iff_left]","premises":[{"full_name":"Nat.succ_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[616,8],"def_end_pos":[616,15]},{"full_name":"add_lt_add_iff_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[104,2],"def_end_pos":[104,13]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]}]}]}
{"url":"Mathlib/Algebra/Module/Defs.lean","commit":"","full_name":"map_intCast_smul","start":[400,0],"end":[403,87],"file_path":"Mathlib/Algebra/Module/Defs.lean","tactics":[{"state_before":"α : Type u_1\nR✝ : Type u_2\nk : Type u_3\nS✝ : Type u_4\nM : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\nι : Type u_8\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\nF : Type u_9\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_10\nS : Type u_11\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nx : ℤ\na : M\n⊢ f (↑x • a) = ↑x • f a","state_after":"no goals","tactic":"simp only [Int.cast_smul_eq_nsmul, map_zsmul]","premises":[{"full_name":"Int.cast_smul_eq_nsmul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[373,6],"def_end_pos":[373,28]},{"full_name":"map_zsmul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[444,2],"def_end_pos":[444,13]}]}]}
{"url":"Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean","commit":"","full_name":"CliffordAlgebraDualNumber.ι_mul_ι","start":[359,0],"end":[362,14],"file_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr₁ r₂ : R\n⊢ (ι 0) r₁ * (ι 0) r₂ = 0","state_after":"no goals","tactic":"rw [← mul_one r₁, ← mul_one r₂, ← smul_eq_mul R, ← smul_eq_mul R, LinearMap.map_smul,\n    LinearMap.map_smul, smul_mul_smul, ι_sq_scalar, QuadraticMap.zero_apply, RingHom.map_zero,\n    smul_zero]","premises":[{"full_name":"CliffordAlgebra.ι_sq_scalar","def_path":"Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean","def_pos":[109,8],"def_end_pos":[109,19]},{"full_name":"LinearMap.map_smul","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[348,18],"def_end_pos":[348,26]},{"full_name":"QuadraticMap.zero_apply","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[420,8],"def_end_pos":[420,18]},{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]},{"full_name":"smul_mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[450,6],"def_end_pos":[450,19]},{"full_name":"smul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[63,8],"def_end_pos":[63,17]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Basic.lean","commit":"","full_name":"Set.countable_infinite_iff_nonempty_denumerable","start":[1508,0],"end":[1510,74],"file_path":"Mathlib/SetTheory/Cardinal/Basic.lean","tactics":[{"state_before":"α✝ β : Type u\nα : Type u_1\ns : Set α\n⊢ s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable ↑s)","state_after":"no goals","tactic":"rw [nonempty_denumerable_iff, ← Set.infinite_coe_iff, countable_coe_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.countable_coe_iff","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[46,8],"def_end_pos":[46,25]},{"full_name":"Set.infinite_coe_iff","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[1158,8],"def_end_pos":[1158,24]},{"full_name":"nonempty_denumerable_iff","def_path":"Mathlib/Logic/Denumerable.lean","def_pos":[338,8],"def_end_pos":[338,32]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Splits.lean","commit":"","full_name":"Polynomial.splits_prod_iff","start":[227,0],"end":[235,94],"file_path":"Mathlib/Algebra/Polynomial/Splits.lean","tactics":[{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt : Finset ι\n⊢ (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))","state_after":"no goals","tactic":"classical\n  refine\n    Finset.induction_on t (fun _ =>\n        ⟨fun _ _ h => by simp only [Finset.not_mem_empty] at h, fun _ => splits_one i⟩)\n      fun a t hat ih ht => ?_\n  rw [Finset.forall_mem_insert] at ht ⊢\n  rw [Finset.prod_insert hat, splits_mul_iff i ht.1 (Finset.prod_ne_zero_iff.2 ht.2), ih ht.2]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.forall_mem_insert","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2532,8],"def_end_pos":[2532,25]},{"full_name":"Finset.induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1076,18],"def_end_pos":[1076,30]},{"full_name":"Finset.not_mem_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[477,8],"def_end_pos":[477,21]},{"full_name":"Finset.prod_insert","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[329,8],"def_end_pos":[329,19]},{"full_name":"Finset.prod_ne_zero_iff","def_path":"Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean","def_pos":[49,6],"def_end_pos":[49,22]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Polynomial.splits_mul_iff","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[223,8],"def_end_pos":[223,22]},{"full_name":"Polynomial.splits_one","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[112,8],"def_end_pos":[112,18]}]}]}
{"url":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","commit":"","full_name":"MeasureTheory.UnifIntegrable.indicator","start":[135,0],"end":[145,41],"file_path":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (i : ι) (s : Set α),\n        MeasurableSet s →\n          μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε✝ : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhε :\n  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (i : ι) (s : Set α),\n        MeasurableSet s →\n          μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε","tactic":"obtain ⟨δ, hδ_pos, hε⟩ := hf hε","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε✝ : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhε :\n  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ,\n    ∃ (_ : 0 < δ),\n      ∀ (i : ι) (s : Set α),\n        MeasurableSet s →\n          μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε✝ : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhε :\n  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε","tactic":"refine ⟨δ, hδ_pos, fun i s hs hμs ↦ ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε✝ : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhε :\n  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε","state_after":"no goals","tactic":"calc\n    eLpNorm (s.indicator (E.indicator (f i))) p μ\n      = eLpNorm (E.indicator (s.indicator (f i))) p μ := by\n      simp only [indicator_indicator, inter_comm]\n    _ ≤ eLpNorm (s.indicator (f i)) p μ := eLpNorm_indicator_le _\n    _ ≤ ENNReal.ofReal ε := hε _ _ hs hμs","premises":[{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]},{"full_name":"MeasureTheory.eLpNorm_indicator_le","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[537,8],"def_end_pos":[537,28]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]},{"full_name":"Set.indicator_indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[171,2],"def_end_pos":[171,13]},{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]}]}]}
{"url":"Mathlib/Topology/ContinuousOn.lean","commit":"","full_name":"mem_closure_ne_iff_frequently_within","start":[51,0],"end":[53,62],"file_path":"Mathlib/Topology/ContinuousOn.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : TopologicalSpace α\nz : α\ns : Set α\n⊢ z ∈ closure (s \\ {z}) ↔ ∃ᶠ (x : α) in 𝓝[≠] z, x ∈ s","state_after":"no goals","tactic":"simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]","premises":[{"full_name":"frequently_nhdsWithin_iff","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[47,8],"def_end_pos":[47,33]},{"full_name":"mem_closure_iff_frequently","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1053,8],"def_end_pos":[1053,34]}]}]}
{"url":"Mathlib/Data/List/Perm.lean","commit":"","full_name":"List.subperm_singleton_iff","start":[128,0],"end":[134,40],"file_path":"Mathlib/Data/List/Perm.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl l₁ l₂ : List α\na : α\n⊢ l <+~ [a] ↔ l = [] ∨ l = [a]","state_after":"case mp\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ : List α\na : α\n⊢ l <+~ [a] → l = [] ∨ l = [a]\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ : List α\na : α\n⊢ l = [] ∨ l = [a] → l <+~ [a]","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","commit":"","full_name":"CategoryTheory.IsPushout.zero_left","start":[881,0],"end":[885,43],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nZ X✝ Y P : C\nf : Z ⟶ X✝\ng : Z ⟶ Y\ninl : X✝ ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ CommSq 0 0 (𝟙 X) 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nZ X✝ Y P : C\nf : Z ⟶ X✝\ng : Z ⟶ Y\ninl : X✝ ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ 𝟙 X ≫ ((coprodZeroIso X).symm ≪≫ (pushoutZeroZeroIso X 0).symm).hom = pushout.inl 0 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nZ X✝ Y P : C\nf : Z ⟶ X✝\ng : Z ⟶ Y\ninl : X✝ ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ 0 ≫ ((coprodZeroIso X).symm ≪≫ (pushoutZeroZeroIso X 0).symm).hom = pushout.inr 0 0","state_after":"no goals","tactic":"simp [eq_iff_true_of_subsingleton]","premises":[{"full_name":"eq_iff_true_of_subsingleton","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[132,8],"def_end_pos":[132,35]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","commit":"","full_name":"Polynomial.div_tendsto_atTop_of_degree_gt'","start":[156,0],"end":[166,7],"file_path":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.degree < P.degree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\n⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop","state_after":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.degree < P.degree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop","tactic":"have hQ : Q ≠ 0 := fun h => by\n    simp only [h, div_zero, leadingCoeff_zero] at hpos\n    exact hpos.false","premises":[{"full_name":"LT.lt.false","def_path":"Mathlib/Order/Basic.lean","def_pos":[264,18],"def_end_pos":[264,23]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.leadingCoeff_zero","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[573,8],"def_end_pos":[573,25]},{"full_name":"div_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[301,8],"def_end_pos":[301,16]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.degree < P.degree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop","state_after":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop","tactic":"rw [← natDegree_lt_natDegree_iff hQ] at hdeg","premises":[{"full_name":"Polynomial.natDegree_lt_natDegree_iff","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[532,8],"def_end_pos":[532,34]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop","state_after":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop","tactic":"refine (isEquivalent_atTop_div P Q).symm.tendsto_atTop ?_","premises":[{"full_name":"Asymptotics.IsEquivalent.symm","def_path":"Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean","def_pos":[99,8],"def_end_pos":[99,25]},{"full_name":"Asymptotics.IsEquivalent.tendsto_atTop","def_path":"Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean","def_pos":[280,8],"def_end_pos":[280,34]},{"full_name":"Polynomial.isEquivalent_atTop_div","def_path":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","def_pos":[112,8],"def_end_pos":[112,30]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop","state_after":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop","tactic":"apply Tendsto.const_mul_atTop hpos","premises":[{"full_name":"Filter.Tendsto.const_mul_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[926,8],"def_end_pos":[926,31]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop","state_after":"case hn\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ 0 < ↑P.natDegree - ↑Q.natDegree","tactic":"apply tendsto_zpow_atTop_atTop","premises":[{"full_name":"Filter.tendsto_zpow_atTop_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[955,6],"def_end_pos":[955,30]}]},{"state_before":"case hn\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ 0 < ↑P.natDegree - ↑Q.natDegree","state_after":"no goals","tactic":"omega","premises":[]}]}
{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Complex.ofReal_cosh_ofReal_re","start":[315,0],"end":[316,54],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x✝ y : ℂ\nx : ℝ\n⊢ (starRingEnd ℂ) (cosh ↑x) = cosh ↑x","state_after":"no goals","tactic":"rw [← cosh_conj, conj_ofReal]","premises":[{"full_name":"Complex.conj_ofReal","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[472,8],"def_end_pos":[472,19]},{"full_name":"Complex.cosh_conj","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[310,8],"def_end_pos":[310,17]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"eq_inv_mul_of_mul_eq","start":[616,0],"end":[617,78],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na b c d : G\nn : ℤ\nh : b * a = c\n⊢ a = b⁻¹ * c","state_after":"no goals","tactic":"simp [h.symm]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/FixedPoint.lean","commit":"","full_name":"Ordinal.deriv_eq_enumOrd","start":[472,0],"end":[475,33],"file_path":"Mathlib/SetTheory/Ordinal/FixedPoint.lean","tactics":[{"state_before":"f : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\n⊢ deriv f = enumOrd (fixedPoints f)","state_after":"case h.e'_3.h.e'_1\nf : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\n⊢ fixedPoints f = ⋂ i, fixedPoints f","tactic":"convert derivFamily_eq_enumOrd fun _ : Unit => H","premises":[{"full_name":"Ordinal.derivFamily_eq_enumOrd","def_path":"Mathlib/SetTheory/Ordinal/FixedPoint.lean","def_pos":[203,8],"def_end_pos":[203,30]},{"full_name":"Unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[142,7],"def_end_pos":[142,11]}]},{"state_before":"case h.e'_3.h.e'_1\nf : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\n⊢ fixedPoints f = ⋂ i, fixedPoints f","state_after":"no goals","tactic":"exact (Set.iInter_const _).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Set.iInter_const","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[378,6],"def_end_pos":[378,18]}]}]}
{"url":"Mathlib/Algebra/Category/Ring/Constructions.lean","commit":"","full_name":"CommRingCat.subsingleton_of_isTerminal","start":[129,0],"end":[131,47],"file_path":"Mathlib/Algebra/Category/Ring/Constructions.lean","tactics":[{"state_before":"X : CommRingCat\nhX : IsTerminal X\n⊢ Subsingleton PUnit.{u_1 + 1}","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Control/Applicative.lean","commit":"","full_name":"Applicative.map_seq_map","start":[29,0],"end":[31,27],"file_path":"Mathlib/Control/Applicative.lean","tactics":[{"state_before":"F : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nα β γ σ : Type u\nf : α → β → γ\ng : σ → β\nx : F α\ny : F σ\n⊢ (Seq.seq (f <$> x) fun x => g <$> y) = Seq.seq (((fun x => x ∘ g) ∘ f) <$> x) fun x => y","state_after":"no goals","tactic":"simp [flip, functor_norm]","premises":[{"full_name":"flip","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[30,14],"def_end_pos":[30,18]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.tendsto_measure_Ico_atTop","start":[1843,0],"end":[1858,79],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\n⊢ Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a)))","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\n⊢ Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a)))","tactic":"haveI : Nonempty α := ⟨a⟩","premises":[{"full_name":"Nonempty","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[709,16],"def_end_pos":[709,24]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\n⊢ Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a)))","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\n⊢ Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a)))","tactic":"have h_mono : Monotone fun x => μ (Ico a x) := fun i j hij => by simp only; gcongr","premises":[{"full_name":"Monotone","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[76,4],"def_end_pos":[76,12]},{"full_name":"Set.Ico","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[46,4],"def_end_pos":[46,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\n⊢ Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a)))","state_after":"case h.e'_5.h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)","tactic":"convert tendsto_atTop_iSup h_mono","premises":[{"full_name":"tendsto_atTop_iSup","def_path":"Mathlib/Topology/Order/MonotoneConvergence.lean","def_pos":[141,8],"def_end_pos":[141,26]}]},{"state_before":"case h.e'_5.h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)","state_after":"case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)","tactic":"obtain ⟨xs, hxs_mono, hxs_tendsto⟩ := exists_seq_monotone_tendsto_atTop_atTop α","premises":[{"full_name":"Filter.exists_seq_monotone_tendsto_atTop_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1671,8],"def_end_pos":[1671,47]}]},{"state_before":"case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)","state_after":"case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = ⋃ n, Ico a (xs n)\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)","tactic":"have h_Ici : Ici a = ⋃ n, Ico a (xs n) := by\n    ext1 x\n    simp only [mem_Ici, mem_iUnion, mem_Ico, exists_and_left, iff_self_and]\n    intro\n    obtain ⟨y, hxy⟩ := NoMaxOrder.exists_gt x\n    obtain ⟨n, hn⟩ := tendsto_atTop_atTop.mp hxs_tendsto y\n    exact ⟨n, hxy.trans_le (hn n le_rfl)⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.tendsto_atTop_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1198,8],"def_end_pos":[1198,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"NoMaxOrder.exists_gt","def_path":"Mathlib/Order/Max.lean","def_pos":[56,2],"def_end_pos":[56,11]},{"full_name":"Set.Ici","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[66,4],"def_end_pos":[66,7]},{"full_name":"Set.Ico","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[46,4],"def_end_pos":[46,7]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Set.mem_Ici","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[122,8],"def_end_pos":[122,15]},{"full_name":"Set.mem_Ico","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[102,8],"def_end_pos":[102,15]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"exists_and_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[288,16],"def_end_pos":[288,31]},{"full_name":"iff_self_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[167,16],"def_end_pos":[167,28]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = ⋃ n, Ico a (xs n)\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)","state_after":"case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = ⋃ n, Ico a (xs n)\n⊢ Directed (fun x x_1 => x ⊆ x_1) fun n => Ico a (xs n)","tactic":"rw [h_Ici, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto]","premises":[{"full_name":"MeasureTheory.measure_iUnion_eq_iSup","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[422,8],"def_end_pos":[422,30]},{"full_name":"iSup_eq_iSup_subseq_of_monotone","def_path":"Mathlib/Topology/Order/MonotoneConvergence.lean","def_pos":[291,8],"def_end_pos":[291,39]}]},{"state_before":"case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = ⋃ n, Ico a (xs n)\n⊢ Directed (fun x x_1 => x ⊆ x_1) fun n => Ico a (xs n)","state_after":"no goals","tactic":"exact Monotone.directed_le fun i j hij => Ico_subset_Ico_right (hxs_mono hij)","premises":[{"full_name":"Monotone.directed_le","def_path":"Mathlib/Order/Directed.lean","def_pos":[174,8],"def_end_pos":[174,28]},{"full_name":"Set.Ico_subset_Ico_right","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[373,8],"def_end_pos":[373,28]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","commit":"","full_name":"WeakFEPair.hf_zero'","start":[158,0],"end":[170,75],"file_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","tactics":[{"state_before":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nP : WeakFEPair E\n⊢ (fun x => P.f x - P.f₀) =O[𝓝[>] 0] fun x => x ^ (-P.k)","state_after":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nP : WeakFEPair E\n⊢ (fun x => P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀ + ((P.ε * ↑(x ^ (-P.k))) • P.g₀ - P.f₀)) =O[𝓝[>] 0] fun x => x ^ (-P.k)","tactic":"simp_rw [← fun x ↦ sub_add_sub_cancel (P.f x) ((P.ε * ↑(x ^ (-P.k))) • P.g₀) P.f₀]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"WeakFEPair.f","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[80,3],"def_end_pos":[80,4]},{"full_name":"WeakFEPair.f₀","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[86,3],"def_end_pos":[86,5]},{"full_name":"WeakFEPair.g₀","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[86,6],"def_end_pos":[86,8]},{"full_name":"WeakFEPair.k","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[82,3],"def_end_pos":[82,4]},{"full_name":"WeakFEPair.ε","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[84,3],"def_end_pos":[84,4]},{"full_name":"sub_add_sub_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[730,29],"def_end_pos":[730,47]}]},{"state_before":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nP : WeakFEPair E\n⊢ (fun x => P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀ + ((P.ε * ↑(x ^ (-P.k))) • P.g₀ - P.f₀)) =O[𝓝[>] 0] fun x => x ^ (-P.k)","state_after":"case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nP : WeakFEPair E\n⊢ (fun x => (P.ε * ↑(x ^ (-P.k))) • P.g₀) =O[𝓝[>] 0] fun x => x ^ (-P.k)\n\ncase refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nP : WeakFEPair E\n⊢ (fun x => P.f₀) =O[𝓝[>] 0] fun x => x ^ (-P.k)","tactic":"refine (P.hf_zero _).add (IsBigO.sub ?_ ?_)","premises":[{"full_name":"Asymptotics.IsBigO.add","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[898,8],"def_end_pos":[898,18]},{"full_name":"Asymptotics.IsBigO.sub","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[934,8],"def_end_pos":[934,18]},{"full_name":"WeakFEPair.hf_zero","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[136,6],"def_end_pos":[136,13]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","commit":"","full_name":"Real.rpow_lt_rpow_of_neg","start":[501,0],"end":[505,22],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","tactics":[{"state_before":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\n⊢ y ^ z < x ^ z","state_after":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ y ^ z < x ^ z","tactic":"have := hx.trans hxy","premises":[]},{"state_before":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ y ^ z < x ^ z","state_after":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ x ^ (-z) < y ^ (-z)\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ y\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase ha\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < x ^ z\n\ncase hb\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < y ^ z","tactic":"rw [← inv_lt_inv, ← rpow_neg, ← rpow_neg]","premises":[{"full_name":"Real.rpow_neg","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[218,8],"def_end_pos":[218,16]},{"full_name":"inv_lt_inv","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[167,8],"def_end_pos":[167,18]}]},{"state_before":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ x ^ (-z) < y ^ (-z)\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ y\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase ha\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < x ^ z\n\ncase hb\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < y ^ z","state_after":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ y\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase ha\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < x ^ z\n\ncase hb\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < y ^ z","tactic":"on_goal 1 => refine rpow_lt_rpow ?_ hxy (neg_pos.2 hz)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Real.rpow_lt_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[480,8],"def_end_pos":[480,20]}]},{"state_before":"x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ y\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase ha\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < x ^ z\n\ncase hb\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < y ^ z","state_after":"no goals","tactic":"all_goals positivity","premises":[]}]}
{"url":"Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean","commit":"","full_name":"AlgebraCat.forget₂_map_associator_hom","start":[58,0],"end":[64,5],"file_path":"Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat R\n⊢ (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n    (α_ ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X) ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)\n        ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Z)).hom","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.nonempty_inter_of_measure_lt_add","start":[400,0],"end":[409,50],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nht : MeasurableSet t\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ u < μ s + μ t\n⊢ (s ∩ t).Nonempty","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nht : MeasurableSet t\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ u < μ s + μ t\n⊢ ¬Disjoint s t","tactic":"rw [← Set.not_disjoint_iff_nonempty_inter]","premises":[{"full_name":"Set.not_disjoint_iff_nonempty_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1224,6],"def_end_pos":[1224,37]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nht : MeasurableSet t\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ u < μ s + μ t\n⊢ ¬Disjoint s t","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nht : MeasurableSet t\nh's : s ⊆ u\nh't : t ⊆ u\nh : Disjoint s t\n⊢ μ s + μ t ≤ μ u","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nht : MeasurableSet t\nh's : s ⊆ u\nh't : t ⊆ u\nh : Disjoint s t\n⊢ μ s + μ t ≤ μ u","state_after":"no goals","tactic":"calc\n    μ s + μ t = μ (s ∪ t) := (measure_union h ht).symm\n    _ ≤ μ u := measure_mono (union_subset h's h't)","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"MeasureTheory.measure_union","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[108,8],"def_end_pos":[108,21]},{"full_name":"Set.union_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[662,8],"def_end_pos":[662,20]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]}]}]}
{"url":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","commit":"","full_name":"CategoryTheory.Functor.isRightKanExtension_iff_isIso","start":[141,0],"end":[152,18],"file_path":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","tactics":[{"state_before":"C : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁷ : Category.{u_9, u_1} C\ninst✝⁶ : Category.{?u.29812, u_2} C'\ninst✝⁵ : Category.{u_8, u_3} H\ninst✝⁴ : Category.{?u.29820, u_4} H'\ninst✝³ : Category.{u_7, u_5} D\ninst✝² : Category.{?u.29828, u_6} D'\nF'✝ : D ⥤ H\nL✝ : C ⥤ D\nF✝ : C ⥤ H\nα✝ : L✝ ⋙ F'✝ ⟶ F✝\ninst✝¹ : F'✝.IsRightKanExtension α✝\nF' F'' : D ⥤ H\nφ : F'' ⟶ F'\nL : C ⥤ D\nF : C ⥤ H\nα : L ⋙ F' ⟶ F\nα' : L ⋙ F'' ⟶ F\ncomm : whiskerLeft L φ ≫ α = α'\ninst✝ : F'.IsRightKanExtension α\n⊢ F''.IsRightKanExtension α' ↔ IsIso φ","state_after":"case mp\nC : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁷ : Category.{u_9, u_1} C\ninst✝⁶ : Category.{?u.29812, u_2} C'\ninst✝⁵ : Category.{u_8, u_3} H\ninst✝⁴ : Category.{?u.29820, u_4} H'\ninst✝³ : Category.{u_7, u_5} D\ninst✝² : Category.{?u.29828, u_6} D'\nF'✝ : D ⥤ H\nL✝ : C ⥤ D\nF✝ : C ⥤ H\nα✝ : L✝ ⋙ F'✝ ⟶ F✝\ninst✝¹ : F'✝.IsRightKanExtension α✝\nF' F'' : D ⥤ H\nφ : F'' ⟶ F'\nL : C ⥤ D\nF : C ⥤ H\nα : L ⋙ F' ⟶ F\nα' : L ⋙ F'' ⟶ F\ncomm : whiskerLeft L φ ≫ α = α'\ninst✝ : F'.IsRightKanExtension α\n⊢ F''.IsRightKanExtension α' → IsIso φ\n\ncase mpr\nC : Type u_1\nC' : Type u_2\nH : Type u_3\nH' : Type u_4\nD : Type u_5\nD' : Type u_6\ninst✝⁷ : Category.{u_9, u_1} C\ninst✝⁶ : Category.{?u.29812, u_2} C'\ninst✝⁵ : Category.{u_8, u_3} H\ninst✝⁴ : Category.{?u.29820, u_4} H'\ninst✝³ : Category.{u_7, u_5} D\ninst✝² : Category.{?u.29828, u_6} D'\nF'✝ : D ⥤ H\nL✝ : C ⥤ D\nF✝ : C ⥤ H\nα✝ : L✝ ⋙ F'✝ ⟶ F✝\ninst✝¹ : F'✝.IsRightKanExtension α✝\nF' F'' : D ⥤ H\nφ : F'' ⟶ F'\nL : C ⥤ D\nF : C ⥤ H\nα : L ⋙ F' ⟶ F\nα' : L ⋙ F'' ⟶ F\ncomm : whiskerLeft L φ ≫ α = α'\ninst✝ : F'.IsRightKanExtension α\n⊢ IsIso φ → F''.IsRightKanExtension α'","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/BigOperators/Fin.lean","commit":"","full_name":"Fin.prod_Ioi_zero","start":[148,0],"end":[151,52],"file_path":"Mathlib/Algebra/BigOperators/Fin.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\nv : Fin n.succ → M\n⊢ ∏ i ∈ Ioi 0, v i = ∏ j : Fin n, v j.succ","state_after":"no goals","tactic":"rw [Ioi_zero_eq_map, Finset.prod_map, val_succEmb]","premises":[{"full_name":"Fin.Ioi_zero_eq_map","def_path":"Mathlib/Data/Fintype/Fin.lean","def_pos":[30,8],"def_end_pos":[30,23]},{"full_name":"Fin.val_succEmb","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[478,8],"def_end_pos":[478,19]},{"full_name":"Finset.prod_map","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[372,8],"def_end_pos":[372,16]}]}]}
{"url":"Mathlib/Order/BooleanAlgebra.lean","commit":"","full_name":"sdiff_lt","start":[290,0],"end":[293,31],"file_path":"Mathlib/Order/BooleanAlgebra.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\n⊢ x \\ y < x","state_after":"α : Type u\nβ : Type u_1\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x \\ y = x\n⊢ y = ⊥","tactic":"refine sdiff_le.lt_of_ne fun h => hy ?_","premises":[{"full_name":"sdiff_le","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[384,8],"def_end_pos":[384,16]}]},{"state_before":"α : Type u\nβ : Type u_1\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x \\ y = x\n⊢ y = ⊥","state_after":"α : Type u\nβ : Type u_1\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x ⊓ y = ⊥\n⊢ y = ⊥","tactic":"rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h","premises":[{"full_name":"disjoint_iff","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[116,8],"def_end_pos":[116,20]},{"full_name":"sdiff_eq_self_iff_disjoint'","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[287,8],"def_end_pos":[287,35]}]},{"state_before":"α : Type u\nβ : Type u_1\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x ⊓ y = ⊥\n⊢ y = ⊥","state_after":"no goals","tactic":"rw [← h, inf_eq_right.mpr hx]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"inf_eq_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[341,8],"def_end_pos":[341,20]}]}]}
{"url":"Mathlib/RingTheory/PowerSeries/Basic.lean","commit":"","full_name":"PowerSeries.span_X_isPrime","start":[711,0],"end":[719,59],"file_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (Ideal.span {X}).IsPrime","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ Ideal.span {X} = RingHom.ker (constantCoeff R)","tactic":"suffices Ideal.span ({X} : Set R⟦X⟧) = RingHom.ker (constantCoeff R) by\n    rw [this]\n    exact RingHom.ker_isPrime _","premises":[{"full_name":"Ideal.span","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[101,4],"def_end_pos":[101,8]},{"full_name":"PowerSeries","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[54,7],"def_end_pos":[54,18]},{"full_name":"PowerSeries.X","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[202,4],"def_end_pos":[202,5]},{"full_name":"PowerSeries.constantCoeff","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[192,4],"def_end_pos":[192,17]},{"full_name":"RingHom.ker","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[520,4],"def_end_pos":[520,7]},{"full_name":"RingHom.ker_isPrime","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[588,8],"def_end_pos":[588,19]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ Ideal.span {X} = RingHom.ker (constantCoeff R)","state_after":"case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ ∀ (x : R⟦X⟧), x ∈ Ideal.span {X} ↔ x ∈ RingHom.ker (constantCoeff R)","tactic":"apply Ideal.ext","premises":[{"full_name":"Ideal.ext","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[69,8],"def_end_pos":[69,11]}]},{"state_before":"case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ ∀ (x : R⟦X⟧), x ∈ Ideal.span {X} ↔ x ∈ RingHom.ker (constantCoeff R)","state_after":"case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nφ : R⟦X⟧\n⊢ φ ∈ Ideal.span {X} ↔ φ ∈ RingHom.ker (constantCoeff R)","tactic":"intro φ","premises":[]},{"state_before":"case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nφ : R⟦X⟧\n⊢ φ ∈ Ideal.span {X} ↔ φ ∈ RingHom.ker (constantCoeff R)","state_after":"no goals","tactic":"rw [RingHom.mem_ker, Ideal.mem_span_singleton, X_dvd_iff]","premises":[{"full_name":"Ideal.mem_span_singleton","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[429,8],"def_end_pos":[429,26]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"PowerSeries.X_dvd_iff","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[474,8],"def_end_pos":[474,17]},{"full_name":"RingHom.mem_ker","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[524,8],"def_end_pos":[524,15]}]}]}
{"url":"Mathlib/Topology/PartitionOfUnity.lean","commit":"","full_name":"PartitionOfUnity.finsupport_subset_fintsupport","start":[249,0],"end":[252,35],"file_path":"Mathlib/Topology/PartitionOfUnity.lean","tactics":[{"state_before":"ι : Type u\nX : Type v\ninst✝⁴ : TopologicalSpace X\nE : Type u_1\ninst✝³ : AddCommMonoid E\ninst✝² : SMulWithZero ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns✝ : Set X\nf : PartitionOfUnity ι X s✝\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\ni : ι\nhi : i ∈ ρ.finsupport x₀\n⊢ i ∈ ρ.fintsupport x₀","state_after":"ι : Type u\nX : Type v\ninst✝⁴ : TopologicalSpace X\nE : Type u_1\ninst✝³ : AddCommMonoid E\ninst✝² : SMulWithZero ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns✝ : Set X\nf : PartitionOfUnity ι X s✝\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\ni : ι\nhi : i ∈ ρ.finsupport x₀\n⊢ x₀ ∈ tsupport ⇑(ρ i)","tactic":"rw [ρ.mem_fintsupport_iff]","premises":[{"full_name":"PartitionOfUnity.mem_fintsupport_iff","def_path":"Mathlib/Topology/PartitionOfUnity.lean","def_pos":[239,8],"def_end_pos":[239,27]}]},{"state_before":"ι : Type u\nX : Type v\ninst✝⁴ : TopologicalSpace X\nE : Type u_1\ninst✝³ : AddCommMonoid E\ninst✝² : SMulWithZero ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns✝ : Set X\nf : PartitionOfUnity ι X s✝\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\ni : ι\nhi : i ∈ ρ.finsupport x₀\n⊢ x₀ ∈ tsupport ⇑(ρ i)","state_after":"case a\nι : Type u\nX : Type v\ninst✝⁴ : TopologicalSpace X\nE : Type u_1\ninst✝³ : AddCommMonoid E\ninst✝² : SMulWithZero ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns✝ : Set X\nf : PartitionOfUnity ι X s✝\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\ni : ι\nhi : i ∈ ρ.finsupport x₀\n⊢ x₀ ∈ support ⇑(ρ i)","tactic":"apply subset_closure","premises":[{"full_name":"subset_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[347,8],"def_end_pos":[347,22]}]},{"state_before":"case a\nι : Type u\nX : Type v\ninst✝⁴ : TopologicalSpace X\nE : Type u_1\ninst✝³ : AddCommMonoid E\ninst✝² : SMulWithZero ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns✝ : Set X\nf : PartitionOfUnity ι X s✝\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\ni : ι\nhi : i ∈ ρ.finsupport x₀\n⊢ x₀ ∈ support ⇑(ρ i)","state_after":"no goals","tactic":"exact (ρ.mem_finsupport x₀).mp hi","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"PartitionOfUnity.mem_finsupport","def_path":"Mathlib/Topology/PartitionOfUnity.lean","def_pos":[186,8],"def_end_pos":[186,22]}]}]}
{"url":"Mathlib/GroupTheory/Index.lean","commit":"","full_name":"AddSubgroup.index_map_dvd","start":[240,0],"end":[244,35],"file_path":"Mathlib/GroupTheory/Index.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nf : G →* G'\nhf : Function.Surjective ⇑f\n⊢ (map f H).index ∣ H.index","state_after":"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nf : G →* G'\nhf : Function.Surjective ⇑f\n⊢ (H ⊔ f.ker).index ∣ H.index","tactic":"rw [index_map, f.range_top_of_surjective hf, index_top, mul_one]","premises":[{"full_name":"MonoidHom.range_top_of_surjective","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2005,8],"def_end_pos":[2005,31]},{"full_name":"Subgroup.index_map","def_path":"Mathlib/GroupTheory/Index.lean","def_pos":[236,8],"def_end_pos":[236,17]},{"full_name":"Subgroup.index_top","def_path":"Mathlib/GroupTheory/Index.lean","def_pos":[178,8],"def_end_pos":[178,17]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nf : G →* G'\nhf : Function.Surjective ⇑f\n⊢ (H ⊔ f.ker).index ∣ H.index","state_after":"no goals","tactic":"exact index_dvd_of_le le_sup_left","premises":[{"full_name":"Subgroup.index_dvd_of_le","def_path":"Mathlib/GroupTheory/Index.lean","def_pos":[90,8],"def_end_pos":[90,23]},{"full_name":"le_sup_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[106,8],"def_end_pos":[106,19]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/Convergence.lean","commit":"","full_name":"LSeries.abscissaOfAbsConv_le_one_of_isBigO_one","start":[116,0],"end":[121,39],"file_path":"Mathlib/NumberTheory/LSeries/Convergence.lean","tactics":[{"state_before":"f : ℕ → ℂ\nh : f =O[atTop] 1\n⊢ abscissaOfAbsConv f ≤ 1","state_after":"case h.e'_4\nf : ℕ → ℂ\nh : f =O[atTop] 1\n⊢ 1 = ↑0 + 1\n\ncase convert_2\nf : ℕ → ℂ\nh : f =O[atTop] 1\n⊢ f =O[atTop] fun n => ↑n ^ 0","tactic":"convert abscissaOfAbsConv_le_of_isBigO_rpow (x := 0) ?_","premises":[{"full_name":"LSeries.abscissaOfAbsConv_le_of_isBigO_rpow","def_path":"Mathlib/NumberTheory/LSeries/Convergence.lean","def_pos":[99,6],"def_end_pos":[99,49]}]}]}
{"url":"Mathlib/Topology/Separation.lean","commit":"","full_name":"SeparationQuotient.t2Space_iff","start":[1337,0],"end":[1340,26],"file_path":"Mathlib/Topology/Separation.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\n⊢ T2Space (SeparationQuotient X) ↔ R1Space X","state_after":"no goals","tactic":"simp only [t2Space_iff_disjoint_nhds, Pairwise, surjective_mk.forall₂, ne_eq, mk_eq_mk,\n    r1Space_iff_inseparable_or_disjoint_nhds, ← disjoint_comap_iff surjective_mk, comap_mk_nhds_mk,\n    ← or_iff_not_imp_left]","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Filter.disjoint_comap_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2210,8],"def_end_pos":[2210,26]},{"full_name":"Function.Surjective.forall₂","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[170,18],"def_end_pos":[170,36]},{"full_name":"Pairwise","def_path":"Mathlib/Logic/Pairwise.lean","def_pos":[31,4],"def_end_pos":[31,12]},{"full_name":"SeparationQuotient.comap_mk_nhds_mk","def_path":"Mathlib/Topology/Inseparable.lean","def_pos":[565,8],"def_end_pos":[565,24]},{"full_name":"SeparationQuotient.mk_eq_mk","def_path":"Mathlib/Topology/Inseparable.lean","def_pos":[525,8],"def_end_pos":[525,16]},{"full_name":"SeparationQuotient.surjective_mk","def_path":"Mathlib/Topology/Inseparable.lean","def_pos":[528,8],"def_end_pos":[528,21]},{"full_name":"ne_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[89,16],"def_end_pos":[89,21]},{"full_name":"r1Space_iff_inseparable_or_disjoint_nhds","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1047,8],"def_end_pos":[1047,48]},{"full_name":"t2Space_iff_disjoint_nhds","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1308,8],"def_end_pos":[1308,33]}]}]}
{"url":"Mathlib/LinearAlgebra/SModEq.lean","commit":"","full_name":"SModEq.sub_mem","start":[38,0],"end":[38,89],"file_path":"Mathlib/LinearAlgebra/SModEq.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁵ : Ring R\nA : Type u_2\ninst✝⁴ : CommRing A\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nU U₁ U₂ : Submodule R M\nx x₁ x₂ y y₁ y₂ z z₁ z₂ : M\nN : Type u_4\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nV V₁ V₂ : Submodule R N\n⊢ x ≡ y [SMOD U] ↔ x - y ∈ U","state_after":"no goals","tactic":"rw [SModEq.def, Submodule.Quotient.eq]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"SModEq.def","def_path":"Mathlib/LinearAlgebra/SModEq.lean","def_pos":[32,18],"def_end_pos":[32,28]},{"full_name":"Submodule.Quotient.eq","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[72,18],"def_end_pos":[72,20]}]}]}
{"url":"Mathlib/RingTheory/Prime.lean","commit":"","full_name":"Prime.neg","start":[61,0],"end":[63,82],"file_path":"Mathlib/RingTheory/Prime.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : CommRing α\np : α\nhp : Prime p\n⊢ Prime (-p)","state_after":"case intro.intro\nα : Type u_1\ninst✝ : CommRing α\np : α\nh1 : p ≠ 0\nh2 : ¬IsUnit p\nh3 : ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b\n⊢ Prime (-p)","tactic":"obtain ⟨h1, h2, h3⟩ := hp","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\ninst✝ : CommRing α\np : α\nh1 : p ≠ 0\nh2 : ¬IsUnit p\nh3 : ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b\n⊢ Prime (-p)","state_after":"no goals","tactic":"exact ⟨neg_ne_zero.mpr h1, by rwa [IsUnit.neg_iff], by simpa [neg_dvd] using h3⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsUnit.neg_iff","def_path":"Mathlib/Algebra/Ring/Units.lean","def_pos":[98,8],"def_end_pos":[98,22]},{"full_name":"neg_dvd","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[82,8],"def_end_pos":[82,15]},{"full_name":"neg_ne_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[466,2],"def_end_pos":[466,13]}]}]}
{"url":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","commit":"","full_name":"tendsto_prod_filter_iff","start":[284,0],"end":[289,5],"file_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nc : β\n⊢ Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nc : β\n⊢ Tendsto (Prod.mk c ∘ ↿F) (p ×ˢ p') (𝓤 β) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'","tactic":"simp_rw [nhds_eq_comap_uniformity, tendsto_comap_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.tendsto_comap_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2686,8],"def_end_pos":[2686,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"nhds_eq_comap_uniformity","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[401,8],"def_end_pos":[401,32]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nc : β\n⊢ Tendsto (Prod.mk c ∘ ↿F) (p ×ˢ p') (𝓤 β) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.filter_filter","start":[2172,0],"end":[2174,89],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns✝ t s : Finset α\na : α\n⊢ a ∈ filter q (filter p s) ↔ a ∈ filter (fun a => p a ∧ q a) s","state_after":"no goals","tactic":"simp only [mem_filter, and_assoc, Bool.decide_and, Bool.decide_coe, Bool.and_eq_true]","premises":[{"full_name":"Bool.and_eq_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[218,16],"def_end_pos":[218,32]},{"full_name":"Bool.decide_and","def_path":".lake/packages/lean4/src/lean/Init/Data/Bool.lean","def_pos":[503,16],"def_end_pos":[503,26]},{"full_name":"Bool.decide_coe","def_path":".lake/packages/lean4/src/lean/Init/Data/Bool.lean","def_pos":[501,18],"def_end_pos":[501,28]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[148,8],"def_end_pos":[148,17]}]}]}
{"url":"Mathlib/Order/WellFoundedSet.lean","commit":"","full_name":"WellFounded.prod_lex_of_wellFoundedOn_fiber","start":[826,0],"end":[837,71],"file_path":"Mathlib/Order/WellFoundedSet.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nrα : α → α → Prop\nrβ : β → β → Prop\nf : γ → α\ng : γ → β\ns : Set γ\nhα : WellFounded (rα on f)\nhβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)\n⊢ WellFounded (Prod.Lex rα rβ on fun c => (f c, g c))","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nrα : α → α → Prop\nrβ : β → β → Prop\nf : γ → α\ng : γ → β\ns : Set γ\nhα : WellFounded (rα on f)\nhβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)\nc c' : γ\nh : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'\n⊢ ((PSigma.Lex (fun a b => rα ↑a ↑b) fun a a_1 b => (rβ on g) ↑a_1 ↑b) on fun c => ⟨⟨f c, ⋯⟩, ⟨c, ⋯⟩⟩) c c'","tactic":"refine ((PSigma.lex_wf (wellFoundedOn_range.2 hα) fun a => hβ a).onFun\n    (f := fun c => ⟨⟨_, c, rfl⟩, c, rfl⟩)).mono fun c c' h => ?_","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"PSigma.lex_wf","def_path":"Mathlib/Init/Data/Sigma/Lex.lean","def_pos":[32,8],"def_end_pos":[32,14]},{"full_name":"PSigma.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[199,2],"def_end_pos":[199,4]},{"full_name":"Set.wellFoundedOn_range","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[95,8],"def_end_pos":[95,27]},{"full_name":"WellFounded.mono","def_path":"Mathlib/Order/WellFounded.lean","def_pos":[36,8],"def_end_pos":[36,12]},{"full_name":"WellFounded.onFun","def_path":"Mathlib/Order/WellFounded.lean","def_pos":[39,8],"def_end_pos":[39,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nrα : α → α → Prop\nrβ : β → β → Prop\nf : γ → α\ng : γ → β\ns : Set γ\nhα : WellFounded (rα on f)\nhβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)\nc c' : γ\nh : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'\n⊢ ((PSigma.Lex (fun a b => rα ↑a ↑b) fun a a_1 b => (rβ on g) ↑a_1 ↑b) on fun c => ⟨⟨f c, ⋯⟩, ⟨c, ⋯⟩⟩) c c'","state_after":"case inl\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nrα : α → α → Prop\nrβ : β → β → Prop\nf : γ → α\ng : γ → β\ns : Set γ\nhα : WellFounded (rα on f)\nhβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)\nc c' : γ\nh : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'\nh' : rα ((fun c => (f c, g c)) c).1 ((fun c => (f c, g c)) c').1\n⊢ ((PSigma.Lex (fun a b => rα ↑a ↑b) fun a a_1 b => (rβ on g) ↑a_1 ↑b) on fun c => ⟨⟨f c, ⋯⟩, ⟨c, ⋯⟩⟩) c c'\n\ncase inr\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nrα : α → α → Prop\nrβ : β → β → Prop\nf : γ → α\ng : γ → β\ns : Set γ\nhα : WellFounded (rα on f)\nhβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)\nc c' : γ\nh : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'\nh' :\n  ((fun c => (f c, g c)) c).1 = ((fun c => (f c, g c)) c').1 ∧\n    rβ ((fun c => (f c, g c)) c).2 ((fun c => (f c, g c)) c').2\n⊢ ((PSigma.Lex (fun a b => rα ↑a ↑b) fun a a_1 b => (rβ on g) ↑a_1 ↑b) on fun c => ⟨⟨f c, ⋯⟩, ⟨c, ⋯⟩⟩) c c'","tactic":"obtain h' | h' := Prod.lex_iff.1 h","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Prod.lex_iff","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[192,6],"def_end_pos":[192,13]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","commit":"","full_name":"Real.arsinh_neg","start":[63,0],"end":[68,44],"file_path":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","tactics":[{"state_before":"x✝ y x : ℝ\n⊢ arsinh (-x) = -arsinh x","state_after":"x✝ y x : ℝ\n⊢ -x + √(1 + (-x) ^ 2) = (x + √(1 + x ^ 2))⁻¹","tactic":"rw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh]","premises":[{"full_name":"Real.exp_arsinh","def_path":"Mathlib/Analysis/SpecialFunctions/Arsinh.lean","def_pos":[54,8],"def_end_pos":[54,18]},{"full_name":"Real.exp_eq_exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1024,8],"def_end_pos":[1024,18]},{"full_name":"Real.exp_neg","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[716,15],"def_end_pos":[716,22]}]},{"state_before":"x✝ y x : ℝ\n⊢ -x + √(1 + (-x) ^ 2) = (x + √(1 + x ^ 2))⁻¹","state_after":"case h\nx✝ y x : ℝ\n⊢ (-x + √(1 + (-x) ^ 2)) * (x + √(1 + x ^ 2)) = 1","tactic":"apply eq_inv_of_mul_eq_one_left","premises":[{"full_name":"eq_inv_of_mul_eq_one_left","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[994,8],"def_end_pos":[994,33]}]},{"state_before":"case h\nx✝ y x : ℝ\n⊢ (-x + √(1 + (-x) ^ 2)) * (x + √(1 + x ^ 2)) = 1","state_after":"case h\nx✝ y x : ℝ\n⊢ 0 ≤ 1 + x ^ 2","tactic":"rw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, ← sq_sub_sq, sq_sqrt, add_sub_cancel_right]","premises":[{"full_name":"Real.sq_sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[170,8],"def_end_pos":[170,15]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"add_sub_cancel_right","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[698,2],"def_end_pos":[698,13]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"neg_add_eq_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[524,2],"def_end_pos":[524,13]},{"full_name":"neg_sq","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[138,6],"def_end_pos":[138,12]},{"full_name":"sq_sub_sq","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[188,6],"def_end_pos":[188,15]}]},{"state_before":"case h\nx✝ y x : ℝ\n⊢ 0 ≤ 1 + x ^ 2","state_after":"no goals","tactic":"exact add_nonneg zero_le_one (sq_nonneg _)","premises":[{"full_name":"sq_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[893,6],"def_end_pos":[893,15]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","commit":"","full_name":"Rat.sub_eq_add_neg","start":[256,0],"end":[257,58],"file_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","tactics":[{"state_before":"a b : Rat\n⊢ a - b = a + -b","state_after":"no goals","tactic":"simp [add_def, sub_def, Int.neg_mul, Int.sub_eq_add_neg]","premises":[{"full_name":"Int.neg_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[447,32],"def_end_pos":[447,39]},{"full_name":"Int.sub_eq_add_neg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[87,18],"def_end_pos":[87,32]},{"full_name":"Rat.add_def","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[188,8],"def_end_pos":[188,15]},{"full_name":"Rat.sub_def","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[238,8],"def_end_pos":[238,15]}]}]}
{"url":"Mathlib/Data/ENNReal/Inv.lean","commit":"","full_name":"ENNReal.inv_top","start":[46,0],"end":[47,100],"file_path":"Mathlib/Data/ENNReal/Inv.lean","tactics":[{"state_before":"a✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nh : 0 < a\n⊢ a ∈ {b | 1 ≤ ⊤ * b}","state_after":"no goals","tactic":"simp [*, h.ne', top_mul]","premises":[{"full_name":"ENNReal.top_mul","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[179,16],"def_end_pos":[179,23]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Spec.lean","commit":"","full_name":"AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app","start":[290,0],"end":[296,86],"file_path":"Mathlib/AlgebraicGeometry/Spec.lean","tactics":[{"state_before":"X Y : CommRingCat\nf : X ⟶ Y\n⊢ (𝟭 CommRingCat).map f ≫ ((fun R => asIso (toSpecΓ R)) Y).hom =\n    ((fun R => asIso (toSpecΓ R)) X).hom ≫ (Spec.toLocallyRingedSpace.rightOp ⋙ Γ).map f","state_after":"no goals","tactic":"convert Spec_Γ_naturality (R := X) (S := Y) f","premises":[{"full_name":"AlgebraicGeometry.Spec_Γ_naturality","def_path":"Mathlib/AlgebraicGeometry/Spec.lean","def_pos":[284,8],"def_end_pos":[284,25]}]}]}
{"url":"Mathlib/Algebra/Polynomial/BigOperators.lean","commit":"","full_name":"Polynomial.multiset_prod_X_sub_C_nextCoeff","start":[230,0],"end":[236,23],"file_path":"Mathlib/Algebra/Polynomial/BigOperators.lean","tactics":[{"state_before":"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\n⊢ (Multiset.map (fun x => X - C x) t).prod.nextCoeff = -t.sum","state_after":"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\n⊢ (Multiset.map (fun i => (X - C i).nextCoeff) t).sum = -t.sum\n\ncase h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\n⊢ ∀ i ∈ t, (X - C i).Monic","tactic":"rw [nextCoeff_multiset_prod]","premises":[{"full_name":"Polynomial.Monic.nextCoeff_multiset_prod","def_path":"Mathlib/Algebra/Polynomial/Monic.lean","def_pos":[251,8],"def_end_pos":[251,37]}]}]}
{"url":"Mathlib/Data/Nat/Choose/Basic.lean","commit":"","full_name":"Nat.choose_symm","start":[169,0],"end":[172,38],"file_path":"Mathlib/Data/Nat/Choose/Basic.lean","tactics":[{"state_before":"n k : ℕ\nhk : k ≤ n\n⊢ n.choose (n - k) = n.choose k","state_after":"no goals","tactic":"rw [choose_eq_factorial_div_factorial hk, choose_eq_factorial_div_factorial (Nat.sub_le _ _),\n    Nat.sub_sub_self hk, Nat.mul_comm]","premises":[{"full_name":"Nat.choose_eq_factorial_div_factorial","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[147,8],"def_end_pos":[147,41]},{"full_name":"Nat.mul_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[217,18],"def_end_pos":[217,26]},{"full_name":"Nat.sub_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[291,16],"def_end_pos":[291,22]},{"full_name":"Nat.sub_sub_self","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[122,18],"def_end_pos":[122,30]}]}]}
{"url":"Mathlib/GroupTheory/Subgroup/Centralizer.lean","commit":"","full_name":"Subgroup.mem_centralizer_iff_commutator_eq_one","start":[35,0],"end":[38,65],"file_path":"Mathlib/GroupTheory/Subgroup/Centralizer.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\ng : G\ns : Set G\n⊢ g ∈ centralizer s ↔ ∀ h ∈ s, h * g * h⁻¹ * g⁻¹ = 1","state_after":"no goals","tactic":"simp only [mem_centralizer_iff, mul_inv_eq_iff_eq_mul, one_mul]","premises":[{"full_name":"Subgroup.mem_centralizer_iff","def_path":"Mathlib/GroupTheory/Subgroup/Centralizer.lean","def_pos":[32,8],"def_end_pos":[32,27]},{"full_name":"mul_inv_eq_iff_eq_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[666,8],"def_end_pos":[666,29]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
{"url":"Mathlib/Algebra/Algebra/Operations.lean","commit":"","full_name":"Submodule.mem_span_singleton_mul","start":[331,0],"end":[334,5],"file_path":"Mathlib/Algebra/Algebra/Operations.lean","tactics":[{"state_before":"ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n x y : A\n⊢ x ∈ map ((LinearMap.mul R A) y) P ↔ ∃ z ∈ P, Mul.mul y z = x","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Complex.cos_sq'","start":[616,0],"end":[616,97],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x y : ℂ\n⊢ cos x ^ 2 = 1 - sin x ^ 2","state_after":"no goals","tactic":"rw [← sin_sq_add_cos_sq x, add_sub_cancel_left]","premises":[{"full_name":"Complex.sin_sq_add_cos_sq","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[597,8],"def_end_pos":[597,25]},{"full_name":"add_sub_cancel_left","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[923,2],"def_end_pos":[923,13]}]}]}
{"url":"Mathlib/Data/Matrix/Basis.lean","commit":"","full_name":"Matrix.col_eq_zero_of_commute_stdBasisMatrix","start":[212,0],"end":[215,7],"file_path":"Mathlib/Data/Matrix/Basis.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ninst✝ : Fintype n\ni j k : n\nM : Matrix n n α\nhM : Commute (stdBasisMatrix i j 1) M\nhki : k ≠ i\n⊢ M k i = 0","state_after":"l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ninst✝ : Fintype n\ni j k : n\nM : Matrix n n α\nhM : Commute (stdBasisMatrix i j 1) M\nhki : k ≠ i\nthis : (stdBasisMatrix i j 1 * M) k j = (M * stdBasisMatrix i j 1) k j\n⊢ M k i = 0","tactic":"have := ext_iff.mpr hM k j","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Matrix.ext_iff","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[67,8],"def_end_pos":[67,15]}]},{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ninst✝ : Fintype n\ni j k : n\nM : Matrix n n α\nhM : Commute (stdBasisMatrix i j 1) M\nhki : k ≠ i\nthis : (stdBasisMatrix i j 1 * M) k j = (M * stdBasisMatrix i j 1) k j\n⊢ M k i = 0","state_after":"no goals","tactic":"aesop","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Side.lean","commit":"","full_name":"AffineSubspace.wOppSide_vadd_right_iff","start":[262,0],"end":[264,62],"file_path":"Mathlib/Analysis/Convex/Side.lean","tactics":[{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v ∈ s.direction\n⊢ s.WOppSide x (v +ᵥ y) ↔ s.WOppSide x y","state_after":"no goals","tactic":"rw [wOppSide_comm, wOppSide_vadd_left_iff hv, wOppSide_comm]","premises":[{"full_name":"AffineSubspace.wOppSide_comm","def_path":"Mathlib/Analysis/Convex/Side.lean","def_pos":[171,8],"def_end_pos":[171,21]},{"full_name":"AffineSubspace.wOppSide_vadd_left_iff","def_path":"Mathlib/Analysis/Convex/Side.lean","def_pos":[251,8],"def_end_pos":[251,30]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Topology/Sets/Opens.lean","commit":"","full_name":"TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion","start":[298,0],"end":[307,13],"file_path":"Mathlib/Topology/Sets/Opens.lean","tactics":[{"state_before":"ι✝ : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, ↑(b i)","state_after":"case hb\nι✝ : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ IsTopologicalBasis (range fun i => (b i).carrier)\n\ncase hb'\nι✝ : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier","tactic":"apply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : ι => (b i).1","premises":[{"full_name":"TopologicalSpace.Opens.carrier","def_path":"Mathlib/Topology/Sets/Opens.lean","def_pos":[63,2],"def_end_pos":[63,9]},{"full_name":"isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[544,8],"def_end_pos":[544,65]}]}]}
{"url":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","commit":"","full_name":"Module.End.eigenspace_div","start":[142,0],"end":[151,95],"file_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","tactics":[{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ f.eigenspace (a / b) = f.eigenspace (b⁻¹ * a)","state_after":"no goals","tactic":"rw [div_eq_mul_inv, mul_comm]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ f.eigenspace (b⁻¹ * a) = LinearMap.ker (f - (b⁻¹ * a) • LinearMap.id)","state_after":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ LinearMap.ker (f - (algebraMap K (End K V)) (b⁻¹ * a)) = LinearMap.ker (f - (b⁻¹ * a) • LinearMap.id)","tactic":"rw [eigenspace]","premises":[{"full_name":"Module.End.eigenspace","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[63,4],"def_end_pos":[63,14]}]},{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ LinearMap.ker (f - (algebraMap K (End K V)) (b⁻¹ * a)) = LinearMap.ker (f - (b⁻¹ * a) • LinearMap.id)","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ LinearMap.ker (f - (b⁻¹ * a) • LinearMap.id) = LinearMap.ker (f - b⁻¹ • a • LinearMap.id)","state_after":"no goals","tactic":"rw [smul_smul]","premises":[{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ LinearMap.ker (f - b⁻¹ • (algebraMap K (End K V)) a) = LinearMap.ker (b • (f - b⁻¹ • (algebraMap K (End K V)) a))","state_after":"no goals","tactic":"rw [LinearMap.ker_smul _ b hb]","premises":[{"full_name":"LinearMap.ker_smul","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[214,8],"def_end_pos":[214,16]}]},{"state_before":"K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\na b : K\nhb : b ≠ 0\n⊢ LinearMap.ker (b • (f - b⁻¹ • (algebraMap K (End K V)) a)) = LinearMap.ker (b • f - (algebraMap K (End K V)) a)","state_after":"no goals","tactic":"rw [smul_sub, smul_inv_smul₀ hb]","premises":[{"full_name":"smul_inv_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[39,8],"def_end_pos":[39,22]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean","commit":"","full_name":"SzemerediRegularity.le_sum_distinctPairs_edgeDensity_sq","start":[121,0],"end":[130,77],"file_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nx : { i // i ∈ P.parts.offDiag }\nhε₁ : ε ≤ 1\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5\n⊢ ↑(G.edgeDensity (↑x).1 (↑x).2) ^ 2 + ((if G.IsUniform ε (↑x).1 (↑x).2 then 0 else ε ^ 4 / 3) - ε ^ 5 / 25) ≤\n    (∑ i ∈ SzemerediRegularity.distinctPairs hP G ε x, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ P.parts.card","state_after":"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nx : { i // i ∈ P.parts.offDiag }\nhε₁ : ε ≤ 1\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5\n⊢ (↑(G.edgeDensity (↑x).1 (↑x).2) ^ 2 - ε ^ 5 / 25 + if G.IsUniform ε (↑x).1 (↑x).2 then 0 else ε ^ 4 / 3) ≤\n    (∑ i ∈ (chunk hP G ε ⋯).parts ×ˢ (chunk hP G ε ⋯).parts, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ P.parts.card","tactic":"rw [distinctPairs, ← add_sub_assoc, add_sub_right_comm]","premises":[{"full_name":"_private.Mathlib.Combinatorics.SimpleGraph.Regularity.Increment.0.SzemerediRegularity.distinctPairs","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean","def_pos":[86,26],"def_end_pos":[86,39]},{"full_name":"add_sub_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[329,2],"def_end_pos":[329,13]},{"full_name":"add_sub_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[551,2],"def_end_pos":[551,13]}]},{"state_before":"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nx : { i // i ∈ P.parts.offDiag }\nhε₁ : ε ≤ 1\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5\n⊢ (↑(G.edgeDensity (↑x).1 (↑x).2) ^ 2 - ε ^ 5 / 25 + if G.IsUniform ε (↑x).1 (↑x).2 then 0 else ε ^ 4 / 3) ≤\n    (∑ i ∈ (chunk hP G ε ⋯).parts ×ˢ (chunk hP G ε ⋯).parts, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ P.parts.card","state_after":"case pos\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nx : { i // i ∈ P.parts.offDiag }\nhε₁ : ε ≤ 1\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nh : G.IsUniform ε (↑x).1 (↑x).2\n⊢ ↑(G.edgeDensity (↑x).1 (↑x).2) ^ 2 - ε ^ 5 / 25 + 0 ≤\n    (∑ i ∈ (chunk hP G ε ⋯).parts ×ˢ (chunk hP G ε ⋯).parts, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ P.parts.card\n\ncase neg\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nx : { i // i ∈ P.parts.offDiag }\nhε₁ : ε ≤ 1\nhPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5\nh : ¬G.IsUniform ε (↑x).1 (↑x).2\n⊢ ↑(G.edgeDensity (↑x).1 (↑x).2) ^ 2 - ε ^ 5 / 25 + ε ^ 4 / 3 ≤\n    (∑ i ∈ (chunk hP G ε ⋯).parts ×ˢ (chunk hP G ε ⋯).parts, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ P.parts.card","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieIdeal.map_comap_le","start":[954,0],"end":[954,75],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\n⊢ map f (comap f J) ≤ J","state_after":"no goals","tactic":"rw [map_le_iff_le_comap]","premises":[{"full_name":"LieIdeal.map_le_iff_le_comap","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[940,8],"def_end_pos":[940,27]}]}]}
{"url":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","commit":"","full_name":"FractionalIdeal.div_one","start":[429,0],"end":[439,35],"file_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\n⊢ I / 1 = I","state_after":"R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\n⊢ ⟨↑I / ↑1, ⋯⟩ = I","tactic":"rw [div_nonzero (one_ne_zero' (FractionalIdeal R₁⁰ K))]","premises":[{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.div_nonzero","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[388,8],"def_end_pos":[388,19]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]},{"full_name":"one_ne_zero'","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[66,6],"def_end_pos":[66,18]}]},{"state_before":"R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\n⊢ ⟨↑I / ↑1, ⋯⟩ = I","state_after":"case a\nR : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\nx✝ : K\n⊢ x✝ ∈ ⟨↑I / ↑1, ⋯⟩ ↔ x✝ ∈ I","tactic":"ext","premises":[]},{"state_before":"case a\nR : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\nx✝ : K\n⊢ x✝ ∈ ⟨↑I / ↑1, ⋯⟩ ↔ x✝ ∈ I","state_after":"case a.mp\nR : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\nx✝ : K\nh : x✝ ∈ ⟨↑I / ↑1, ⋯⟩\n⊢ x✝ ∈ I\n\ncase a.mpr\nR : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_3\ninst✝³ : CommRing R₁\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J I : FractionalIdeal R₁⁰ K\nx✝ : K\nh : x✝ ∈ I\n⊢ x✝ ∈ ⟨↑I / ↑1, ⋯⟩","tactic":"constructor <;> intro h","premises":[]}]}
{"url":"Mathlib/Data/Bool/Basic.lean","commit":"","full_name":"Bool.decide_iff","start":[71,0],"end":[71,80],"file_path":"Mathlib/Data/Bool/Basic.lean","tactics":[{"state_before":"p : Prop\nd : Decidable p\n⊢ decide p = true ↔ p","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/TrivSqZeroExt.lean","commit":"","full_name":"TrivSqZeroExt.fst_map","start":[984,0],"end":[986,61],"file_path":"Mathlib/Algebra/TrivSqZeroExt.lean","tactics":[{"state_before":"S : Type u_1\nR R' : Type u\nM : Type v\ninst✝²⁶ : CommSemiring S\ninst✝²⁵ : Semiring R\ninst✝²⁴ : CommSemiring R'\ninst✝²³ : AddCommMonoid M\ninst✝²² : Algebra S R\ninst✝²¹ : Algebra S R'\ninst✝²⁰ : Module S M\ninst✝¹⁹ : Module R M\ninst✝¹⁸ : Module Rᵐᵒᵖ M\ninst✝¹⁷ : SMulCommClass R Rᵐᵒᵖ M\ninst✝¹⁶ : IsScalarTower S R M\ninst✝¹⁵ : IsScalarTower S Rᵐᵒᵖ M\ninst✝¹⁴ : Module R' M\ninst✝¹³ : Module R'ᵐᵒᵖ M\ninst✝¹² : IsCentralScalar R' M\ninst✝¹¹ : IsScalarTower S R' M\nA : Type u_2\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra S A\ninst✝⁸ : Algebra R' A\nN : Type u_3\nP : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R' N\ninst✝⁵ : Module R'ᵐᵒᵖ N\ninst✝⁴ : IsCentralScalar R' N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R' P\ninst✝¹ : Module R'ᵐᵒᵖ P\ninst✝ : IsCentralScalar R' P\nf : M →ₗ[R'] N\nx : tsze R' M\n⊢ ((map f) x).fst = x.fst","state_after":"no goals","tactic":"simp [map, lift_def, Algebra.ofId_apply, algebraMap_eq_inl]","premises":[{"full_name":"Algebra.ofId_apply","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[450,8],"def_end_pos":[450,18]},{"full_name":"TrivSqZeroExt.algebraMap_eq_inl","def_path":"Mathlib/Algebra/TrivSqZeroExt.lean","def_pos":[801,8],"def_end_pos":[801,25]},{"full_name":"TrivSqZeroExt.lift_def","def_path":"Mathlib/Algebra/TrivSqZeroExt.lean","def_pos":[876,8],"def_end_pos":[876,16]},{"full_name":"TrivSqZeroExt.map","def_path":"Mathlib/Algebra/TrivSqZeroExt.lean","def_pos":[973,4],"def_end_pos":[973,7]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.toMeasure_apply₀","start":[684,0],"end":[691,25],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace α\ns✝ t : Set α\nm : OuterMeasure α\nh : ms ≤ m.caratheodory\ns : Set α\nhs : NullMeasurableSet s (m.toMeasure h)\n⊢ (m.toMeasure h) s = m s","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace α\ns✝ t : Set α\nm : OuterMeasure α\nh : ms ≤ m.caratheodory\ns : Set α\nhs : NullMeasurableSet s (m.toMeasure h)\n⊢ (m.toMeasure h) s ≤ m s","tactic":"refine le_antisymm ?_ (le_toMeasure_apply _ _ _)","premises":[{"full_name":"MeasureTheory.le_toMeasure_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[680,8],"def_end_pos":[680,26]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace α\ns✝ t : Set α\nm : OuterMeasure α\nh : ms ≤ m.caratheodory\ns : Set α\nhs : NullMeasurableSet s (m.toMeasure h)\n⊢ (m.toMeasure h) s ≤ m s","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace α\ns✝ t✝ : Set α\nm : OuterMeasure α\nh : ms ≤ m.caratheodory\ns : Set α\nhs : NullMeasurableSet s (m.toMeasure h)\nt : Set α\nhts : t ⊆ s\nhtm : MeasurableSet t\nheq : t =ᶠ[ae (m.toMeasure h)] s\n⊢ (m.toMeasure h) s ≤ m s","tactic":"rcases hs.exists_measurable_subset_ae_eq with ⟨t, hts, htm, heq⟩","premises":[{"full_name":"MeasureTheory.NullMeasurableSet.exists_measurable_subset_ae_eq","def_path":"Mathlib/MeasureTheory/Measure/NullMeasurable.lean","def_pos":[201,8],"def_end_pos":[201,38]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace α\ns✝ t✝ : Set α\nm : OuterMeasure α\nh : ms ≤ m.caratheodory\ns : Set α\nhs : NullMeasurableSet s (m.toMeasure h)\nt : Set α\nhts : t ⊆ s\nhtm : MeasurableSet t\nheq : t =ᶠ[ae (m.toMeasure h)] s\n⊢ (m.toMeasure h) s ≤ m s","state_after":"no goals","tactic":"calc\n    m.toMeasure h s = m.toMeasure h t := measure_congr heq.symm\n    _ = m t := toMeasure_apply m h htm\n    _ ≤ m s := m.mono hts","premises":[{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"MeasureTheory.OuterMeasure.mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Defs.lean","def_pos":[46,12],"def_end_pos":[46,16]},{"full_name":"MeasureTheory.OuterMeasure.toMeasure","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[663,4],"def_end_pos":[663,26]},{"full_name":"MeasureTheory.measure_congr","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[228,8],"def_end_pos":[228,21]},{"full_name":"MeasureTheory.toMeasure_apply","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[676,8],"def_end_pos":[676,23]}]}]}
{"url":"Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean","commit":"","full_name":"legendreSym.at_two","start":[57,0],"end":[60,85],"file_path":"Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\n⊢ legendreSym p 2 = χ₈ ↑p","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\nthis : 2 = ↑2\n⊢ legendreSym p 2 = χ₈ ↑p","tactic":"have : (2 : ZMod p) = (2 : ℤ) := by norm_cast","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\nthis : 2 = ↑2\n⊢ legendreSym p 2 = χ₈ ↑p","state_after":"no goals","tactic":"rw [legendreSym, ← this, quadraticChar_two ((ringChar_zmod_n p).substr hp), card p]","premises":[{"full_name":"Eq.substr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[679,8],"def_end_pos":[679,17]},{"full_name":"ZMod.card","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[113,8],"def_end_pos":[113,12]},{"full_name":"ZMod.ringChar_zmod_n","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[124,8],"def_end_pos":[124,23]},{"full_name":"legendreSym","def_path":"Mathlib/NumberTheory/LegendreSymbol/Basic.lean","def_pos":[104,4],"def_end_pos":[104,15]},{"full_name":"quadraticChar_two","def_path":"Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean","def_pos":[33,8],"def_end_pos":[33,25]}]}]}
{"url":"Mathlib/Algebra/Group/Int.lean","commit":"","full_name":"Int.isUnit_iff","start":[105,0],"end":[109,47],"file_path":"Mathlib/Algebra/Group/Int.lean","tactics":[{"state_before":"u v : ℤ\n⊢ IsUnit u ↔ u = 1 ∨ u = -1","state_after":"u v : ℤ\nh : u = 1 ∨ u = -1\n⊢ IsUnit u","tactic":"refine ⟨fun h ↦ isUnit_eq_one_or h, fun h ↦ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Int.isUnit_eq_one_or","def_path":"Mathlib/Algebra/Group/Int.lean","def_pos":[96,6],"def_end_pos":[96,22]}]},{"state_before":"u v : ℤ\nh : u = 1 ∨ u = -1\n⊢ IsUnit u","state_after":"case inl\nv : ℤ\n⊢ IsUnit 1\n\ncase inr\nv : ℤ\n⊢ IsUnit (-1)","tactic":"rcases h with (rfl | rfl)","premises":[]}]}
{"url":"Mathlib/Data/Fintype/Order.lean","commit":"","full_name":"Directed.finite_set_le","start":[177,0],"end":[179,63],"file_path":"Mathlib/Data/Fintype/Order.lean","tactics":[{"state_before":"α : Type u_1\nr : α → α → Prop\ninst✝² : IsTrans α r\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Nonempty γ\nf : γ → α\ninst✝ : Finite β\nD : Directed r f\ns : Set γ\nhs : s.Finite\n⊢ ∃ z, ∀ i ∈ s, r (f i) (f z)","state_after":"case h.e'_2.h.h.h.a\nα : Type u_1\nr : α → α → Prop\ninst✝² : IsTrans α r\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Nonempty γ\nf : γ → α\ninst✝ : Finite β\nD : Directed r f\ns : Set γ\nhs : s.Finite\nx✝ a✝ : γ\n⊢ a✝ ∈ s ↔ a✝ ∈ hs.toFinset","tactic":"convert D.finset_le hs.toFinset","premises":[{"full_name":"Directed.finset_le","def_path":"Mathlib/Data/Finset/Order.lean","def_pos":[17,8],"def_end_pos":[17,26]},{"full_name":"Set.Finite.toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[91,28],"def_end_pos":[91,43]}]},{"state_before":"case h.e'_2.h.h.h.a\nα : Type u_1\nr : α → α → Prop\ninst✝² : IsTrans α r\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Nonempty γ\nf : γ → α\ninst✝ : Finite β\nD : Directed r f\ns : Set γ\nhs : s.Finite\nx✝ a✝ : γ\n⊢ a✝ ∈ s ↔ a✝ ∈ hs.toFinset","state_after":"no goals","tactic":"rw [Set.Finite.mem_toFinset]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.Finite.mem_toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[146,18],"def_end_pos":[146,30]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","commit":"","full_name":"CategoryTheory.Limits.CokernelCofork.condition","start":[515,0],"end":[517,34],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ns : CokernelCofork f\n⊢ f ≫ Cofork.π s = 0","state_after":"no goals","tactic":"rw [Cofork.condition, zero_comp]","premises":[{"full_name":"CategoryTheory.Limits.Cofork.condition","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[353,8],"def_end_pos":[353,24]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]}]}]}
{"url":"Mathlib/CategoryTheory/Shift/Basic.lean","commit":"","full_name":"CategoryTheory.ShiftMkCore.assoc_inv_app","start":[89,0],"end":[98,5],"file_path":"Mathlib/CategoryTheory/Shift/Basic.lean","tactics":[{"state_before":"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ (h.F (m₁ + (m₂ + m₃))).obj X = (h.F (m₁ + m₂ + m₃)).obj X","state_after":"no goals","tactic":"rw [add_assoc]","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ (h.F m₃).map ((h.add m₁ m₂).inv.app X) ≫ (h.add (m₁ + m₂) m₃).inv.app X =\n    (h.add m₂ m₃).inv.app ((h.F m₁).obj X) ≫ (h.add m₁ (m₂ + m₃)).inv.app X ≫ eqToHom ⋯","state_after":"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ 𝟙 ((h.F m₃).obj ((h.F m₁ ⋙ h.F m₂).obj X)) = 𝟙 ((h.F m₂ ⋙ h.F m₃).obj ((h.F m₁).obj X))","tactic":"rw [← cancel_mono ((h.add (m₁ + m₂) m₃).hom.app X ≫ (h.F m₃).map ((h.add m₁ m₂).hom.app X)),\n    Category.assoc, Category.assoc, Category.assoc, Iso.inv_hom_id_app_assoc, ← Functor.map_comp,\n    Iso.inv_hom_id_app, Functor.map_id, h.assoc_hom_app, eqToHom_trans_assoc, eqToHom_refl,\n    Category.id_comp, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv_hom_id_app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[64,8],"def_end_pos":[64,22]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.ShiftMkCore.F","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[66,2],"def_end_pos":[66,3]},{"full_name":"CategoryTheory.ShiftMkCore.add","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[70,2],"def_end_pos":[70,5]},{"full_name":"CategoryTheory.ShiftMkCore.assoc_hom_app","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[72,2],"def_end_pos":[72,15]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]},{"full_name":"CategoryTheory.eqToHom_refl","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[44,8],"def_end_pos":[44,20]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ 𝟙 ((h.F m₃).obj ((h.F m₁ ⋙ h.F m₂).obj X)) = 𝟙 ((h.F m₂ ⋙ h.F m₃).obj ((h.F m₁).obj X))","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/Convolution.lean","commit":"","full_name":"MeasureTheory.continuousOn_convolution_right_with_param_comp","start":[586,0],"end":[597,76],"file_path":"Mathlib/Analysis/Convolution.lean","tactics":[{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng✝ g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddGroup G\ninst✝³ : TopologicalSpace G\ninst✝² : TopologicalAddGroup G\ninst✝¹ : BorelSpace G\ninst✝ : TopologicalSpace P\ns : Set P\nv : P → G\nhv : ContinuousOn v s\ng : P → G → E'\nk : Set G\nhk : IsCompact k\nhgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0\nhf : LocallyIntegrable f μ\nhg : ContinuousOn (↿g) (s ×ˢ univ)\n⊢ ContinuousOn (fun x => (f ⋆[L, μ] g x) (v x)) s","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng✝ g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddGroup G\ninst✝³ : TopologicalSpace G\ninst✝² : TopologicalAddGroup G\ninst✝¹ : BorelSpace G\ninst✝ : TopologicalSpace P\ns : Set P\nv : P → G\nhv : ContinuousOn v s\ng : P → G → E'\nk : Set G\nhk : IsCompact k\nhgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0\nhf : LocallyIntegrable f μ\nhg : ContinuousOn (↿g) (s ×ˢ univ)\n⊢ MapsTo (fun x => (_root_.id x, v x)) s (s ×ˢ univ)","tactic":"apply\n    (continuousOn_convolution_right_with_param L hk hgs hf hg).comp (continuousOn_id.prod hv)","premises":[{"full_name":"ContinuousOn.comp","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[789,8],"def_end_pos":[789,25]},{"full_name":"ContinuousOn.prod","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[969,8],"def_end_pos":[969,25]},{"full_name":"MeasureTheory.continuousOn_convolution_right_with_param","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[539,8],"def_end_pos":[539,49]},{"full_name":"continuousOn_id","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[897,8],"def_end_pos":[897,23]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng✝ g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddGroup G\ninst✝³ : TopologicalSpace G\ninst✝² : TopologicalAddGroup G\ninst✝¹ : BorelSpace G\ninst✝ : TopologicalSpace P\ns : Set P\nv : P → G\nhv : ContinuousOn v s\ng : P → G → E'\nk : Set G\nhk : IsCompact k\nhgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0\nhf : LocallyIntegrable f μ\nhg : ContinuousOn (↿g) (s ×ˢ univ)\n⊢ MapsTo (fun x => (_root_.id x, v x)) s (s ×ˢ univ)","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddGroup G\ninst✝³ : TopologicalSpace G\ninst✝² : TopologicalAddGroup G\ninst✝¹ : BorelSpace G\ninst✝ : TopologicalSpace P\ns : Set P\nv : P → G\nhv : ContinuousOn v s\ng : P → G → E'\nk : Set G\nhk : IsCompact k\nhgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0\nhf : LocallyIntegrable f μ\nhg : ContinuousOn (↿g) (s ×ˢ univ)\nx : P\nhx : x ∈ s\n⊢ (fun x => (_root_.id x, v x)) x ∈ s ×ˢ univ","tactic":"intro x hx","premises":[]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddGroup G\ninst✝³ : TopologicalSpace G\ninst✝² : TopologicalAddGroup G\ninst✝¹ : BorelSpace G\ninst✝ : TopologicalSpace P\ns : Set P\nv : P → G\nhv : ContinuousOn v s\ng : P → G → E'\nk : Set G\nhk : IsCompact k\nhgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0\nhf : LocallyIntegrable f μ\nhg : ContinuousOn (↿g) (s ×ˢ univ)\nx : P\nhx : x ∈ s\n⊢ (fun x => (_root_.id x, v x)) x ∈ s ×ˢ univ","state_after":"no goals","tactic":"simp only [hx, prod_mk_mem_set_prod_eq, mem_univ, and_self_iff, _root_.id]","premises":[{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"Set.prod_mk_mem_set_prod_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[187,8],"def_end_pos":[187,31]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"inner_add_add_self","start":[557,0],"end":[559,51],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\n⊢ ⟪x + y, x + y⟫_𝕜 = ⟪x, x⟫_𝕜 + ⟪x, y⟫_𝕜 + ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\n⊢ ⟪x, x⟫_𝕜 + ⟪y, x⟫_𝕜 + (⟪x, y⟫_𝕜 + ⟪y, y⟫_𝕜) = ⟪x, x⟫_𝕜 + ⟪x, y⟫_𝕜 + ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜","tactic":"simp only [inner_add_left, inner_add_right]","premises":[{"full_name":"inner_add_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[402,8],"def_end_pos":[402,22]},{"full_name":"inner_add_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[405,8],"def_end_pos":[405,23]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : E\n⊢ ⟪x, x⟫_𝕜 + ⟪y, x⟫_𝕜 + (⟪x, y⟫_𝕜 + ⟪y, y⟫_𝕜) = ⟪x, x⟫_𝕜 + ⟪x, y⟫_𝕜 + ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜","state_after":"no goals","tactic":"ring","premises":[]}]}
{"url":"Mathlib/RingTheory/FiniteType.lean","commit":"","full_name":"Algebra.FiniteType.mvPolynomial","start":[83,0],"end":[88,43],"file_path":"Mathlib/RingTheory/FiniteType.lean","tactics":[{"state_before":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nM : Type uM\nN : Type uN\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Semiring A\ninst✝⁸ : Semiring B\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\n⊢ FiniteType R (MvPolynomial ι R)","state_after":"case intro\nR : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nM : Type uM\nN : Type uN\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Semiring A\ninst✝⁸ : Semiring B\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ FiniteType R (MvPolynomial ι R)","tactic":"cases nonempty_fintype ι","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nR : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nM : Type uM\nN : Type uN\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Semiring A\ninst✝⁸ : Semiring B\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ FiniteType R (MvPolynomial ι R)","state_after":"no goals","tactic":"exact\n    ⟨⟨Finset.univ.image MvPolynomial.X, by\n        rw [Finset.coe_image, Finset.coe_univ, Set.image_univ]\n        exact MvPolynomial.adjoin_range_X⟩⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.coe_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[354,8],"def_end_pos":[354,17]},{"full_name":"Finset.coe_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[84,8],"def_end_pos":[84,16]},{"full_name":"Finset.image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[289,4],"def_end_pos":[289,9]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"MvPolynomial.X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[178,4],"def_end_pos":[178,5]},{"full_name":"MvPolynomial.adjoin_range_X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[449,8],"def_end_pos":[449,22]},{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]}]}]}
{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.eq_or_mem_of_mem_cons","start":[104,0],"end":[111,17],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn : ℕ\nh : (fun b => a = b) ((b :: s).get n)\n⊢ a = b ∨ a ∈ s","state_after":"case zero\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nh : a = (b :: s).get 0\n⊢ a = b ∨ a ∈ s\n\ncase succ\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = (b :: s).get (n' + 1)\n⊢ a = b ∨ a ∈ s","tactic":"cases' n with n'","premises":[]}]}
{"url":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","commit":"","full_name":"Padic.exi_rat_seq_conv_cauchy","start":[644,0],"end":[669,23],"file_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","tactic":"have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)","premises":[{"full_name":"div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[45,6],"def_end_pos":[45,13]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","tactic":"let ⟨N, hN⟩ := exi_rat_seq_conv f hε3","premises":[{"full_name":"Padic.exi_rat_seq_conv","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[633,8],"def_end_pos":[633,24]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","tactic":"let ⟨N2, hN2⟩ := f.cauchy₂ hε3","premises":[{"full_name":"CauSeq.cauchy₂","def_path":"Mathlib/Algebra/Order/CauSeq/Basic.lean","def_pos":[189,8],"def_end_pos":[189,15]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\n⊢ ∀ j ≥ max N N2, padicNorm p (limSeq f j - limSeq f (max N N2)) < ε","tactic":"exists max N N2","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Max.max","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1129,2],"def_end_pos":[1129,5]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\n⊢ ∀ j ≥ max N N2, padicNorm p (limSeq f j - limSeq f (max N N2)) < ε","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ padicNorm p (limSeq f j - limSeq f (max N N2)) < ε","tactic":"intro j hj","premises":[]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ padicNorm p (limSeq f j - limSeq f (max N N2)) < ε","state_after":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2) + (↑f (max N N2) - ↑(limSeq f (max N N2)))) < ε","tactic":"suffices\n    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by\n    ring_nf at this ⊢\n    rw [← padicNormE.eq_padic_norm']\n    exact mod_cast this","premises":[{"full_name":"Max.max","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1129,2],"def_end_pos":[1129,5]},{"full_name":"Padic","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[429,4],"def_end_pos":[429,9]},{"full_name":"Padic.limSeq","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[630,4],"def_end_pos":[630,10]},{"full_name":"padicNormE","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[527,4],"def_end_pos":[527,14]},{"full_name":"padicNormE.eq_padic_norm'","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[583,8],"def_end_pos":[583,22]}]},{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2) + (↑f (max N N2) - ↑(limSeq f (max N N2)))) < ε","state_after":"case a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2) + (↑f (max N N2) - ↑(limSeq f (max N N2)))) ≤ ?b\n\ncase a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ ?b < ε\n\ncase b\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\n⊢ ℚ","tactic":"apply lt_of_le_of_lt","premises":[{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]}]}]}
{"url":"Mathlib/Data/Set/Pairwise/Lattice.lean","commit":"","full_name":"Set.PairwiseDisjoint.prod_left","start":[80,0],"end":[94,44],"file_path":"Mathlib/Data/Set/Pairwise/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\n⊢ (s ×ˢ t).PairwiseDisjoint f","state_after":"case mk.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i') ∈ s ×ˢ t\nj : ι\nj' : ι'\nhj : (j, j') ∈ s ×ˢ t\nh : (i, i') ≠ (j, j')\n⊢ (Disjoint on f) (i, i') (j, j')","tactic":"rintro ⟨i, i'⟩ hi ⟨j, j'⟩ hj h","premises":[]},{"state_before":"case mk.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i') ∈ s ×ˢ t\nj : ι\nj' : ι'\nhj : (j, j') ∈ s ×ˢ t\nh : (i, i') ≠ (j, j')\n⊢ (Disjoint on f) (i, i') (j, j')","state_after":"case mk.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t\nj : ι\nj' : ι'\nhj : (j, j').1 ∈ s ∧ (j, j').2 ∈ t\nh : (i, i') ≠ (j, j')\n⊢ (Disjoint on f) (i, i') (j, j')","tactic":"rw [mem_prod] at hi hj","premises":[{"full_name":"Set.mem_prod","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[184,8],"def_end_pos":[184,16]}]},{"state_before":"case mk.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t\nj : ι\nj' : ι'\nhj : (j, j').1 ∈ s ∧ (j, j').2 ∈ t\nh : (i, i') ≠ (j, j')\n⊢ (Disjoint on f) (i, i') (j, j')","state_after":"case mk.mk.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t\nj' : ι'\nhj : (i, j').1 ∈ s ∧ (i, j').2 ∈ t\nh : (i, i') ≠ (i, j')\n⊢ (Disjoint on f) (i, i') (i, j')\n\ncase mk.mk.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t\nj : ι\nj' : ι'\nhj : (j, j').1 ∈ s ∧ (j, j').2 ∈ t\nh : (i, i') ≠ (j, j')\nhij : i ≠ j\n⊢ (Disjoint on f) (i, i') (j, j')","tactic":"obtain rfl | hij := eq_or_ne i j","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
{"url":"Mathlib/Analysis/Normed/Module/Span.lean","commit":"","full_name":"LinearIsometryEquiv.toSpanUnitSingleton_apply","start":[109,0],"end":[111,5],"file_path":"Mathlib/Analysis/Normed/Module/Span.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nx : E\nhx : ‖x‖ = 1\nr : 𝕜\n⊢ r • x ∈ Submodule.span 𝕜 {x}","state_after":"no goals","tactic":"aesop","premises":[]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nx : E\nhx : ‖x‖ = 1\nr : 𝕜\n⊢ (toSpanUnitSingleton x hx) r = ⟨r • x, ⋯⟩","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/Bornology/Constructions.lean","commit":"","full_name":"boundedSpace_induced_iff","start":[144,0],"end":[146,79],"file_path":"Mathlib/Topology/Bornology/Constructions.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ✝ : Type u_2\nι : Type u_3\nπ : ι → Type u_4\ninst✝³ : Bornology α✝\ninst✝² : Bornology β✝\ninst✝¹ : (i : ι) → Bornology (π i)\nα : Type u_5\nβ : Type u_6\ninst✝ : Bornology β\nf : α → β\n⊢ BoundedSpace α ↔ IsBounded (range f)","state_after":"no goals","tactic":"rw [← @isBounded_univ _ (Bornology.induced f), isBounded_induced, image_univ]","premises":[{"full_name":"Bornology.induced","def_path":"Mathlib/Topology/Bornology/Constructions.lean","def_pos":[34,7],"def_end_pos":[34,24]},{"full_name":"Bornology.isBounded_induced","def_path":"Mathlib/Topology/Bornology/Constructions.lean","def_pos":[121,8],"def_end_pos":[121,25]},{"full_name":"Bornology.isBounded_univ","def_path":"Mathlib/Topology/Bornology/Basic.lean","def_pos":[300,8],"def_end_pos":[300,22]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]}]}]}
{"url":"Mathlib/Combinatorics/Additive/ETransform.lean","commit":"","full_name":"Finset.mulETransformLeft_one","start":[115,0],"end":[116,88],"file_path":"Mathlib/Combinatorics/Additive/ETransform.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Group α\ne : α\nx : Finset α × Finset α\n⊢ mulETransformLeft 1 x = x","state_after":"no goals","tactic":"simp [mulETransformLeft]","premises":[{"full_name":"Finset.mulETransformLeft","def_path":"Mathlib/Combinatorics/Additive/ETransform.lean","def_pos":[105,4],"def_end_pos":[105,21]}]}]}
{"url":"Mathlib/LinearAlgebra/Span.lean","commit":"","full_name":"Submodule.span_induction","start":[154,0],"end":[160,81],"file_path":"Mathlib/LinearAlgebra/Span.lean","tactics":[{"state_before":"R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nM : Type u_4\nM₂ : Type u_5\nV : Type u_6\nS : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\nF : Type u_8\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F σ₁₂ M M₂\ns✝ t : Set M\np : M → Prop\nh : x ∈ span R s✝\nmem : ∀ x ∈ s✝, p x\nzero : p 0\nadd : ∀ (x y : M), p x → p y → p (x + y)\nsmul : ∀ (a : R) (x : M), p x → p (a • x)\ns : Set M\n⊢ ∀ {a b : M}, a ∈ p → b ∈ p → a + b ∈ p","state_after":"R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nM : Type u_4\nM₂ : Type u_5\nV : Type u_6\nS : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx✝ : M\np✝ p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\nF : Type u_8\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F σ₁₂ M M₂\ns✝ t : Set M\np : M → Prop\nh : x✝ ∈ span R s✝\nmem : ∀ x ∈ s✝, p x\nzero : p 0\nadd : ∀ (x y : M), p x → p y → p (x + y)\nsmul : ∀ (a : R) (x : M), p x → p (a • x)\ns : Set M\nx y : M\n⊢ x ∈ p → y ∈ p → x + y ∈ p","tactic":"intros x y","premises":[]},{"state_before":"R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nM : Type u_4\nM₂ : Type u_5\nV : Type u_6\nS : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx✝ : M\np✝ p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\nF : Type u_8\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F σ₁₂ M M₂\ns✝ t : Set M\np : M → Prop\nh : x✝ ∈ span R s✝\nmem : ∀ x ∈ s✝, p x\nzero : p 0\nadd : ∀ (x y : M), p x → p y → p (x + y)\nsmul : ∀ (a : R) (x : M), p x → p (a • x)\ns : Set M\nx y : M\n⊢ x ∈ p → y ∈ p → x + y ∈ p","state_after":"no goals","tactic":"exact add x y","premises":[]}]}
{"url":"Mathlib/Order/Interval/Set/OrderIso.lean","commit":"","full_name":"OrderIso.preimage_Iic","start":[21,0],"end":[24,22],"file_path":"Mathlib/Order/Interval/Set/OrderIso.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\n⊢ ⇑e ⁻¹' Iic b = Iic (e.symm b)","state_after":"case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ⇑e ⁻¹' Iic b ↔ x ∈ Iic (e.symm b)","tactic":"ext x","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ⇑e ⁻¹' Iic b ↔ x ∈ Iic (e.symm b)","state_after":"no goals","tactic":"simp [← e.le_iff_le]","premises":[{"full_name":"OrderIso.le_iff_le","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[900,8],"def_end_pos":[900,17]}]}]}
{"url":"Mathlib/Algebra/Module/Submodule/Range.lean","commit":"","full_name":"LinearMap.mem_submoduleImage_of_le","start":[366,0],"end":[373,28],"file_path":"Mathlib/Algebra/Module/Submodule/Range.lean","tactics":[{"state_before":"R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nK : Type u_4\nK₂ : Type u_5\nM : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nV : Type u_9\nV₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nM' : Type u_11\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nO : Submodule R M\nϕ : ↥O →ₗ[R] M'\nN : Submodule R M\nhNO : N ≤ O\nx : M'\n⊢ x ∈ ϕ.submoduleImage N ↔ ∃ y, ∃ (yN : y ∈ N), ϕ ⟨y, ⋯⟩ = x","state_after":"case refine_1\nR : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nK : Type u_4\nK₂ : Type u_5\nM : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nV : Type u_9\nV₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nM' : Type u_11\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nO : Submodule R M\nϕ : ↥O →ₗ[R] M'\nN : Submodule R M\nhNO : N ≤ O\nx : M'\n⊢ (∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x) → ∃ y, ∃ (yN : y ∈ N), ϕ ⟨y, ⋯⟩ = x\n\ncase refine_2\nR : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nK : Type u_4\nK₂ : Type u_5\nM : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nV : Type u_9\nV₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nM' : Type u_11\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nO : Submodule R M\nϕ : ↥O →ₗ[R] M'\nN : Submodule R M\nhNO : N ≤ O\nx : M'\n⊢ (∃ y, ∃ (yN : y ∈ N), ϕ ⟨y, ⋯⟩ = x) → ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x","tactic":"refine mem_submoduleImage.trans ⟨?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"LinearMap.mem_submoduleImage","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[357,8],"def_end_pos":[357,26]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"innerSL_apply_norm","start":[1562,0],"end":[1574,58],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : E\n⊢ ‖(innerSL 𝕜) x‖ = ‖x‖","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : E\n⊢ ‖x‖ ≤ ‖(innerSL 𝕜) x‖","tactic":"refine\n    le_antisymm ((innerSL 𝕜 x).opNorm_le_bound (norm_nonneg _) fun y => norm_inner_le_norm _ _) ?_","premises":[{"full_name":"ContinuousLinearMap.opNorm_le_bound","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean","def_pos":[142,8],"def_end_pos":[142,23]},{"full_name":"innerSL","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1550,4],"def_end_pos":[1550,11]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"norm_inner_le_norm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[931,8],"def_end_pos":[931,26]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : E\n⊢ ‖x‖ ≤ ‖(innerSL 𝕜) x‖","state_after":"case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\n⊢ ‖0‖ ≤ ‖(innerSL 𝕜) 0‖\n\ncase inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : E\nh : x ≠ 0\n⊢ ‖x‖ ≤ ‖(innerSL 𝕜) x‖","tactic":"rcases eq_or_ne x 0 with (rfl | h)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Mirror.lean","commit":"","full_name":"Polynomial.mirror_eval_one","start":[89,0],"end":[108,39],"file_path":"Mathlib/Algebra/Polynomial/Mirror.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ eval 1 p.mirror = eval 1 p","state_after":"R : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ ∑ x ∈ Finset.range (p.natDegree + 1), p.mirror.coeff x = ∑ x ∈ Finset.range (p.natDegree + 1), p.coeff x","tactic":"simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Polynomial.eval_eq_sum_range","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[273,8],"def_end_pos":[273,25]},{"full_name":"Polynomial.mirror_natDegree","def_path":"Mathlib/Algebra/Polynomial/Mirror.lean","def_pos":[57,8],"def_end_pos":[57,24]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]}]},{"state_before":"R : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ ∑ x ∈ Finset.range (p.natDegree + 1), p.mirror.coeff x = ∑ x ∈ Finset.range (p.natDegree + 1), p.coeff x","state_after":"case refine_1\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ (a : ℕ) → a ∈ Finset.range (p.natDegree + 1) → p.mirror.coeff a ≠ 0 → ℕ\n\ncase refine_2\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ ∀ (a : ℕ) (h₁ : a ∈ Finset.range (p.natDegree + 1)) (h₂ : p.mirror.coeff a ≠ 0),\n    ?refine_1 a h₁ h₂ ∈ Finset.range (p.natDegree + 1)\n\ncase refine_3\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ ∀ (a₁ : ℕ) (h₁₁ : a₁ ∈ Finset.range (p.natDegree + 1)) (h₁₂ : p.mirror.coeff a₁ ≠ 0) (a₂ : ℕ)\n    (h₂₁ : a₂ ∈ Finset.range (p.natDegree + 1)) (h₂₂ : p.mirror.coeff a₂ ≠ 0),\n    ?refine_1 a₁ h₁₁ h₁₂ = ?refine_1 a₂ h₂₁ h₂₂ → a₁ = a₂\n\ncase refine_4\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ ∀ b ∈ Finset.range (p.natDegree + 1),\n    p.coeff b ≠ 0 → ∃ a, ∃ (h₁ : a ∈ Finset.range (p.natDegree + 1)) (h₂ : p.mirror.coeff a ≠ 0), ?refine_1 a h₁ h₂ = b\n\ncase refine_5\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ ∀ (a : ℕ) (h₁ : a ∈ Finset.range (p.natDegree + 1)) (h₂ : p.mirror.coeff a ≠ 0),\n    p.mirror.coeff a = p.coeff (?refine_1 a h₁ h₂)","tactic":"refine Finset.sum_bij_ne_zero ?_ ?_ ?_ ?_ ?_","premises":[{"full_name":"Finset.sum_bij_ne_zero","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1250,2],"def_end_pos":[1250,13]}]}]}
{"url":"Mathlib/Data/Nat/Prime/Defs.lean","commit":"","full_name":"Nat.Prime.eq_two_or_odd","start":[180,0],"end":[182,87],"file_path":"Mathlib/Data/Nat/Prime/Defs.lean","tactics":[{"state_before":"n p : ℕ\nhp : Prime p\nh : p % 2 = 0\n⊢ ¬2 = 1","state_after":"no goals","tactic":"decide","premises":[]}]}
{"url":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","commit":"","full_name":"Metric.mem_sphere'","start":[414,0],"end":[414,84],"file_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\n⊢ y ∈ sphere x ε ↔ dist x y = ε","state_after":"no goals","tactic":"rw [dist_comm, mem_sphere]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Metric.mem_sphere","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[412,16],"def_end_pos":[412,26]},{"full_name":"dist_comm","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[166,8],"def_end_pos":[166,17]}]}]}
{"url":"Mathlib/Topology/Category/CompactlyGenerated.lean","commit":"","full_name":"uCompactlyGeneratedSpace_of_continuous_maps","start":[77,0],"end":[90,34],"file_path":"Mathlib/Topology/Category/CompactlyGenerated.lean","tactics":[{"state_before":"X : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\n⊢ t ≤ compactlyGenerated X","state_after":"X : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\n⊢ Continuous id","tactic":"suffices Continuous[t, compactlyGenerated.{u} X] (id : X → X) by\n      rwa [← continuous_id_iff_le]","premises":[{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"TopologicalSpace.compactlyGenerated","def_path":"Mathlib/Topology/Category/CompactlyGenerated.lean","def_pos":[37,4],"def_end_pos":[37,39]},{"full_name":"continuous_id_iff_le","def_path":"Mathlib/Topology/Order.lean","def_pos":[713,8],"def_end_pos":[713,28]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"X : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\n⊢ Continuous id","state_after":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\n⊢ ∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (id ∘ ⇑g)","tactic":"apply h (tY := compactlyGenerated.{u} X)","premises":[{"full_name":"TopologicalSpace.compactlyGenerated","def_path":"Mathlib/Topology/Category/CompactlyGenerated.lean","def_pos":[37,4],"def_end_pos":[37,39]}]},{"state_before":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\n⊢ ∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (id ∘ ⇑g)","state_after":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\n⊢ Continuous (id ∘ ⇑g)","tactic":"intro S g","premises":[]},{"state_before":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\n⊢ Continuous (id ∘ ⇑g)","state_after":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\nf : (i : (T : CompHaus) × C(↑T.toTop, X)) × ↑i.fst.toTop → X :=\n  fun x =>\n    match x with\n    | ⟨⟨fst, i⟩, s⟩ => i s\n⊢ Continuous (id ∘ ⇑g)","tactic":"let f : (Σ (i : (T : CompHaus.{u}) × C(T, X)), i.fst) → X := fun ⟨⟨_, i⟩, s⟩ ↦ i s","premises":[{"full_name":"CompHaus","def_path":"Mathlib/Topology/Category/CompHaus/Basic.lean","def_pos":[38,7],"def_end_pos":[38,15]},{"full_name":"ContinuousMap","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[28,10],"def_end_pos":[28,23]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.fst","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[175,2],"def_end_pos":[175,5]}]},{"state_before":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\nf : (i : (T : CompHaus) × C(↑T.toTop, X)) × ↑i.fst.toTop → X :=\n  fun x =>\n    match x with\n    | ⟨⟨fst, i⟩, s⟩ => i s\n⊢ Continuous (id ∘ ⇑g)","state_after":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\nf : (i : (T : CompHaus) × C(↑T.toTop, X)) × ↑i.fst.toTop → X :=\n  fun x =>\n    match x with\n    | ⟨⟨fst, i⟩, s⟩ => i s\n⊢ ∀ (i : (T : CompHaus) × C(↑T.toTop, X)), Continuous fun a => f ⟨i, a⟩","tactic":"suffices ∀ (i : (T : CompHaus.{u}) × C(T, X)),\n      Continuous[inferInstance, compactlyGenerated X] (fun (a : i.fst) ↦ f ⟨i, a⟩) from this ⟨S, g⟩","premises":[{"full_name":"CompHaus","def_path":"Mathlib/Topology/Category/CompHaus/Basic.lean","def_pos":[38,7],"def_end_pos":[38,15]},{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"ContinuousMap","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[28,10],"def_end_pos":[28,23]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.fst","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[175,2],"def_end_pos":[175,5]},{"full_name":"Sigma.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[173,2],"def_end_pos":[173,4]},{"full_name":"TopologicalSpace.compactlyGenerated","def_path":"Mathlib/Topology/Category/CompactlyGenerated.lean","def_pos":[37,4],"def_end_pos":[37,39]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\nf : (i : (T : CompHaus) × C(↑T.toTop, X)) × ↑i.fst.toTop → X :=\n  fun x =>\n    match x with\n    | ⟨⟨fst, i⟩, s⟩ => i s\n⊢ ∀ (i : (T : CompHaus) × C(↑T.toTop, X)), Continuous fun a => f ⟨i, a⟩","state_after":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\nf : (i : (T : CompHaus) × C(↑T.toTop, X)) × ↑i.fst.toTop → X :=\n  fun x =>\n    match x with\n    | ⟨⟨fst, i⟩, s⟩ => i s\n⊢ Continuous f","tactic":"rw [← @continuous_sigma_iff]","premises":[{"full_name":"continuous_sigma_iff","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1465,8],"def_end_pos":[1465,28]}]},{"state_before":"case a\nX : Type w\nt : TopologicalSpace X\nh :\n  ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),\n    (∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)) → Continuous f\nS : CompHaus\ng : C(↑S.toTop, X)\nf : (i : (T : CompHaus) × C(↑T.toTop, X)) × ↑i.fst.toTop → X :=\n  fun x =>\n    match x with\n    | ⟨⟨fst, i⟩, s⟩ => i s\n⊢ Continuous f","state_after":"no goals","tactic":"apply continuous_coinduced_rng","premises":[{"full_name":"continuous_coinduced_rng","def_path":"Mathlib/Topology/Order.lean","def_pos":[629,8],"def_end_pos":[629,32]}]}]}
{"url":"Mathlib/Data/Complex/FiniteDimensional.lean","commit":"","full_name":"Complex.nonempty_linearEquiv_real","start":[70,0],"end":[73,55],"file_path":"Mathlib/Data/Complex/FiniteDimensional.lean","tactics":[{"state_before":"⊢ Module.rank ℚ ℂ = Module.rank ℚ ℝ","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","commit":"","full_name":"CategoryTheory.Limits.MulticospanIndex.toPiForkFunctor_map_hom","start":[445,0],"end":[459,16],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nI : MulticospanIndex C\ninst✝¹ : HasProduct I.left\ninst✝ : HasProduct I.right\nK₁ K₂ : Multifork I\nf : K₁ ⟶ K₂\n⊢ ∀ (j : WalkingParallelPair), f.hom ≫ K₂.toPiFork.π.app j = K₁.toPiFork.π.app j","state_after":"case zero\nC : Type u\ninst✝² : Category.{v, u} C\nI : MulticospanIndex C\ninst✝¹ : HasProduct I.left\ninst✝ : HasProduct I.right\nK₁ K₂ : Multifork I\nf : K₁ ⟶ K₂\n⊢ f.hom ≫ K₂.toPiFork.π.app WalkingParallelPair.zero = K₁.toPiFork.π.app WalkingParallelPair.zero\n\ncase one\nC : Type u\ninst✝² : Category.{v, u} C\nI : MulticospanIndex C\ninst✝¹ : HasProduct I.left\ninst✝ : HasProduct I.right\nK₁ K₂ : Multifork I\nf : K₁ ⟶ K₂\n⊢ f.hom ≫ K₂.toPiFork.π.app WalkingParallelPair.one = K₁.toPiFork.π.app WalkingParallelPair.one","tactic":"rintro (_ | _)","premises":[]}]}
{"url":"Mathlib/Topology/MetricSpace/Infsep.lean","commit":"","full_name":"Finset.coe_infsep_of_offDiag_empty","start":[428,0],"end":[431,36],"file_path":"Mathlib/Topology/MetricSpace/Infsep.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns✝ t : Set α\ninst✝ : DecidableEq α\ns : Finset α\nhs : s.offDiag = ∅\n⊢ (↑s).infsep = 0","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns✝ t : Set α\ninst✝ : DecidableEq α\ns : Finset α\nhs : ¬s.offDiag.Nonempty\n⊢ (↑s).infsep = 0","tactic":"rw [← Finset.not_nonempty_iff_eq_empty] at hs","premises":[{"full_name":"Finset.not_nonempty_iff_eq_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[525,8],"def_end_pos":[525,33]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns✝ t : Set α\ninst✝ : DecidableEq α\ns : Finset α\nhs : ¬s.offDiag.Nonempty\n⊢ (↑s).infsep = 0","state_after":"no goals","tactic":"rw [Finset.coe_infsep, dif_neg hs]","premises":[{"full_name":"Finset.coe_infsep","def_path":"Mathlib/Topology/MetricSpace/Infsep.lean","def_pos":[416,8],"def_end_pos":[416,32]},{"full_name":"dif_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[954,8],"def_end_pos":[954,15]}]}]}
{"url":"Mathlib/Algebra/Algebra/Operations.lean","commit":"","full_name":"Submodule.map_one","start":[108,0],"end":[111,6],"file_path":"Mathlib/Algebra/Algebra/Operations.lean","tactics":[{"state_before":"ι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\nA' : Type u_1\ninst✝¹ : Semiring A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\n⊢ map f.toLinearMap 1 = 1","state_after":"case h\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\nA' : Type u_1\ninst✝¹ : Semiring A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\nx✝ : A'\n⊢ x✝ ∈ map f.toLinearMap 1 ↔ x✝ ∈ 1","tactic":"ext","premises":[]},{"state_before":"case h\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\nA' : Type u_1\ninst✝¹ : Semiring A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\nx✝ : A'\n⊢ x✝ ∈ map f.toLinearMap 1 ↔ x✝ ∈ 1","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/GroupTheory/Perm/Fin.lean","commit":"","full_name":"Fin.cycleRange_apply","start":[190,0],"end":[195,74],"file_path":"Mathlib/GroupTheory/Perm/Fin.lean","tactics":[{"state_before":"n : ℕ\ni j : Fin n.succ\n⊢ i.cycleRange j = if j < i then j + 1 else if j = i then 0 else j","state_after":"case pos\nn : ℕ\ni j : Fin n.succ\nh₁ : j < i\n⊢ i.cycleRange j = j + 1\n\ncase pos\nn : ℕ\ni j : Fin n.succ\nh₁ : ¬j < i\nh₂ : j = i\n⊢ i.cycleRange j = 0\n\ncase neg\nn : ℕ\ni j : Fin n.succ\nh₁ : ¬j < i\nh₂ : ¬j = i\n⊢ i.cycleRange j = j","tactic":"split_ifs with h₁ h₂","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Finmap.lean","commit":"","full_name":"Finmap.lookup_singleton_eq","start":[266,0],"end":[268,81],"file_path":"Mathlib/Data/Finmap.lean","tactics":[{"state_before":"α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\n⊢ lookup a (singleton a b) = some b","state_after":"no goals","tactic":"rw [singleton, lookup_toFinmap, AList.singleton, AList.lookup, dlookup_cons_eq]","premises":[{"full_name":"AList.lookup","def_path":"Mathlib/Data/List/AList.lean","def_pos":[137,4],"def_end_pos":[137,10]},{"full_name":"AList.singleton","def_path":"Mathlib/Data/List/AList.lean","def_pos":[118,4],"def_end_pos":[118,13]},{"full_name":"Finmap.lookup_toFinmap","def_path":"Mathlib/Data/Finmap.lean","def_pos":[235,8],"def_end_pos":[235,23]},{"full_name":"Finmap.singleton","def_path":"Mathlib/Data/Finmap.lean","def_pos":[210,4],"def_end_pos":[210,13]},{"full_name":"List.dlookup_cons_eq","def_path":"Mathlib/Data/List/Sigma.lean","def_pos":[151,8],"def_end_pos":[151,23]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Notation.lean","commit":"","full_name":"ONote.NFBelow.snd","start":[206,0],"end":[207,47],"file_path":"Mathlib/SetTheory/Ordinal/Notation.lean","tactics":[{"state_before":"e : ONote\nn : ℕ+\na : ONote\nb : Ordinal.{0}\nh : (e.oadd n a).NFBelow b\n⊢ a.NFBelow e.repr","state_after":"case oadd'\ne : ONote\nn : ℕ+\na : ONote\nb eb : Ordinal.{0}\nh₁ : e.NFBelow eb\nh₂ : a.NFBelow e.repr\nh₃ : e.repr < b\n⊢ a.NFBelow e.repr","tactic":"cases' h with _ _ _ _ eb _ h₁ h₂ h₃","premises":[]},{"state_before":"case oadd'\ne : ONote\nn : ℕ+\na : ONote\nb eb : Ordinal.{0}\nh₁ : e.NFBelow eb\nh₂ : a.NFBelow e.repr\nh₃ : e.repr < b\n⊢ a.NFBelow e.repr","state_after":"no goals","tactic":"exact h₂","premises":[]}]}
{"url":"Mathlib/RingTheory/MvPolynomial/Symmetric.lean","commit":"","full_name":"MvPolynomial.esymm_eq_multiset_esymm","start":[170,0],"end":[173,50],"file_path":"Mathlib/RingTheory/MvPolynomial/Symmetric.lean","tactics":[{"state_before":"σ : Type u_1\nR : Type u_2\nτ : Type u_3\nS : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\n⊢ esymm σ R = (Multiset.map X univ.val).esymm","state_after":"no goals","tactic":"exact funext fun n => (esymm_map_val X _ n).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset.esymm_map_val","def_path":"Mathlib/RingTheory/MvPolynomial/Symmetric.lean","def_pos":[60,8],"def_end_pos":[60,35]},{"full_name":"MvPolynomial.X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[178,4],"def_end_pos":[178,5]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]}]}
{"url":"Mathlib/Topology/Constructions.lean","commit":"","full_name":"Prod.tendsto_iff","start":[524,0],"end":[527,45],"file_path":"Mathlib/Topology/Constructions.lean","tactics":[{"state_before":"X✝ : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nX : Type u_5\nseq : X → Y × Z\nf : Filter X\np : Y × Z\n⊢ Tendsto seq f (𝓝 p) ↔ Tendsto (fun n => (seq n).1) f (𝓝 p.1) ∧ Tendsto (fun n => (seq n).2) f (𝓝 p.2)","state_after":"no goals","tactic":"rw [nhds_prod_eq, Filter.tendsto_prod_iff']","premises":[{"full_name":"Filter.tendsto_prod_iff'","def_path":"Mathlib/Order/Filter/Prod.lean","def_pos":[413,8],"def_end_pos":[413,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"nhds_prod_eq","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[476,8],"def_end_pos":[476,20]}]}]}
{"url":"Mathlib/Algebra/GCDMonoid/Multiset.lean","commit":"","full_name":"Multiset.normalize_gcd","start":[144,0],"end":[146,55],"file_path":"Mathlib/Algebra/GCDMonoid/Multiset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns : Multiset α\n⊢ normalize (gcd 0) = gcd 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns✝ : Multiset α\na : α\ns : Multiset α\nx✝ : normalize s.gcd = s.gcd\n⊢ normalize (a ::ₘ s).gcd = (a ::ₘ s).gcd","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/SetTheory/Game/Birthday.lean","commit":"","full_name":"SetTheory.PGame.birthday_def","start":[44,0],"end":[48,29],"file_path":"Mathlib/SetTheory/Game/Birthday.lean","tactics":[{"state_before":"x : PGame\n⊢ x.birthday = max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)","state_after":"case mk\nα✝ β✝ : Type u\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ (mk α✝ β✝ a✝¹ a✝).birthday =\n    max (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday)","tactic":"cases x","premises":[]},{"state_before":"case mk\nα✝ β✝ : Type u\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ (mk α✝ β✝ a✝¹ a✝).birthday =\n    max (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday)","state_after":"case mk\nα✝ β✝ : Type u\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ max (lsub fun i => (a✝¹ i).birthday) (lsub fun i => (a✝ i).birthday) =\n    max (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday)","tactic":"rw [birthday]","premises":[{"full_name":"SetTheory.PGame.birthday","def_path":"Mathlib/SetTheory/Game/Birthday.lean","def_pos":[40,18],"def_end_pos":[40,26]}]},{"state_before":"case mk\nα✝ β✝ : Type u\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ max (lsub fun i => (a✝¹ i).birthday) (lsub fun i => (a✝ i).birthday) =\n    max (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","commit":"","full_name":"MeasureTheory.IsFundamentalDomain.fundamentalInterior","start":[610,0],"end":[625,95],"file_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ fundamentalInterior G s","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ μ (⋃ i, i⁻¹ • fundamentalInterior G s)ᶜ = 0","tactic":"simp_rw [ae_iff, not_exists, ← mem_inv_smul_set_iff, setOf_forall, ← compl_setOf,\n      setOf_mem_eq, ← compl_iUnion]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.ae_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[73,8],"def_end_pos":[73,14]},{"full_name":"Set.compl_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[389,8],"def_end_pos":[389,20]},{"full_name":"Set.compl_setOf","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1296,8],"def_end_pos":[1296,19]},{"full_name":"Set.mem_inv_smul_set_iff","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[709,8],"def_end_pos":[709,28]},{"full_name":"Set.setOf_forall","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[210,8],"def_end_pos":[210,20]},{"full_name":"Set.setOf_mem_eq","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[215,8],"def_end_pos":[215,20]},{"full_name":"not_exists","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[254,16],"def_end_pos":[254,26]}]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ μ (⋃ i, i⁻¹ • fundamentalInterior G s)ᶜ = 0","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ⊆ ⋃ g, g⁻¹ • fundamentalInterior G s\n⊢ μ (⋃ i, i⁻¹ • fundamentalInterior G s)ᶜ = 0","tactic":"have :\n      ((⋃ g : G, g⁻¹ • s) \\ ⋃ g : G, g⁻¹ • fundamentalFrontier G s) ⊆\n        ⋃ g : G, g⁻¹ • fundamentalInterior G s := by\n      simp_rw [diff_subset_iff, ← iUnion_union_distrib, ← smul_set_union (α := G) (β := α),\n        fundamentalFrontier_union_fundamentalInterior]; rfl","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.fundamentalFrontier","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[498,4],"def_end_pos":[498,23]},{"full_name":"MeasureTheory.fundamentalFrontier_union_fundamentalInterior","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[540,8],"def_end_pos":[540,53]},{"full_name":"MeasureTheory.fundamentalInterior","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[504,4],"def_end_pos":[504,23]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"Set.diff_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1517,8],"def_end_pos":[1517,23]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Set.iUnion_union_distrib","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[420,8],"def_end_pos":[420,28]},{"full_name":"Set.smul_set_union","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[270,8],"def_end_pos":[270,22]}]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ⊆ ⋃ g, g⁻¹ • fundamentalInterior G s\n⊢ μ (⋃ i, i⁻¹ • fundamentalInterior G s)ᶜ = 0","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ⊆ ⋃ g, g⁻¹ • fundamentalInterior G s\n⊢ μ.measureOf ((⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s)ᶜ = ⊥","tactic":"refine eq_bot_mono (μ.mono <| compl_subset_compl.2 this) ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.OuterMeasure.mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Defs.lean","def_pos":[46,12],"def_end_pos":[46,16]},{"full_name":"Set.compl_subset_compl","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1376,8],"def_end_pos":[1376,26]},{"full_name":"eq_bot_mono","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[285,8],"def_end_pos":[285,19]}]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ⊆ ⋃ g, g⁻¹ • fundamentalInterior G s\n⊢ μ.measureOf ((⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s)ᶜ = ⊥","state_after":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ⊆ ⋃ g, g⁻¹ • fundamentalInterior G s\n⊢ μ.measureOf ((⋃ g, g • fundamentalFrontier G s) ∪ {a | ∃ g, g • a ∈ s}ᶜ) = 0","tactic":"simp only [iUnion_inv_smul, compl_sdiff, ENNReal.bot_eq_zero, himp_eq, sup_eq_union,\n      @iUnion_smul_eq_setOf_exists _ _ _ _ s]","premises":[{"full_name":"ENNReal.bot_eq_zero","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[440,8],"def_end_pos":[440,19]},{"full_name":"Set.iUnion_inv_smul","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[795,8],"def_end_pos":[795,23]},{"full_name":"Set.iUnion_smul_eq_setOf_exists","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[799,8],"def_end_pos":[799,35]},{"full_name":"Set.sup_eq_union","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[97,8],"def_end_pos":[97,20]},{"full_name":"compl_sdiff","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[653,8],"def_end_pos":[653,19]},{"full_name":"himp_eq","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[535,8],"def_end_pos":[535,15]}]},{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ⊆ ⋃ g, g⁻¹ • fundamentalInterior G s\n⊢ μ.measureOf ((⋃ g, g • fundamentalFrontier G s) ∪ {a | ∃ g, g • a ∈ s}ᶜ) = 0","state_after":"no goals","tactic":"exact measure_union_null\n      (measure_iUnion_null fun _ => measure_smul_null hs.measure_fundamentalFrontier _) hs.ae_covers","premises":[{"full_name":"MeasureTheory.IsFundamentalDomain.ae_covers","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[70,12],"def_end_pos":[70,21]},{"full_name":"MeasureTheory.IsFundamentalDomain.measure_fundamentalFrontier","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[596,8],"def_end_pos":[596,35]},{"full_name":"MeasureTheory.measure_smul_null","def_path":"Mathlib/MeasureTheory/Group/Action.lean","def_pos":[124,8],"def_end_pos":[124,25]},{"full_name":"MeasureTheory.measure_union_null","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[113,8],"def_end_pos":[113,26]}]}]}
{"url":"Mathlib/Analysis/Convex/Between.lean","commit":"","full_name":"sbtw_lineMap_iff","start":[354,0],"end":[359,46],"file_path":"Mathlib/Analysis/Convex/Between.lean","tactics":[{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\n⊢ Sbtw R x ((lineMap x y) r) y ↔ x ≠ y ∧ r ∈ Set.Ioo 0 1","state_after":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\n⊢ x ≠ y → ((lineMap x y) r ∈ ⇑(lineMap x y) '' Set.Ioo 0 1 ↔ r ∈ Set.Ioo 0 1)","tactic":"rw [sbtw_iff_mem_image_Ioo_and_ne, and_comm, and_congr_right]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"and_congr_right","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[142,8],"def_end_pos":[142,23]},{"full_name":"sbtw_iff_mem_image_Ioo_and_ne","def_path":"Mathlib/Analysis/Convex/Between.lean","def_pos":[281,8],"def_end_pos":[281,37]}]},{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\n⊢ x ≠ y → ((lineMap x y) r ∈ ⇑(lineMap x y) '' Set.Ioo 0 1 ↔ r ∈ Set.Ioo 0 1)","state_after":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x ≠ y\n⊢ (lineMap x y) r ∈ ⇑(lineMap x y) '' Set.Ioo 0 1 ↔ r ∈ Set.Ioo 0 1","tactic":"intro hxy","premises":[]},{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x ≠ y\n⊢ (lineMap x y) r ∈ ⇑(lineMap x y) '' Set.Ioo 0 1 ↔ r ∈ Set.Ioo 0 1","state_after":"no goals","tactic":"rw [(lineMap_injective R hxy).mem_set_image]","premises":[{"full_name":"AffineMap.lineMap_injective","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[513,8],"def_end_pos":[513,25]},{"full_name":"Function.Injective.mem_set_image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[187,8],"def_end_pos":[187,47]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Analysis/Calculus/SmoothSeries.lean","commit":"","full_name":"contDiff_tsum_of_eventually","start":[242,0],"end":[282,62],"file_path":"Mathlib/Analysis/Calculus/SmoothSeries.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\ng g' : α → 𝕜 → F\nv : ℕ → α → ℝ\ns : Set E\nt : Set 𝕜\nx₀ x : E\ny₀ y : 𝕜\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ContDiff 𝕜 N fun x => ∑' (i : α), f i x","state_after":"no goals","tactic":"classical\n    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_\n    let t : Set α :=\n      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }\n    have ht : Set.Finite t :=\n      haveI A :\n        ∀ᶠ i in (Filter.cofinite : Filter α),\n          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by\n        rw [eventually_all_finset]\n        intro i hi\n        apply h'f\n        simp only [Finset.mem_range_succ_iff] at hi\n        exact (WithTop.coe_le_coe.2 hi).trans hm\n      eventually_cofinite.2 A\n    let T : Finset α := ht.toFinset\n    have : (fun x => ∑' i, f i x) = (fun x => ∑ i ∈ T, f i x) +\n        fun x => ∑' i : { i // i ∉ T }, f i x := by\n      ext1 x\n      refine (sum_add_tsum_subtype_compl ?_ T).symm\n      refine .of_norm_bounded_eventually _ (hv 0 (zero_le _)) ?_\n      filter_upwards [h'f 0 (zero_le _)] with i hi\n      simpa only [norm_iteratedFDeriv_zero] using hi x\n    rw [this]\n    apply (ContDiff.sum fun i _ => (hf i).of_le hm).add\n    have h'u : ∀ k : ℕ, (k : ℕ∞) ≤ m → Summable (v k ∘ ((↑) : { i // i ∉ T } → α)) := fun k hk =>\n      (hv k (hk.trans hm)).subtype _\n    refine contDiff_tsum (fun i => (hf i).of_le hm) h'u ?_\n    rintro k ⟨i, hi⟩ x hk\n    simp only [t, T, Finite.mem_toFinset, mem_setOf_eq, Finset.mem_range, not_forall, not_le,\n      exists_prop, not_exists, not_and, not_lt] at hi\n    exact hi k (Nat.lt_succ_iff.2 (WithTop.coe_le_coe.1 hk)) x","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Classical.not_forall","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[137,20],"def_end_pos":[137,30]},{"full_name":"ContDiff.add","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1160,8],"def_end_pos":[1160,20]},{"full_name":"ContDiff.of_le","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1365,8],"def_end_pos":[1365,22]},{"full_name":"ContDiff.sum","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1298,8],"def_end_pos":[1298,20]},{"full_name":"ENat","def_path":"Mathlib/Data/ENat/Basic.lean","def_pos":[28,4],"def_end_pos":[28,8]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Filter","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[88,10],"def_end_pos":[88,16]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.cofinite","def_path":"Mathlib/Order/Filter/Cofinite.lean","def_pos":[30,4],"def_end_pos":[30,12]},{"full_name":"Filter.eventually_all_finset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1037,16],"def_end_pos":[1037,37]},{"full_name":"Filter.eventually_cofinite","def_path":"Mathlib/Order/Filter/Cofinite.lean","def_pos":[38,8],"def_end_pos":[38,27]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.mem_range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2450,8],"def_end_pos":[2450,17]},{"full_name":"Finset.mem_range_succ_iff","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2487,8],"def_end_pos":[2487,26]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.lt_succ_iff","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[573,18],"def_end_pos":[573,29]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.Finite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[56,14],"def_end_pos":[56,20]},{"full_name":"Set.Finite.mem_toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[146,18],"def_end_pos":[146,30]},{"full_name":"Set.Finite.toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[91,28],"def_end_pos":[91,43]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"Summable","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[88,2],"def_end_pos":[88,13]},{"full_name":"Summable.of_norm_bounded_eventually","def_path":"Mathlib/Analysis/Normed/Group/InfiniteSum.lean","def_pos":[147,8],"def_end_pos":[147,43]},{"full_name":"Summable.subtype","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Group.lean","def_pos":[266,2],"def_end_pos":[266,13]},{"full_name":"WithTop.coe_le_coe","def_path":"Mathlib/Order/WithBot.lean","def_pos":[794,8],"def_end_pos":[794,18]},{"full_name":"contDiff_iff_forall_nat_le","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1381,8],"def_end_pos":[1381,34]},{"full_name":"contDiff_tsum","def_path":"Mathlib/Analysis/Calculus/SmoothSeries.lean","def_pos":[219,8],"def_end_pos":[219,21]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"iteratedFDeriv","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1401,18],"def_end_pos":[1401,32]},{"full_name":"norm_iteratedFDeriv_zero","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1421,8],"def_end_pos":[1421,32]},{"full_name":"not_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[116,16],"def_end_pos":[116,23]},{"full_name":"not_exists","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[254,16],"def_end_pos":[254,26]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]},{"full_name":"sum_add_tsum_subtype_compl","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Group.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"tsum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[94,2],"def_end_pos":[94,13]},{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]}]}]}
{"url":"Mathlib/Data/Matrix/RowCol.lean","commit":"","full_name":"Matrix.map_updateColumn","start":[220,0],"end":[224,25],"file_path":"Mathlib/Data/Matrix/RowCol.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\n⊢ (M.updateColumn j c).map f = (M.map f).updateColumn j (f ∘ c)","state_after":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\ni✝ : m\nj✝ : n\n⊢ (M.updateColumn j c).map f i✝ j✝ = (M.map f).updateColumn j (f ∘ c) i✝ j✝","tactic":"ext","premises":[]},{"state_before":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\ni✝ : m\nj✝ : n\n⊢ (M.updateColumn j c).map f i✝ j✝ = (M.map f).updateColumn j (f ∘ c) i✝ j✝","state_after":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\ni✝ : m\nj✝ : n\n⊢ f (if j✝ = j then c i✝ else M i✝ j✝) = if j✝ = j then (f ∘ c) i✝ else f (M i✝ j✝)","tactic":"rw [updateColumn_apply, map_apply, map_apply, updateColumn_apply]","premises":[{"full_name":"Matrix.map_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Matrix.updateColumn_apply","def_path":"Mathlib/Data/Matrix/RowCol.lean","def_pos":[196,8],"def_end_pos":[196,26]}]},{"state_before":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\ni✝ : m\nj✝ : n\n⊢ f (if j✝ = j then c i✝ else M i✝ j✝) = if j✝ = j then (f ∘ c) i✝ else f (M i✝ j✝)","state_after":"no goals","tactic":"exact apply_ite f _ _ _","premises":[{"full_name":"apply_ite","def_path":".lake/packages/lean4/src/lean/Init/ByCases.lean","def_pos":[36,8],"def_end_pos":[36,17]}]}]}
{"url":"Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean","commit":"","full_name":"mdifferentiableAt_atlas","start":[81,0],"end":[98,94],"file_path":"Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\n⊢ MDifferentiableAt I I (↑e) x","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\n⊢ ContinuousAt (↑e) x ∧ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"rw [mdifferentiableAt_iff]","premises":[{"full_name":"mdifferentiableAt_iff","def_path":"Mathlib/Geometry/Manifold/MFDeriv/Defs.lean","def_pos":[229,8],"def_end_pos":[229,29]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\n⊢ ContinuousAt (↑e) x ∧ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"refine ⟨(e.continuousOn x hx).continuousAt (e.open_source.mem_nhds hx), ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"ContinuousWithinAt.continuousAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[760,8],"def_end_pos":[760,39]},{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"PartialHomeomorph.continuousOn","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[94,18],"def_end_pos":[94,30]},{"full_name":"PartialHomeomorph.open_source","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[55,2],"def_end_pos":[55,13]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"have mem :\n    I ((chartAt H x : M → H) x) ∈ I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range I := by\n    simp only [hx, mfld_simps]","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ModelWithCorners.symm","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[176,14],"def_end_pos":[176,18]},{"full_name":"PartialEquiv.source","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[124,2],"def_end_pos":[124,8]},{"full_name":"PartialHomeomorph.symm","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[78,14],"def_end_pos":[78,18]},{"full_name":"PartialHomeomorph.trans","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[707,14],"def_end_pos":[707,19]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"chartAt","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[563,7],"def_end_pos":[563,14]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"have : (chartAt H x).symm.trans e ∈ contDiffGroupoid ∞ I :=\n    HasGroupoid.compatible (chart_mem_atlas H x) h","premises":[{"full_name":"ENat","def_path":"Mathlib/Data/ENat/Basic.lean","def_pos":[28,4],"def_end_pos":[28,8]},{"full_name":"HasGroupoid.compatible","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[923,2],"def_end_pos":[923,12]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"PartialHomeomorph.symm","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[78,14],"def_end_pos":[78,18]},{"full_name":"PartialHomeomorph.trans","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[707,14],"def_end_pos":[707,19]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"chartAt","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[563,7],"def_end_pos":[563,14]},{"full_name":"chart_mem_atlas","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[573,6],"def_end_pos":[573,21]},{"full_name":"contDiffGroupoid","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[519,4],"def_end_pos":[519,20]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"have A :\n    ContDiffOn 𝕜 ∞ (I ∘ (chartAt H x).symm.trans e ∘ I.symm)\n      (I.symm ⁻¹' ((chartAt H x).symm.trans e).source ∩ range I) :=\n    this.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"ContDiffOn","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[596,4],"def_end_pos":[596,14]},{"full_name":"ENat","def_path":"Mathlib/Data/ENat/Basic.lean","def_pos":[28,4],"def_end_pos":[28,8]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"ModelWithCorners.symm","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[176,14],"def_end_pos":[176,18]},{"full_name":"PartialEquiv.source","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[124,2],"def_end_pos":[124,8]},{"full_name":"PartialHomeomorph.symm","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[78,14],"def_end_pos":[78,18]},{"full_name":"PartialHomeomorph.trans","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[707,14],"def_end_pos":[707,19]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"chartAt","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[563,7],"def_end_pos":[563,14]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB :\n  DifferentiableWithinAt 𝕜 (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm)\n    (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I) (↑I (↑(chartAt H x) x))\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"have B := A.differentiableOn le_top (I ((chartAt H x : M → H) x)) mem","premises":[{"full_name":"ContDiffOn.differentiableOn","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[670,8],"def_end_pos":[670,35]},{"full_name":"chartAt","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[563,7],"def_end_pos":[563,14]},{"full_name":"le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[62,8],"def_end_pos":[62,14]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB :\n  DifferentiableWithinAt 𝕜 (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm)\n    (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I) (↑I (↑(chartAt H x) x))\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB :\n  DifferentiableWithinAt 𝕜 (↑I ∘ (↑e ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)\n    (↑I.symm ⁻¹' (chartAt H x).target ∩ ↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' e.source) ∩ range ↑I)\n    (↑I (↑(chartAt H x) x))\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","tactic":"simp only [mfld_simps] at B","premises":[]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB :\n  DifferentiableWithinAt 𝕜 (↑I ∘ (↑e ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)\n    (↑I.symm ⁻¹' (chartAt H x).target ∩ ↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' e.source) ∩ range ↑I)\n    (↑I (↑(chartAt H x) x))\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)","state_after":"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB : DifferentiableWithinAt 𝕜 (↑I ∘ (↑e ∘ ↑(chartAt H x).symm) ∘ ↑I.symm) (range ↑I) (↑I (↑(chartAt H x) x))\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)\n\n𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB :\n  DifferentiableWithinAt 𝕜 (↑I ∘ (↑e ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)\n    (range ↑I ∩ (↑I.symm ⁻¹' (chartAt H x).target ∩ ↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' e.source)))\n    (↑I (↑(chartAt H x) x))\n⊢ ↑I.symm ⁻¹' (chartAt H x).target ∩ ↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' e.source) ∈ 𝓝 (↑I (↑(chartAt H x) x))","tactic":"rw [inter_comm, differentiableWithinAt_inter] at B","premises":[{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]},{"full_name":"differentiableWithinAt_inter","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[568,8],"def_end_pos":[568,36]}]}]}
{"url":"Mathlib/Order/Interval/Set/Image.lean","commit":"","full_name":"_private.Mathlib.Order.Interval.Set.Image.0.Set.image_subtype_val_Ixx_Ixi","start":[242,0],"end":[246,71],"file_path":"Mathlib/Order/Interval/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r (↑c) x → p a x\n⊢ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} = {y | r (↑c) y ∧ q y b}","state_after":"case h\nα : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r (↑c) x → p a x\nx✝ : α\n⊢ x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} ↔ x✝ ∈ {y | r (↑c) y ∧ q y b}","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r (↑c) x → p a x\nx✝ : α\n⊢ x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} ↔ x✝ ∈ {y | r (↑c) y ∧ q y b}","state_after":"no goals","tactic":"simp (config := { contextual := true }) [@and_comm (r _ _), h]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","commit":"","full_name":"Polynomial.degree_pow_le","start":[712,0],"end":[718,79],"file_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\n⊢ (p ^ 0).degree ≤ 0 • p.degree","state_after":"R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\n⊢ degree 1 ≤ 0","tactic":"rw [pow_zero, zero_nsmul]","premises":[{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]},{"full_name":"zero_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[562,14],"def_end_pos":[562,24]}]},{"state_before":"R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\n⊢ degree 1 ≤ 0","state_after":"no goals","tactic":"exact degree_one_le","premises":[{"full_name":"Polynomial.degree_one_le","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[219,8],"def_end_pos":[219,21]}]},{"state_before":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ (n + 1)).degree ≤ (p ^ n).degree + p.degree","state_after":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n * p).degree ≤ (p ^ n).degree + p.degree","tactic":"rw [pow_succ]","premises":[{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]},{"state_before":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n * p).degree ≤ (p ^ n).degree + p.degree","state_after":"no goals","tactic":"exact degree_mul_le _ _","premises":[{"full_name":"Polynomial.degree_mul_le","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[704,8],"def_end_pos":[704,21]}]},{"state_before":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n).degree + p.degree ≤ (n + 1) • p.degree","state_after":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n).degree + p.degree ≤ n • p.degree + p.degree","tactic":"rw [succ_nsmul]","premises":[{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]},{"state_before":"R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n).degree + p.degree ≤ n • p.degree + p.degree","state_after":"no goals","tactic":"exact add_le_add_right (degree_pow_le _ _) _","premises":[{"full_name":"add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[74,31],"def_end_pos":[74,47]}]}]}
{"url":"Mathlib/Data/ENNReal/Basic.lean","commit":"","full_name":"ENNReal.iInter_Ici_coe_nat","start":[590,0],"end":[592,74],"file_path":"Mathlib/Data/ENNReal/Basic.lean","tactics":[{"state_before":"α : Type u_1\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ⋂ n, Ici ↑n = {⊤}","state_after":"no goals","tactic":"simp only [← compl_Iio, ← compl_iUnion, iUnion_Iio_coe_nat, compl_compl]","premises":[{"full_name":"ENNReal.iUnion_Iio_coe_nat","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[564,8],"def_end_pos":[564,26]},{"full_name":"Set.compl_Iio","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[884,8],"def_end_pos":[884,17]},{"full_name":"Set.compl_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[389,8],"def_end_pos":[389,20]},{"full_name":"compl_compl","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[583,8],"def_end_pos":[583,19]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"WellFounded.rank_eq","start":[2256,0],"end":[2259,5],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u\nr : α → α → Prop\na b : α\nhwf : WellFounded r\n⊢ hwf.rank a = sup fun b => succ (hwf.rank ↑b)","state_after":"α : Type u\nr : α → α → Prop\na b : α\nhwf : WellFounded r\n⊢ (sup fun b => succ ⋯.rank) = sup fun b => succ (hwf.rank ↑b)","tactic":"rw [rank, Acc.rank_eq]","premises":[{"full_name":"Acc.rank_eq","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[2232,8],"def_end_pos":[2232,15]},{"full_name":"WellFounded.rank","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[2253,18],"def_end_pos":[2253,22]}]},{"state_before":"α : Type u\nr : α → α → Prop\na b : α\nhwf : WellFounded r\n⊢ (sup fun b => succ ⋯.rank) = sup fun b => succ (hwf.rank ↑b)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/SetTheory/Cardinal/Divisibility.lean","commit":"","full_name":"Cardinal.nat_is_prime_iff","start":[103,0],"end":[126,83],"file_path":"Mathlib/SetTheory/Cardinal/Divisibility.lean","tactics":[{"state_before":"a b : Cardinal.{u}\nn m : ℕ\n⊢ Prime ↑n ↔ Nat.Prime n","state_after":"a b : Cardinal.{u}\nn m : ℕ\n⊢ (↑n ≠ 0 ∧ ¬IsUnit ↑n ∧ ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b) ↔\n    n ≠ 0 ∧ ¬IsUnit n ∧ ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b","tactic":"simp only [Prime, Nat.prime_iff]","premises":[{"full_name":"Nat.prime_iff","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[414,8],"def_end_pos":[414,17]},{"full_name":"Prime","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[40,4],"def_end_pos":[40,9]}]},{"state_before":"a b : Cardinal.{u}\nn m : ℕ\n⊢ (↑n ≠ 0 ∧ ¬IsUnit ↑n ∧ ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b) ↔\n    n ≠ 0 ∧ ¬IsUnit n ∧ ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b","state_after":"case refine_1\na b : Cardinal.{u}\nn m : ℕ\n⊢ ¬IsUnit ↑n ↔ ¬IsUnit n\n\ncase refine_2\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b\nb c : ℕ\nhbc : n ∣ b * c\n⊢ n ∣ b ∨ n ∣ c\n\ncase refine_3\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\n⊢ ↑n ∣ b ∨ ↑n ∣ c","tactic":"refine and_congr (by simp) (and_congr ?_ ⟨fun h b c hbc => ?_, fun h b c hbc => ?_⟩)","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"and_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[43,8],"def_end_pos":[43,17]}]},{"state_before":"case refine_3\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\n⊢ ↑n ∣ b ∨ ↑n ∣ c","state_after":"case refine_3.inl\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : b * c < ℵ₀\n⊢ ↑n ∣ b ∨ ↑n ∣ c\n\ncase refine_3.inr\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\n⊢ ↑n ∣ b ∨ ↑n ∣ c","tactic":"cases' lt_or_le (b * c) ℵ₀ with h' h'","premises":[{"full_name":"Cardinal.aleph0","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1071,4],"def_end_pos":[1071,10]},{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"case refine_3.inr\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\n⊢ ↑n ∣ b ∨ ↑n ∣ c","state_after":"case refine_3.inr.intro.intro\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\n⊢ ↑n ∣ b ∨ ↑n ∣ c","tactic":"rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩","premises":[{"full_name":"Cardinal.aleph0_le_mul_iff","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1461,8],"def_end_pos":[1461,25]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case refine_3.inr.intro.intro\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\n⊢ ↑n ∣ b ∨ ↑n ∣ c","state_after":"case refine_3.inr.intro.intro\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\n⊢ ↑n ∣ b ∨ ↑n ∣ c","tactic":"have hn : (n : Cardinal) ≠ 0 := by\n    intro h\n    rw [h, zero_dvd_iff, mul_eq_zero] at hbc\n    cases hbc <;> contradiction","premises":[{"full_name":"Cardinal","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[107,4],"def_end_pos":[107,12]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"mul_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[248,8],"def_end_pos":[248,19]},{"full_name":"zero_dvd_iff","def_path":"Mathlib/Algebra/GroupWithZero/Divisibility.lean","def_pos":[31,8],"def_end_pos":[31,20]}]},{"state_before":"case refine_3.inr.intro.intro\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\n⊢ ↑n ∣ b ∨ ↑n ∣ c","state_after":"case refine_3.inr.intro.intro.inr\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nthis :\n  ∀ {a b : Cardinal.{u}} {n : ℕ} {m : ℕ},\n    (∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b) →\n      ∀ (b c : Cardinal.{u_1}),\n        ↑n ∣ b * c → ℵ₀ ≤ b * c → b ≠ 0 → c ≠ 0 → ℵ₀ ≤ b ∨ ℵ₀ ≤ c → ↑n ≠ 0 → ℵ₀ ≤ b → ↑n ∣ b ∨ ↑n ∣ c\nhℵ₀b : ¬ℵ₀ ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c\n\nn✝ : ℕ\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nhℵ₀b : ℵ₀ ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c","tactic":"wlog hℵ₀b : ℵ₀ ≤ b","premises":[{"full_name":"Cardinal.aleph0","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1071,4],"def_end_pos":[1071,10]}]},{"state_before":"case refine_3.inr.intro.intro.inr\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nthis :\n  ∀ {a b : Cardinal.{u}} {n : ℕ} {m : ℕ},\n    (∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b) →\n      ∀ (b c : Cardinal.{u_1}),\n        ↑n ∣ b * c → ℵ₀ ≤ b * c → b ≠ 0 → c ≠ 0 → ℵ₀ ≤ b ∨ ℵ₀ ≤ c → ↑n ≠ 0 → ℵ₀ ≤ b → ↑n ∣ b ∨ ↑n ∣ c\nhℵ₀b : ¬ℵ₀ ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c\n\nn✝ : ℕ\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nhℵ₀b : ℵ₀ ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c","state_after":"a b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nthis :\n  ∀ {a b : Cardinal.{u}} {n : ℕ} {m : ℕ},\n    (∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b) →\n      ∀ (b c : Cardinal.{u_1}),\n        ↑n ∣ b * c → ℵ₀ ≤ b * c → b ≠ 0 → c ≠ 0 → ℵ₀ ≤ b ∨ ℵ₀ ≤ c → ↑n ≠ 0 → ℵ₀ ≤ b → ↑n ∣ b ∨ ↑n ∣ c\nhℵ₀b : ¬ℵ₀ ≤ b\n⊢ ↑n ∣ c * b\n\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nthis :\n  ∀ {a b : Cardinal.{u}} {n : ℕ} {m : ℕ},\n    (∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b) →\n      ∀ (b c : Cardinal.{u_1}),\n        ↑n ∣ b * c → ℵ₀ ≤ b * c → b ≠ 0 → c ≠ 0 → ℵ₀ ≤ b ∨ ℵ₀ ≤ c → ↑n ≠ 0 → ℵ₀ ≤ b → ↑n ∣ b ∨ ↑n ∣ c\nhℵ₀b : ¬ℵ₀ ≤ b\n⊢ ℵ₀ ≤ c * b\n\nn✝ : ℕ\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nhℵ₀b : ℵ₀ ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c","tactic":"apply (this h c b _ _ hc hb hℵ₀.symm hn (hℵ₀.resolve_left hℵ₀b)).symm <;> try assumption","premises":[{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]},{"full_name":"Or.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[821,8],"def_end_pos":[821,15]}]}]}
{"url":"Mathlib/RingTheory/Coprime/Lemmas.lean","commit":"","full_name":"Nat.isCoprime_iff_coprime","start":[40,0],"end":[41,60],"file_path":"Mathlib/RingTheory/Coprime/Lemmas.lean","tactics":[{"state_before":"R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\nm n : ℕ\n⊢ IsCoprime ↑m ↑n ↔ m.Coprime n","state_after":"no goals","tactic":"rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Int.gcd_natCast_natCast","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[152,35],"def_end_pos":[152,54]},{"full_name":"Int.isCoprime_iff_gcd_eq_one","def_path":"Mathlib/RingTheory/Coprime/Lemmas.lean","def_pos":[31,8],"def_end_pos":[31,36]}]}]}
{"url":"Mathlib/Order/Interval/Set/OrderEmbedding.lean","commit":"","full_name":"OrderEmbedding.preimage_Ioc","start":[35,0],"end":[35,78],"file_path":"Mathlib/Order/Interval/Set/OrderEmbedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ↪o β\nx y : α\n⊢ ⇑e ⁻¹' Ioc (e x) (e y) = Ioc x y","state_after":"case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ↪o β\nx y x✝ : α\n⊢ x✝ ∈ ⇑e ⁻¹' Ioc (e x) (e y) ↔ x✝ ∈ Ioc x y","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ↪o β\nx y x✝ : α\n⊢ x✝ ∈ ⇑e ⁻¹' Ioc (e x) (e y) ↔ x✝ ∈ Ioc x y","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Order/MinMax.lean","commit":"","full_name":"Monotone.map_max","start":[214,0],"end":[215,51],"file_path":"Mathlib/Order/MinMax.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\ns : Set α\na b c d : α\nhf : Monotone f\n⊢ f (max a b) = max (f a) (f b)","state_after":"no goals","tactic":"rcases le_total a b with h | h <;> simp [h, hf h]","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]}]}
{"url":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","commit":"","full_name":"CategoryTheory.Localization.homEquiv_map","start":[151,0],"end":[153,23],"file_path":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","tactics":[{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{u_8, u_1} C\ninst✝⁸ : Category.{?u.46756, u_2} C₁\ninst✝⁷ : Category.{?u.46760, u_3} C₂\ninst✝⁶ : Category.{?u.46764, u_4} C₃\ninst✝⁵ : Category.{u_10, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.46776, u_7} D₃\nW : MorphismProperty C\nL₁ : C ⥤ D₁\ninst✝² : L₁.IsLocalization W\nL₂ : C ⥤ D₂\ninst✝¹ : L₂.IsLocalization W\nL₃ : C ⥤ D₃\ninst✝ : L₃.IsLocalization W\nX Y Z : C\nf : X ⟶ Y\n⊢ (homEquiv W L₁ L₂) (L₁.map f) = L₂.map f","state_after":"no goals","tactic":"simp [homEquiv_apply]","premises":[{"full_name":"CategoryTheory.Localization.homEquiv_apply","def_path":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","def_pos":[116,32],"def_end_pos":[116,37]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.cons_merge_cons_neg","start":[1582,0],"end":[1584,32],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\na b : α\ns : α → α → Bool\nl r : List α\nh : ¬s a b = true\n⊢ merge s (a :: l) (b :: r) = b :: merge s (a :: l) r","state_after":"no goals","tactic":"rw [cons_merge_cons, if_neg h]","premises":[{"full_name":"List.cons_merge_cons","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[1574,8],"def_end_pos":[1574,23]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]}]}]}
{"url":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","commit":"","full_name":"LinearMap.BilinForm.exists_bilinForm_self_ne_zero","start":[1175,0],"end":[1182,61],"file_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","tactics":[{"state_before":"S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nhB₁ : B ≠ 0\nhB₂ : IsSymm B\n⊢ ∃ x, ¬IsOrtho B x x","state_after":"case intro\nS : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nQ : QuadraticMap R M R\nhB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0\n⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x","tactic":"lift B to QuadraticForm R M using hB₂ with Q","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"QuadraticForm","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[152,7],"def_end_pos":[152,20]}]},{"state_before":"case intro\nS : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nQ : QuadraticMap R M R\nhB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0\n⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x","state_after":"case intro.intro\nS : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nQ : QuadraticMap R M R\nhB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0\nx : M\nhx : Q x ≠ 0\n⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x","tactic":"obtain ⟨x, hx⟩ := QuadraticMap.exists_quadraticForm_ne_zero hB₁","premises":[{"full_name":"QuadraticMap.exists_quadraticForm_ne_zero","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[892,8],"def_end_pos":[892,36]}]},{"state_before":"case intro.intro\nS : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nQ : QuadraticMap R M R\nhB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0\nx : M\nhx : Q x ≠ 0\n⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x","state_after":"no goals","tactic":"exact ⟨x, fun h => hx (Q.associated_eq_self_apply ℕ x ▸ h)⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"QuadraticMap.associated_eq_self_apply","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[865,8],"def_end_pos":[865,32]}]}]}
{"url":"Mathlib/CategoryTheory/Iso.lean","commit":"","full_name":"CategoryTheory.IsIso.inv_id","start":[347,0],"end":[350,6],"file_path":"Mathlib/CategoryTheory/Iso.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ inv (𝟙 X) = 𝟙 X","state_after":"case hom_inv_id\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X","tactic":"apply inv_eq_of_hom_inv_id","premises":[{"full_name":"CategoryTheory.IsIso.inv_eq_of_hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[310,8],"def_end_pos":[310,28]}]},{"state_before":"case hom_inv_id\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Complex/Basic.lean","commit":"","full_name":"Complex.norm_exp_ofReal_mul_I","start":[58,0],"end":[59,47],"file_path":"Mathlib/Analysis/Complex/Basic.lean","tactics":[{"state_before":"z : ℂ\nt : ℝ\n⊢ ‖cexp (↑t * I)‖ = 1","state_after":"no goals","tactic":"simp only [norm_eq_abs, abs_exp_ofReal_mul_I]","premises":[{"full_name":"Complex.abs_exp_ofReal_mul_I","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1474,8],"def_end_pos":[1474,28]},{"full_name":"Complex.norm_eq_abs","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[53,8],"def_end_pos":[53,19]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","commit":"","full_name":"CategoryTheory.ShortComplex.LeftHomologyData.exact_iff","start":[65,0],"end":[69,36],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{?u.2051, u_2} D\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ninst✝ : S.HasHomology\nh : S.LeftHomologyData\n⊢ S.Exact ↔ IsZero h.H","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{?u.2051, u_2} D\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ninst✝ : S.HasHomology\nh : S.LeftHomologyData\n⊢ IsZero S.homology ↔ IsZero h.H","tactic":"rw [S.exact_iff_isZero_homology]","premises":[{"full_name":"CategoryTheory.ShortComplex.exact_iff_isZero_homology","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[55,6],"def_end_pos":[55,31]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{?u.2051, u_2} D\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ninst✝ : S.HasHomology\nh : S.LeftHomologyData\n⊢ IsZero S.homology ↔ IsZero h.H","state_after":"no goals","tactic":"exact Iso.isZero_iff h.homologyIso","premises":[{"full_name":"CategoryTheory.Iso.isZero_iff","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean","def_pos":[123,8],"def_end_pos":[123,22]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyData.homologyIso","def_path":"Mathlib/Algebra/Homology/ShortComplex/Homology.lean","def_pos":[387,18],"def_end_pos":[387,46]}]}]}
{"url":"Mathlib/Order/Cover.lean","commit":"","full_name":"WCovBy.image","start":[102,0],"end":[106,19],"file_path":"Mathlib/Order/Cover.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\n⊢ f a ⩿ f b","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : β\nha : f a < c\nhb : c < f b\n⊢ False","tactic":"refine ⟨f.monotone hab.le, fun c ha hb => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"OrderEmbedding.monotone","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[552,18],"def_end_pos":[552,26]},{"full_name":"WCovBy.le","def_path":"Mathlib/Order/Cover.lean","def_pos":[43,8],"def_end_pos":[43,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : β\nha : f a < c\nhb : c < f b\n⊢ False","state_after":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : α\nha : f a < f c\nhb : f c < f b\n⊢ False","tactic":"obtain ⟨c, rfl⟩ := h.out (mem_range_self _) (mem_range_self _) ⟨ha.le, hb.le⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Set.OrdConnected.out","def_path":"Mathlib/Order/Interval/Set/OrdConnected.lean","def_pos":[40,8],"def_end_pos":[40,24]},{"full_name":"Set.mem_range_self","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[148,22],"def_end_pos":[148,36]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : α\nha : f a < f c\nhb : f c < f b\n⊢ False","state_after":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : α\nha : a < c\nhb : c < b\n⊢ False","tactic":"rw [f.lt_iff_lt] at ha hb","premises":[{"full_name":"OrderEmbedding.lt_iff_lt","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[546,8],"def_end_pos":[546,17]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : α\nha : a < c\nhb : c < b\n⊢ False","state_after":"no goals","tactic":"exact hab.2 ha hb","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.sup_eq_sup","start":[1226,0],"end":[1230,40],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal.{u}\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\n⊢ range (familyOfBFamily' r ho f) = range (familyOfBFamily' r' ho' f)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","commit":"","full_name":"CategoryTheory.braiding_inv_tensorUnit_right","start":[342,0],"end":[346,11],"file_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (β_ X (𝟙_ C)).inv = (λ_ X).hom ≫ (ρ_ X).inv","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)","tactic":"rw [Iso.inv_ext]","premises":[{"full_name":"CategoryTheory.Iso.inv_ext","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[445,8],"def_end_pos":[445,15]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((ρ_ X).hom ≫ (λ_ X).inv) ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)","tactic":"rw [braiding_tensorUnit_right]","premises":[{"full_name":"CategoryTheory.braiding_tensorUnit_right","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[339,8],"def_end_pos":[339,33]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((ρ_ X).hom ≫ (λ_ X).inv) ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)","state_after":"no goals","tactic":"coherence","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CategoryTheory.BicategoricalCoherence.hom","def_path":"Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean","def_pos":[29,2],"def_end_pos":[29,5]},{"full_name":"CategoryTheory.Bicategory.comp_whiskerLeft","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[81,2],"def_end_pos":[81,18]},{"full_name":"CategoryTheory.Bicategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[89,2],"def_end_pos":[89,19]},{"full_name":"CategoryTheory.Bicategory.id_whiskerLeft","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[77,2],"def_end_pos":[77,16]},{"full_name":"CategoryTheory.Bicategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[87,2],"def_end_pos":[87,17]},{"full_name":"CategoryTheory.Bicategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[73,2],"def_end_pos":[73,18]},{"full_name":"CategoryTheory.Bicategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[71,2],"def_end_pos":[71,16]},{"full_name":"CategoryTheory.Bicategory.whiskerRight_comp","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[97,2],"def_end_pos":[97,19]},{"full_name":"CategoryTheory.Bicategory.whiskerRight_id","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[93,2],"def_end_pos":[93,17]},{"full_name":"CategoryTheory.Bicategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[103,2],"def_end_pos":[103,15]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.FreeMonoidalCategory.projectMap","def_path":"Mathlib/CategoryTheory/Monoidal/Free/Basic.lean","def_pos":[312,4],"def_end_pos":[312,14]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"CategoryTheory.MonoidalCategory.id_tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[222,8],"def_end_pos":[222,20]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[237,8],"def_end_pos":[237,22]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[172,2],"def_end_pos":[172,17]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_def","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[157,2],"def_end_pos":[157,15]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[227,8],"def_end_pos":[227,20]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[242,8],"def_end_pos":[242,26]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[170,2],"def_end_pos":[170,16]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerRight_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[254,8],"def_end_pos":[254,23]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerRight_tensor","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[259,8],"def_end_pos":[259,27]},{"full_name":"CategoryTheory.MonoidalCategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[266,8],"def_end_pos":[266,21]},{"full_name":"CategoryTheory.bicategoricalComp","def_path":"Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean","def_pos":[47,4],"def_end_pos":[47,21]},{"full_name":"CategoryTheory.monoidalComp","def_path":"Mathlib/Tactic/CategoryTheory/MonoidalComp.lean","def_pos":[67,4],"def_end_pos":[67,16]},{"full_name":"Mathlib.Tactic.Coherence.LiftHom.lift","def_path":"Mathlib/Tactic/CategoryTheory/Coherence.lean","def_pos":[54,12],"def_end_pos":[54,16]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
{"url":"Mathlib/Data/Fin/Basic.lean","commit":"","full_name":"Fin.succAbove_pred_of_lt","start":[982,0],"end":[984,62],"file_path":"Mathlib/Data/Fin/Basic.lean","tactics":[{"state_before":"n m : ℕ\np✝ : Fin (n + 1)\ni✝ j : Fin n\np i : Fin (n + 1)\nh : p < i\nhi : optParam (i ≠ 0) ⋯\n⊢ p.succAbove (i.pred hi) = i","state_after":"no goals","tactic":"rw [succAbove_of_lt_succ _ _ (succ_pred _ _ ▸ h), succ_pred]","premises":[{"full_name":"Fin.succAbove_of_lt_succ","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[961,6],"def_end_pos":[961,26]},{"full_name":"Fin.succ_pred","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[521,16],"def_end_pos":[521,25]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Basic.lean","commit":"","full_name":"Cardinal.card_le_iff","start":[1208,0],"end":[1209,26],"file_path":"Mathlib/SetTheory/Ordinal/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\nγ : Type u_2\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal.{u_3}\nc : Cardinal.{u_3}\n⊢ o.card ≤ c ↔ o < (succ c).ord","state_after":"no goals","tactic":"rw [lt_ord, lt_succ_iff]","premises":[{"full_name":"Cardinal.lt_ord","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[1188,8],"def_end_pos":[1188,14]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Order.lt_succ_iff","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[319,8],"def_end_pos":[319,19]}]}]}
{"url":"Mathlib/Order/Filter/Pi.lean","commit":"","full_name":"Filter.mem_of_pi_mem_pi","start":[84,0],"end":[92,27],"file_path":"Mathlib/Order/Filter/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\n⊢ s i ∈ f i","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\n⊢ s i ∈ f i","tactic":"rcases mem_pi.1 h with ⟨I', -, t, htf, hts⟩","premises":[{"full_name":"Filter.mem_pi","def_path":"Mathlib/Order/Filter/Pi.lean","def_pos":[70,8],"def_end_pos":[70,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\n⊢ s i ∈ f i","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\n⊢ x ∈ s i","tactic":"refine mem_of_superset (htf i) fun x hx => ?_","premises":[{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\n⊢ x ∈ s i","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\nthis : ∀ (i : ι), (t i).Nonempty\n⊢ x ∈ s i","tactic":"have : ∀ i, (t i).Nonempty := fun i => nonempty_of_mem (htf i)","premises":[{"full_name":"Filter.nonempty_of_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[623,8],"def_end_pos":[623,23]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\nthis : ∀ (i : ι), (t i).Nonempty\n⊢ x ∈ s i","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀ (i : ι), g i ∈ t i\n⊢ x ∈ s i","tactic":"choose g hg using this","premises":[]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀ (i : ι), g i ∈ t i\n⊢ x ∈ s i","state_after":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀ (i : ι), g i ∈ t i\nthis : update g i x ∈ I'.pi t\n⊢ x ∈ s i","tactic":"have : update g i x ∈ I'.pi t := fun j _ => by\n    rcases eq_or_ne j i with (rfl | hne) <;> simp [*]","premises":[{"full_name":"Function.update","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[462,4],"def_end_pos":[462,10]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.pi","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[218,4],"def_end_pos":[218,6]},{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀ (i : ι), g i ∈ t i\nthis : update g i x ∈ I'.pi t\n⊢ x ∈ s i","state_after":"no goals","tactic":"simpa using hts this i hi","premises":[]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.filterMap_eq_filter","start":[1876,0],"end":[1880,27],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ List.filterMap (Option.guard p) l = List.filter (fun b => decide (p b)) l","state_after":"α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ List.filterMap (Option.guard p) l = List.filterMap (Option.guard fun x => decide (p x) = true) l","tactic":"rw [← List.filterMap_eq_filter]","premises":[{"full_name":"List.filterMap_eq_filter","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[972,8],"def_end_pos":[972,27]}]},{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ List.filterMap (Option.guard p) l = List.filterMap (Option.guard fun x => decide (p x) = true) l","state_after":"case e_f\nα : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ Option.guard p = Option.guard fun x => decide (p x) = true","tactic":"congr","premises":[]},{"state_before":"case e_f\nα : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ Option.guard p = Option.guard fun x => decide (p x) = true","state_after":"case e_f.h\nα : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\na : α\n⊢ Option.guard p a = Option.guard (fun x => decide (p x) = true) a","tactic":"funext a","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case e_f.h\nα : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\na : α\n⊢ Option.guard p a = Option.guard (fun x => decide (p x) = true) a","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Int.floor_nonneg","start":[616,0],"end":[616,73],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ 0 ≤ ⌊a⌋ ↔ 0 ≤ a","state_after":"no goals","tactic":"rw [le_floor, Int.cast_zero]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Int.cast_zero","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[53,8],"def_end_pos":[53,17]},{"full_name":"Int.le_floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[607,8],"def_end_pos":[607,16]}]}]}
{"url":"Mathlib/Data/Finset/NoncommProd.lean","commit":"","full_name":"Finset.noncommProd_eq_prod","start":[356,0],"end":[361,17],"file_path":"Mathlib/Data/Finset/NoncommProd.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\n⊢ s.noncommProd f ⋯ = s.prod f","state_after":"case h₁\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\nf : α → β\n⊢ ∅.noncommProd f ⋯ = ∅.prod f\n\ncase h₂\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\nf : α → β\na : α\ns : Finset α\nha : a ∉ s\nIH : s.noncommProd f ⋯ = s.prod f\n⊢ (cons a s ha).noncommProd f ⋯ = (cons a s ha).prod f","tactic":"induction' s using Finset.cons_induction_on with a s ha IH","premises":[{"full_name":"Finset.cons_induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1061,8],"def_end_pos":[1061,25]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Exponential.lean","commit":"","full_name":"hasStrictFDerivAt_exp_zero_of_radius_pos","start":[63,0],"end":[70,42],"file_path":"Mathlib/Analysis/SpecialFunctions/Exponential.lean","tactics":[{"state_before":"𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\n⊢ HasStrictFDerivAt (exp 𝕂) 1 0","state_after":"case h.e'_10\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\n⊢ 1 = (continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)","tactic":"convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt","premises":[{"full_name":"HasFPowerSeriesAt.hasStrictFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","def_pos":[37,8],"def_end_pos":[37,43]},{"full_name":"NormedSpace.hasFPowerSeriesAt_exp_zero_of_radius_pos","def_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","def_pos":[240,8],"def_end_pos":[240,48]}]},{"state_before":"case h.e'_10\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\n⊢ 1 = (continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)","state_after":"case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\nx : 𝔸\n⊢ 1 x = ((continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)) x","tactic":"ext x","premises":[]},{"state_before":"case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\nx : 𝔸\n⊢ 1 x = ((continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)) x","state_after":"case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\nx : 𝔸\n⊢ x = (expSeries 𝕂 𝔸 1) fun x_1 => x","tactic":"change x = expSeries 𝕂 𝔸 1 fun _ => x","premises":[{"full_name":"NormedSpace.expSeries","def_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","def_pos":[99,4],"def_end_pos":[99,13]}]},{"state_before":"case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\nx : 𝔸\n⊢ x = (expSeries 𝕂 𝔸 1) fun x_1 => x","state_after":"no goals","tactic":"simp [expSeries_apply_eq, Nat.factorial]","premises":[{"full_name":"Nat.factorial","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[29,4],"def_end_pos":[29,13]},{"full_name":"NormedSpace.expSeries_apply_eq","def_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","def_pos":[115,8],"def_end_pos":[115,26]}]}]}
{"url":"Mathlib/Algebra/Homology/ComplexShapeSigns.lean","commit":"","full_name":"ComplexShape.ε₁_ε₂","start":[96,0],"end":[105,35],"file_path":"Mathlib/Algebra/Homology/ComplexShapeSigns.lean","tactics":[{"state_before":"I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁, i₂) * 1 = -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')","state_after":"I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (c₁.ε₂ c₂ c₁₂ (i₁, i₂) * (c₁.ε₁ c₂ c₁₂ (i₁, i₂') * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))) =\n    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')","tactic":"rw [← Int.units_mul_self (ComplexShape.ε₁ c₁ c₂ c₁₂ (i₁, i₂')), mul_assoc]","premises":[{"full_name":"ComplexShape.ε₁","def_path":"Mathlib/Algebra/Homology/ComplexShapeSigns.lean","def_pos":[56,7],"def_end_pos":[56,9]},{"full_name":"Int.units_mul_self","def_path":"Mathlib/Data/Int/Order/Units.lean","def_pos":[27,8],"def_end_pos":[27,22]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (c₁.ε₂ c₂ c₁₂ (i₁, i₂) * (c₁.ε₁ c₂ c₁₂ (i₁, i₂') * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))) =\n    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')","state_after":"I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =\n    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')","tactic":"conv_lhs =>\n      arg 2\n      rw [← mul_assoc, ε₂_ε₁ c₁₂ h₁ h₂]","premises":[{"full_name":"ComplexShape.ε₂_ε₁","def_path":"Mathlib/Algebra/Homology/ComplexShapeSigns.lean","def_pos":[91,6],"def_end_pos":[91,11]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =\n    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')","state_after":"no goals","tactic":"rw [neg_mul, neg_mul, neg_mul, mul_neg, neg_inj, ← mul_assoc, ← mul_assoc,\n      Int.units_mul_self, one_mul]","premises":[{"full_name":"Int.units_mul_self","def_path":"Mathlib/Data/Int/Order/Units.lean","def_pos":[27,8],"def_end_pos":[27,22]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]},{"full_name":"neg_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[294,2],"def_end_pos":[294,13]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
{"url":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","commit":"","full_name":"MeasureTheory.addFundamentalFrontier_vadd","start":[553,0],"end":[556,97],"file_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ns : Set α\nx : α\ninst✝² : Group H\ninst✝¹ : MulAction H α\ninst✝ : SMulCommClass H G α\ng : H\n⊢ fundamentalFrontier G (g • s) = g • fundamentalFrontier G s","state_after":"no goals","tactic":"simp_rw [fundamentalFrontier, smul_set_inter, smul_set_iUnion, smul_comm g (_ : G) (_ : Set α)]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.fundamentalFrontier","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[498,4],"def_end_pos":[498,23]},{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.smul_set_iUnion","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[278,8],"def_end_pos":[278,23]},{"full_name":"Set.smul_set_inter","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[736,8],"def_end_pos":[736,22]}]}]}
{"url":"Mathlib/Computability/Primrec.lean","commit":"","full_name":"Primrec.option_getD","start":[544,0],"end":[546,19],"file_path":"Mathlib/Computability/Primrec.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nσ : Type u_5\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nx✝ : Option α × α\no : Option α\na : α\n⊢ (Option.casesOn ((fun a x => a) (o, a).1 (o, a).2) ((fun x b => b) (o, a).1 (o, a).2) fun b => b) =\n    (o, a).1.getD (o, a).2","state_after":"no goals","tactic":"cases o <;> rfl","premises":[]}]}
{"url":"Mathlib/Topology/Homeomorph.lean","commit":"","full_name":"Equiv.toHomeomorph_symm","start":[720,0],"end":[721,92],"file_path":"Mathlib/Topology/Homeomorph.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ✝ : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ne : X ≃ Y\nhe : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s\ns : Set X\n⊢ IsOpen (⇑e.symm ⁻¹' s) ↔ IsOpen s","state_after":"case h.e'_2.h.e'_3\nX : Type u_1\nY : Type u_2\nZ✝ : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ne : X ≃ Y\nhe : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s\ns : Set X\n⊢ s = ⇑e ⁻¹' (⇑e.symm ⁻¹' s)","tactic":"convert (he _).symm","premises":[{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]}]},{"state_before":"case h.e'_2.h.e'_3\nX : Type u_1\nY : Type u_2\nZ✝ : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ne : X ≃ Y\nhe : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s\ns : Set X\n⊢ s = ⇑e ⁻¹' (⇑e.symm ⁻¹' s)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","commit":"","full_name":"MeasureTheory.Measure.measure_eq_zero_of_subset_diff_everywherePosSubset","start":[96,0],"end":[106,63],"file_path":"Mathlib/MeasureTheory/Measure/EverywherePos.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns k : Set α\nhk : IsCompact k\nh'k : k ⊆ s \\ μ.everywherePosSubset s\n⊢ μ k = 0","state_after":"case he\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns k : Set α\nhk : IsCompact k\nh'k : k ⊆ s \\ μ.everywherePosSubset s\n⊢ μ ∅ = 0\n\ncase hmono\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns k : Set α\nhk : IsCompact k\nh'k : k ⊆ s \\ μ.everywherePosSubset s\n⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → μ t = 0 → μ s = 0\n\ncase hunion\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns k : Set α\nhk : IsCompact k\nh'k : k ⊆ s \\ μ.everywherePosSubset s\n⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0\n\ncase hnhds\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns k : Set α\nhk : IsCompact k\nh'k : k ⊆ s \\ μ.everywherePosSubset s\n⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0","tactic":"apply hk.induction_on (p := fun t ↦ μ t = 0)","premises":[{"full_name":"IsCompact.induction_on","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[68,8],"def_end_pos":[68,30]}]}]}
{"url":"Mathlib/Analysis/Calculus/MeanValue.lean","commit":"","full_name":"Convex.norm_image_sub_le_of_norm_hasDerivWithin_le","start":[606,0],"end":[612,58],"file_path":"Mathlib/Analysis/Calculus/MeanValue.lean","tactics":[{"state_before":"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_3\nG : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf f' : 𝕜 → G\ns : Set 𝕜\nx✝ y : 𝕜\nC : ℝ\nhf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nbound : ∀ x ∈ s, ‖f' x‖ ≤ C\nhs : Convex ℝ s\nxs : x✝ ∈ s\nys : y ∈ s\nx : 𝕜\nhx : x ∈ s\n⊢ ‖smulRight 1 (f' x)‖ ≤ ‖f' x‖","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Antidiag/Pi.lean","commit":"","full_name":"Finset.piAntidiag_univ_fin_eq_antidiagonalTuple","start":[205,0],"end":[207,39],"file_path":"Mathlib/Algebra/Order/Antidiag/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝ : DecidableEq ι\nn k : ℕ\n⊢ univ.piAntidiag n = Nat.antidiagonalTuple k n","state_after":"case a\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝ : DecidableEq ι\nn k : ℕ\na✝ : Fin k → ℕ\n⊢ a✝ ∈ univ.piAntidiag n ↔ a✝ ∈ Nat.antidiagonalTuple k n","tactic":"ext","premises":[]},{"state_before":"case a\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝ : DecidableEq ι\nn k : ℕ\na✝ : Fin k → ℕ\n⊢ a✝ ∈ univ.piAntidiag n ↔ a✝ ∈ Nat.antidiagonalTuple k n","state_after":"no goals","tactic":"simp [Nat.mem_antidiagonalTuple]","premises":[{"full_name":"Finset.Nat.mem_antidiagonalTuple","def_path":"Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean","def_pos":[217,8],"def_end_pos":[217,29]}]}]}
{"url":"Mathlib/Data/List/Range.lean","commit":"","full_name":"List.indexOf_finRange","start":[158,0],"end":[162,77],"file_path":"Mathlib/Data/List/Range.lean","tactics":[{"state_before":"α : Type u\nk : ℕ\ni : Fin k\n⊢ indexOf i (finRange k) = ↑i","state_after":"α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\n⊢ indexOf i (finRange k) = ↑i","tactic":"have : (finRange k).indexOf i < (finRange k).length := indexOf_lt_length.mpr (by simp)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"List.finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[87,4],"def_end_pos":[87,12]},{"full_name":"List.indexOf","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[94,4],"def_end_pos":[94,11]},{"full_name":"List.indexOf_lt_length","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[842,8],"def_end_pos":[842,25]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]}]},{"state_before":"α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\n⊢ indexOf i (finRange k) = ↑i","state_after":"α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\nh₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i\n⊢ indexOf i (finRange k) = ↑i","tactic":"have h₁ : (finRange k).get ⟨(finRange k).indexOf i, this⟩ = i := indexOf_get this","premises":[{"full_name":"Fin.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1833,2],"def_end_pos":[1833,4]},{"full_name":"List.finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[87,4],"def_end_pos":[87,12]},{"full_name":"List.get","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2337,4],"def_end_pos":[2337,12]},{"full_name":"List.indexOf","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[94,4],"def_end_pos":[94,11]},{"full_name":"List.indexOf_get","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[970,8],"def_end_pos":[970,19]}]},{"state_before":"α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\nh₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i\n⊢ indexOf i (finRange k) = ↑i","state_after":"α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\nh₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i\nh₂ : (finRange k).get ⟨↑i, ⋯⟩ = i\n⊢ indexOf i (finRange k) = ↑i","tactic":"have h₂ : (finRange k).get ⟨i, by simp⟩ = i := get_finRange _","premises":[{"full_name":"Fin.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1833,2],"def_end_pos":[1833,4]},{"full_name":"List.finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[87,4],"def_end_pos":[87,12]},{"full_name":"List.get","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2337,4],"def_end_pos":[2337,12]},{"full_name":"List.get_finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[144,8],"def_end_pos":[144,20]}]},{"state_before":"α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\nh₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i\nh₂ : (finRange k).get ⟨↑i, ⋯⟩ = i\n⊢ indexOf i (finRange k) = ↑i","state_after":"no goals","tactic":"simpa using (Nodup.get_inj_iff (nodup_finRange k)).mp (Eq.trans h₁ h₂.symm)","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"List.Nodup.get_inj_iff","def_path":"Mathlib/Data/List/Nodup.lean","def_pos":[102,8],"def_end_pos":[102,25]},{"full_name":"List.nodup_finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[101,8],"def_end_pos":[101,22]}]}]}
{"url":"Mathlib/Order/LiminfLimsup.lean","commit":"","full_name":"Monotone.isBoundedUnder_le_comp_iff","start":[1383,0],"end":[1389,72],"file_path":"Mathlib/Order/LiminfLimsup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) → IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","tactic":"refine ⟨?_, fun h => h.isBoundedUnder (α := β) hg⟩","premises":[{"full_name":"Filter.IsBounded.isBoundedUnder","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[100,8],"def_end_pos":[100,32]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) → IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","tactic":"rintro ⟨c, hc⟩","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","tactic":"rw [eventually_map] at hc","premises":[{"full_name":"Filter.eventually_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1650,8],"def_end_pos":[1650,22]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\nb : β\nhb : ∀ a ≥ b, c < g a\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","tactic":"obtain ⟨b, hb⟩ : ∃ b, ∀ a ≥ b, c < g a := eventually_atTop.1 (hg'.eventually_gt_atTop c)","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Filter.Tendsto.eventually_gt_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[176,18],"def_end_pos":[176,45]},{"full_name":"Filter.eventually_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[155,8],"def_end_pos":[155,24]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\nb : β\nhb : ∀ a ≥ b, c < g a\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f","state_after":"no goals","tactic":"exact ⟨b, hc.mono fun x hx => not_lt.1 fun h => (hb _ h.le).not_le hx⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite","start":[1865,0],"end":[1882,89],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","tactic":"set s : ℕ → Set α := disjointed (spanningSets μ)","premises":[{"full_name":"MeasureTheory.spanningSets","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[577,4],"def_end_pos":[577,16]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"disjointed","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[47,4],"def_end_pos":[47,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","tactic":"have : ∀ n, μ (s n) < ∞ := fun n =>\n    (measure_mono <| disjointed_subset _ _).trans_lt (measure_spanningSets_lt_top μ n)","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"MeasureTheory.measure_spanningSets_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[587,8],"def_end_pos":[587,35]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"disjointed_subset","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[150,8],"def_end_pos":[150,25]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","tactic":"obtain ⟨δ, δpos, δsum⟩ : ∃ δ : ℕ → ℝ≥0, (∀ i, 0 < δ i) ∧ (∑' i, μ (s i) * δ i) < ε :=\n    ENNReal.exists_pos_tsum_mul_lt_of_countable ε0 _ fun n => (this n).ne","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"ENNReal.exists_pos_tsum_mul_lt_of_countable","def_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","def_pos":[555,8],"def_end_pos":[555,43]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"tsum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[94,2],"def_end_pos":[94,13]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","tactic":"set N : α → ℕ := spanningSetsIndex μ","premises":[{"full_name":"MeasureTheory.spanningSetsIndex","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[600,18],"def_end_pos":[600,35]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","tactic":"have hN_meas : Measurable N := measurable_spanningSetsIndex μ","premises":[{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"MeasureTheory.measurable_spanningSetsIndex","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[604,8],"def_end_pos":[604,36]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\nhNs : ∀ (n : ℕ), N ⁻¹' {n} = s n\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","tactic":"have hNs : ∀ n, N ⁻¹' {n} = s n := preimage_spanningSetsIndex_singleton μ","premises":[{"full_name":"MeasureTheory.preimage_spanningSetsIndex_singleton","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[609,8],"def_end_pos":[609,44]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\nhNs : ∀ (n : ℕ), N ⁻¹' {n} = s n\n⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\nhNs : ∀ (n : ℕ), N ⁻¹' {n} = s n\n⊢ ∫⁻ (x : α), ↑((δ ∘ N) x) ∂μ < ε","tactic":"refine ⟨δ ∘ N, fun x => δpos _, measurable_from_nat.comp hN_meas, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Measurable.comp","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[498,18],"def_end_pos":[498,33]},{"full_name":"measurable_from_nat","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[364,8],"def_end_pos":[364,27]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\nhNs : ∀ (n : ℕ), N ⁻¹' {n} = s n\n⊢ ∫⁻ (x : α), ↑((δ ∘ N) x) ∂μ < ε","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\nhNs : ∀ (n : ℕ), N ⁻¹' {n} = s n\n⊢ ∫⁻ (a : ℕ), ↑(δ a) ∂Measure.map N μ < ε","tactic":"erw [lintegral_comp measurable_from_nat.coe_nnreal_ennreal hN_meas]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Measurable.coe_nnreal_ennreal","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean","def_pos":[168,8],"def_end_pos":[168,37]},{"full_name":"MeasureTheory.lintegral_comp","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1384,8],"def_end_pos":[1384,22]},{"full_name":"measurable_from_nat","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[364,8],"def_end_pos":[364,27]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\ns : ℕ → Set α := disjointed (spanningSets μ)\nthis : ∀ (n : ℕ), μ (s n) < ⊤\nδ : ℕ → ℝ≥0\nδpos : ∀ (i : ℕ), 0 < δ i\nδsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε\nN : α → ℕ := spanningSetsIndex μ\nhN_meas : Measurable N\nhNs : ∀ (n : ℕ), N ⁻¹' {n} = s n\n⊢ ∫⁻ (a : ℕ), ↑(δ a) ∂Measure.map N μ < ε","state_after":"no goals","tactic":"simpa [N, hNs, lintegral_countable', measurable_spanningSetsIndex, mul_comm] using δsum","premises":[{"full_name":"MeasureTheory.lintegral_countable'","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1595,8],"def_end_pos":[1595,28]},{"full_name":"MeasureTheory.measurable_spanningSetsIndex","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[604,8],"def_end_pos":[604,36]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Data/Set/Basic.lean","commit":"","full_name":"Set.strictMonoOn_iff_strictMono","start":[1913,0],"end":[1915,33],"file_path":"Mathlib/Data/Set/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\n⊢ StrictMonoOn f s ↔ StrictMono fun a => f ↑a","state_after":"no goals","tactic":"simp [StrictMono, StrictMonoOn]","premises":[{"full_name":"StrictMono","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[92,4],"def_end_pos":[92,14]},{"full_name":"StrictMonoOn","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[101,4],"def_end_pos":[101,16]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"sub_zero","start":[353,0],"end":[354,63],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivInvOneMonoid G\na : G\n⊢ a / 1 = a","state_after":"no goals","tactic":"simp [div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","commit":"","full_name":"Polynomial.Chebyshev.T_neg_two","start":[125,0],"end":[125,63],"file_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ T R (-2) = 2 * X ^ 2 - 1","state_after":"no goals","tactic":"simp [T_two]","premises":[{"full_name":"Polynomial.Chebyshev.T_two","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[105,8],"def_end_pos":[105,13]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","commit":"","full_name":"SimpleGraph.Walk.support_concat","start":[475,0],"end":[478,38],"file_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : G.Walk u v\nh : G.Adj v w\n⊢ (p.concat h).support = p.support.concat w","state_after":"no goals","tactic":"induction p <;> simp [*, concat_nil]","premises":[{"full_name":"SimpleGraph.Walk.concat_nil","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[224,8],"def_end_pos":[224,18]}]}]}
{"url":"Mathlib/Probability/Martingale/Centering.lean","commit":"","full_name":"MeasureTheory.predictablePart_add_ae_eq","start":[143,0],"end":[149,37],"file_path":"Mathlib/Probability/Martingale/Centering.lean","tactics":[{"state_before":"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\n⊢ predictablePart (f + g) ℱ μ n =ᶠ[ae μ] g n","state_after":"case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\nω : Ω\nhω : martingalePart (f + g) ℱ μ n ω = f n ω\n⊢ predictablePart (f + g) ℱ μ n ω = g n ω","tactic":"filter_upwards [martingalePart_add_ae_eq hf hg hg0 hgint n] with ω hω","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"MeasureTheory.martingalePart_add_ae_eq","def_path":"Mathlib/Probability/Martingale/Centering.lean","def_pos":[123,8],"def_end_pos":[123,32]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\nω : Ω\nhω : martingalePart (f + g) ℱ μ n ω = f n ω\n⊢ predictablePart (f + g) ℱ μ n ω = g n ω","state_after":"case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\nω : Ω\nhω : martingalePart (f + g) ℱ μ n ω = f n ω\n⊢ f n ω + predictablePart (f + g) ℱ μ n ω = f n ω + g n ω","tactic":"rw [← add_right_inj (f n ω)]","premises":[{"full_name":"add_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[63,2],"def_end_pos":[63,13]}]},{"state_before":"case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\nω : Ω\nhω : martingalePart (f + g) ℱ μ n ω = f n ω\n⊢ f n ω + predictablePart (f + g) ℱ μ n ω = f n ω + g n ω","state_after":"case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\nω : Ω\nhω : martingalePart (f + g) ℱ μ n ω = f n ω\n⊢ f n ω + predictablePart (f + g) ℱ μ n ω = (martingalePart (f + g) ℱ μ + predictablePart (f + g) ℱ μ) n ω","tactic":"conv_rhs => rw [← Pi.add_apply, ← Pi.add_apply, ← martingalePart_add_predictablePart ℱ μ (f + g)]","premises":[{"full_name":"MeasureTheory.martingalePart_add_predictablePart","def_path":"Mathlib/Probability/Martingale/Centering.lean","def_pos":[63,8],"def_end_pos":[63,42]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]}]},{"state_before":"case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn✝ : ℕ\ninst✝ : SigmaFiniteFiltration μ ℱ\nf g : ℕ → Ω → E\nhf : Martingale f ℱ μ\nhg : Adapted ℱ fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : ∀ (n : ℕ), Integrable (g n) μ\nn : ℕ\nω : Ω\nhω : martingalePart (f + g) ℱ μ n ω = f n ω\n⊢ f n ω + predictablePart (f + g) ℱ μ n ω = (martingalePart (f + g) ℱ μ + predictablePart (f + g) ℱ μ) n ω","state_after":"no goals","tactic":"rw [Pi.add_apply, Pi.add_apply, hω]","premises":[{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]}]}]}
{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean","commit":"","full_name":"pow_lt_pow_right'","start":[74,0],"end":[79,65],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean","tactics":[{"state_before":"β : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn m : ℕ\nha : 1 < a\nh : n < m\n⊢ a ^ n < a ^ m","state_after":"case intro\nβ : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : ℕ\nha : 1 < a\nk : ℕ\nh : n < n.succ + k\n⊢ a ^ n < a ^ (n.succ + k)","tactic":"rcases Nat.le.dest h with ⟨k, rfl⟩","premises":[{"full_name":"Nat.le.dest","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[421,8],"def_end_pos":[421,15]}]},{"state_before":"case intro\nβ : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : ℕ\nha : 1 < a\nk : ℕ\nh : n < n.succ + k\n⊢ a ^ n < a ^ (n.succ + k)","state_after":"case intro\nβ : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : ℕ\nha : 1 < a\nk : ℕ\n⊢ a ^ n < a ^ (n.succ + k)","tactic":"clear h","premises":[]},{"state_before":"case intro\nβ : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : ℕ\nha : 1 < a\nk : ℕ\n⊢ a ^ n < a ^ (n.succ + k)","state_after":"case intro\nβ : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : ℕ\nha : 1 < a\nk : ℕ\n⊢ a ^ n < a ^ n * a ^ (k + 1)","tactic":"rw [pow_add, pow_succ, mul_assoc, ← pow_succ']","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"case intro\nβ : Type u_1\nG : Type u_2\nM : Type u_3\ninst✝³ : Monoid M\ninst✝² : Preorder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nx : M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : ℕ\nha : 1 < a\nk : ℕ\n⊢ a ^ n < a ^ n * a ^ (k + 1)","state_after":"no goals","tactic":"exact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)","premises":[{"full_name":"Nat.succ_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[708,16],"def_end_pos":[708,28]},{"full_name":"lt_mul_of_one_lt_right'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[412,8],"def_end_pos":[412,31]},{"full_name":"one_lt_pow'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean","def_pos":[62,8],"def_end_pos":[62,19]}]}]}
{"url":"Mathlib/CategoryTheory/Adjunction/Whiskering.lean","commit":"","full_name":"CategoryTheory.Adjunction.whiskerLeft_unit_app_app","start":[42,0],"end":[58,5],"file_path":"Mathlib/CategoryTheory/Adjunction/Whiskering.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\n⊢ ∀ ⦃X Y : D ⥤ C⦄ (f : X ⟶ Y),\n    (𝟭 (D ⥤ C)).map f ≫\n        (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) Y =\n      (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) X ≫\n        ((whiskeringLeft E D C).obj G ⋙ (whiskeringLeft D E C).obj F).map f","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : D ⥤ C\nf✝ : X✝ ⟶ Y✝\n⊢ (𝟭 (D ⥤ C)).map f✝ ≫\n      (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) Y✝ =\n    (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) X✝ ≫\n      ((whiskeringLeft E D C).obj G ⋙ (whiskeringLeft D E C).obj F).map f✝","tactic":"intros","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : D ⥤ C\nf✝ : X✝ ⟶ Y✝\n⊢ (𝟭 (D ⥤ C)).map f✝ ≫\n      (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) Y✝ =\n    (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) X✝ ≫\n      ((whiskeringLeft E D C).obj G ⋙ (whiskeringLeft D E C).obj F).map f✝","state_after":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : D ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : D\n⊢ ((𝟭 (D ⥤ C)).map f✝ ≫\n          (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) Y✝).app\n      x✝ =\n    ((fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) X✝ ≫\n          ((whiskeringLeft E D C).obj G ⋙ (whiskeringLeft D E C).obj F).map f✝).app\n      x✝","tactic":"ext","premises":[]},{"state_before":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : D ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : D\n⊢ ((𝟭 (D ⥤ C)).map f✝ ≫\n          (fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) Y✝).app\n      x✝ =\n    ((fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom) X✝ ≫\n          ((whiskeringLeft E D C).obj G ⋙ (whiskeringLeft D E C).obj F).map f✝).app\n      x✝","state_after":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : D ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : D\n⊢ f✝.app x✝ ≫ 𝟙 (Y✝.obj x✝) ≫ Y✝.map (adj.unit.app x✝) ≫ 𝟙 (Y✝.obj (G.obj (F.obj x✝))) =\n    (𝟙 (X✝.obj x✝) ≫ X✝.map (adj.unit.app x✝) ≫ 𝟙 (X✝.obj (G.obj (F.obj x✝)))) ≫ f✝.app (G.obj (F.obj x✝))","tactic":"dsimp","premises":[]},{"state_before":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : D ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : D\n⊢ f✝.app x✝ ≫ 𝟙 (Y✝.obj x✝) ≫ Y✝.map (adj.unit.app x✝) ≫ 𝟙 (Y✝.obj (G.obj (F.obj x✝))) =\n    (𝟙 (X✝.obj x✝) ≫ X✝.map (adj.unit.app x✝) ≫ 𝟙 (X✝.obj (G.obj (F.obj x✝)))) ≫ f✝.app (G.obj (F.obj x✝))","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\n⊢ ∀ ⦃X Y : E ⥤ C⦄ (f : X ⟶ Y),\n    ((whiskeringLeft D E C).obj F ⋙ (whiskeringLeft E D C).obj G).map f ≫\n        (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) Y =\n      (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) X ≫ (𝟭 (E ⥤ C)).map f","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : E ⥤ C\nf✝ : X✝ ⟶ Y✝\n⊢ ((whiskeringLeft D E C).obj F ⋙ (whiskeringLeft E D C).obj G).map f✝ ≫\n      (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) Y✝ =\n    (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) X✝ ≫ (𝟭 (E ⥤ C)).map f✝","tactic":"intros","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : E ⥤ C\nf✝ : X✝ ⟶ Y✝\n⊢ ((whiskeringLeft D E C).obj F ⋙ (whiskeringLeft E D C).obj G).map f✝ ≫\n      (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) Y✝ =\n    (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) X✝ ≫ (𝟭 (E ⥤ C)).map f✝","state_after":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : E ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : E\n⊢ (((whiskeringLeft D E C).obj F ⋙ (whiskeringLeft E D C).obj G).map f✝ ≫\n          (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) Y✝).app\n      x✝ =\n    ((fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) X✝ ≫ (𝟭 (E ⥤ C)).map f✝).app x✝","tactic":"ext","premises":[]},{"state_before":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : E ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : E\n⊢ (((whiskeringLeft D E C).obj F ⋙ (whiskeringLeft E D C).obj G).map f✝ ≫\n          (fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) Y✝).app\n      x✝ =\n    ((fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom) X✝ ≫ (𝟭 (E ⥤ C)).map f✝).app x✝","state_after":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : E ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : E\n⊢ f✝.app (F.obj (G.obj x✝)) ≫ 𝟙 (Y✝.obj (F.obj (G.obj x✝))) ≫ Y✝.map (adj.counit.app x✝) ≫ 𝟙 (Y✝.obj x✝) =\n    (𝟙 (X✝.obj (F.obj (G.obj x✝))) ≫ X✝.map (adj.counit.app x✝) ≫ 𝟙 (X✝.obj x✝)) ≫ f✝.app x✝","tactic":"dsimp","premises":[]},{"state_before":"case w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX✝ Y✝ : E ⥤ C\nf✝ : X✝ ⟶ Y✝\nx✝ : E\n⊢ f✝.app (F.obj (G.obj x✝)) ≫ 𝟙 (Y✝.obj (F.obj (G.obj x✝))) ≫ Y✝.map (adj.counit.app x✝) ≫ 𝟙 (Y✝.obj x✝) =\n    (𝟙 (X✝.obj (F.obj (G.obj x✝))) ≫ X✝.map (adj.counit.app x✝) ≫ 𝟙 (X✝.obj x✝)) ≫ f✝.app x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\n⊢ whiskerRight\n        { app := fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n          naturality := ⋯ }\n        ((whiskeringLeft E D C).obj G) ≫\n      (((whiskeringLeft E D C).obj G).associator ((whiskeringLeft D E C).obj F) ((whiskeringLeft E D C).obj G)).hom ≫\n        whiskerLeft ((whiskeringLeft E D C).obj G)\n          { app := fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom, naturality := ⋯ } =\n    NatTrans.id (𝟭 (D ⥤ C) ⋙ (whiskeringLeft E D C).obj G)","state_after":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : D ⥤ C\nx✝ : E\n⊢ ((whiskerRight\n                { app := fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n                  naturality := ⋯ }\n                ((whiskeringLeft E D C).obj G) ≫\n              (((whiskeringLeft E D C).obj G).associator ((whiskeringLeft D E C).obj F)\n                    ((whiskeringLeft E D C).obj G)).hom ≫\n                whiskerLeft ((whiskeringLeft E D C).obj G)\n                  { app := fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom,\n                    naturality := ⋯ }).app\n          x).app\n      x✝ =\n    ((NatTrans.id (𝟭 (D ⥤ C) ⋙ (whiskeringLeft E D C).obj G)).app x).app x✝","tactic":"ext x","premises":[]},{"state_before":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : D ⥤ C\nx✝ : E\n⊢ ((whiskerRight\n                { app := fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n                  naturality := ⋯ }\n                ((whiskeringLeft E D C).obj G) ≫\n              (((whiskeringLeft E D C).obj G).associator ((whiskeringLeft D E C).obj F)\n                    ((whiskeringLeft E D C).obj G)).hom ≫\n                whiskerLeft ((whiskeringLeft E D C).obj G)\n                  { app := fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom,\n                    naturality := ⋯ }).app\n          x).app\n      x✝ =\n    ((NatTrans.id (𝟭 (D ⥤ C) ⋙ (whiskeringLeft E D C).obj G)).app x).app x✝","state_after":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : D ⥤ C\nx✝ : E\n⊢ (𝟙 (x.obj (G.obj x✝)) ≫ x.map (adj.unit.app (G.obj x✝)) ≫ 𝟙 (x.obj (G.obj (F.obj (G.obj x✝))))) ≫\n      𝟙 (x.obj (G.obj (F.obj (G.obj x✝)))) ≫\n        𝟙 (x.obj (G.obj (F.obj (G.obj x✝)))) ≫ x.map (G.map (adj.counit.app x✝)) ≫ 𝟙 (x.obj (G.obj x✝)) =\n    𝟙 (x.obj (G.obj x✝))","tactic":"dsimp","premises":[]},{"state_before":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : D ⥤ C\nx✝ : E\n⊢ (𝟙 (x.obj (G.obj x✝)) ≫ x.map (adj.unit.app (G.obj x✝)) ≫ 𝟙 (x.obj (G.obj (F.obj (G.obj x✝))))) ≫\n      𝟙 (x.obj (G.obj (F.obj (G.obj x✝)))) ≫\n        𝟙 (x.obj (G.obj (F.obj (G.obj x✝)))) ≫ x.map (G.map (adj.counit.app x✝)) ≫ 𝟙 (x.obj (G.obj x✝)) =\n    𝟙 (x.obj (G.obj x✝))","state_after":"no goals","tactic":"simp [Category.id_comp, Category.comp_id, ← x.map_comp]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\n⊢ whiskerLeft ((whiskeringLeft D E C).obj F)\n        { app := fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n          naturality := ⋯ } ≫\n      (((whiskeringLeft D E C).obj F).associator ((whiskeringLeft E D C).obj G) ((whiskeringLeft D E C).obj F)).inv ≫\n        whiskerRight\n          { app := fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom, naturality := ⋯ }\n          ((whiskeringLeft D E C).obj F) =\n    NatTrans.id ((whiskeringLeft D E C).obj F ⋙ 𝟭 (D ⥤ C))","state_after":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : E ⥤ C\nx✝ : D\n⊢ ((whiskerLeft ((whiskeringLeft D E C).obj F)\n                { app := fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n                  naturality := ⋯ } ≫\n              (((whiskeringLeft D E C).obj F).associator ((whiskeringLeft E D C).obj G)\n                    ((whiskeringLeft D E C).obj F)).inv ≫\n                whiskerRight\n                  { app := fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom,\n                    naturality := ⋯ }\n                  ((whiskeringLeft D E C).obj F)).app\n          x).app\n      x✝ =\n    ((NatTrans.id ((whiskeringLeft D E C).obj F ⋙ 𝟭 (D ⥤ C))).app x).app x✝","tactic":"ext x","premises":[]},{"state_before":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : E ⥤ C\nx✝ : D\n⊢ ((whiskerLeft ((whiskeringLeft D E C).obj F)\n                { app := fun X => ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n                  naturality := ⋯ } ≫\n              (((whiskeringLeft D E C).obj F).associator ((whiskeringLeft E D C).obj G)\n                    ((whiskeringLeft D E C).obj F)).inv ≫\n                whiskerRight\n                  { app := fun X => (G.associator F X).inv ≫ whiskerRight adj.counit X ≫ X.leftUnitor.hom,\n                    naturality := ⋯ }\n                  ((whiskeringLeft D E C).obj F)).app\n          x).app\n      x✝ =\n    ((NatTrans.id ((whiskeringLeft D E C).obj F ⋙ 𝟭 (D ⥤ C))).app x).app x✝","state_after":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : E ⥤ C\nx✝ : D\n⊢ (𝟙 (x.obj (F.obj x✝)) ≫ x.map (F.map (adj.unit.app x✝)) ≫ 𝟙 (x.obj (F.obj (G.obj (F.obj x✝))))) ≫\n      𝟙 (x.obj (F.obj (G.obj (F.obj x✝)))) ≫\n        𝟙 (x.obj (F.obj (G.obj (F.obj x✝)))) ≫ x.map (adj.counit.app (F.obj x✝)) ≫ 𝟙 (x.obj (F.obj x✝)) =\n    𝟙 (x.obj (F.obj x✝))","tactic":"dsimp","premises":[]},{"state_before":"case w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{?u.20707, u_1} C\ninst✝¹ : Category.{?u.20711, u_2} D\ninst✝ : Category.{?u.20715, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nx : E ⥤ C\nx✝ : D\n⊢ (𝟙 (x.obj (F.obj x✝)) ≫ x.map (F.map (adj.unit.app x✝)) ≫ 𝟙 (x.obj (F.obj (G.obj (F.obj x✝))))) ≫\n      𝟙 (x.obj (F.obj (G.obj (F.obj x✝)))) ≫\n        𝟙 (x.obj (F.obj (G.obj (F.obj x✝)))) ≫ x.map (adj.counit.app (F.obj x✝)) ≫ 𝟙 (x.obj (F.obj x✝)) =\n    𝟙 (x.obj (F.obj x✝))","state_after":"no goals","tactic":"simp [Category.id_comp, Category.comp_id, ← x.map_comp]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Basic.lean","commit":"","full_name":"MvPolynomial.adjoin_range_X","start":[448,0],"end":[455,80],"file_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","tactics":[{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\n⊢ Algebra.adjoin R (range X) = ⊤","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\n⊢ S = ⊤","tactic":"set S := Algebra.adjoin R (range (X : σ → MvPolynomial σ R))","premises":[{"full_name":"Algebra.adjoin","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[618,4],"def_end_pos":[618,10]},{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"MvPolynomial.X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[178,4],"def_end_pos":[178,5]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\n⊢ S = ⊤","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\nhp : p ∈ ⊤\n⊢ p ∈ S","tactic":"refine top_unique fun p hp => ?_","premises":[{"full_name":"top_unique","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[113,8],"def_end_pos":[113,18]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\nhp : p ∈ ⊤\n⊢ p ∈ S","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\n⊢ p ∈ S","tactic":"clear hp","premises":[]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\n⊢ p ∈ S","state_after":"no goals","tactic":"induction p using MvPolynomial.induction_on with\n  | h_C => exact S.algebraMap_mem _\n  | h_add p q hp hq => exact S.add_mem hp hq\n  | h_X p i hp => exact S.mul_mem hp (Algebra.subset_adjoin <| mem_range_self _)","premises":[{"full_name":"Algebra.subset_adjoin","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[44,8],"def_end_pos":[44,21]},{"full_name":"MvPolynomial.induction_on","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[399,8],"def_end_pos":[399,20]},{"full_name":"Set.mem_range_self","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[148,22],"def_end_pos":[148,36]},{"full_name":"Subalgebra.add_mem","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[120,18],"def_end_pos":[120,25]},{"full_name":"Subalgebra.algebraMap_mem","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[94,18],"def_end_pos":[94,32]},{"full_name":"Subalgebra.mul_mem","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[111,18],"def_end_pos":[111,25]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.exists_of_set","start":[408,0],"end":[410,54],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nn : Nat\na' : α\nl : List α\nh : n < l.length\n⊢ ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂","state_after":"α : Type u_1\nn : Nat\na' : α\nl : List α\nh : n < l.length\n⊢ ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ modifyNth (fun x => a') n l = l₁ ++ a' :: l₂","tactic":"rw [set_eq_modifyNth]","premises":[{"full_name":"List.set_eq_modifyNth","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[388,8],"def_end_pos":[388,24]}]},{"state_before":"α : Type u_1\nn : Nat\na' : α\nl : List α\nh : n < l.length\n⊢ ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ modifyNth (fun x => a') n l = l₁ ++ a' :: l₂","state_after":"no goals","tactic":"exact exists_of_modifyNth _ h","premises":[{"full_name":"List.exists_of_modifyNth","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[366,8],"def_end_pos":[366,27]}]}]}
{"url":"Mathlib/Data/Nat/Factorization/Defs.lean","commit":"","full_name":"Nat.factorization_le_iff_dvd","start":[150,0],"end":[160,8],"file_path":"Mathlib/Data/Nat/Factorization/Defs.lean","tactics":[{"state_before":"a b m n✝ p d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\n⊢ d.factorization ≤ n.factorization ↔ d ∣ n","state_after":"case mp\na b m n✝ p d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\n⊢ d.factorization ≤ n.factorization → d ∣ n\n\ncase mpr\na b m n✝ p d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\n⊢ d ∣ n → d.factorization ≤ n.factorization","tactic":"constructor","premises":[]}]}
{"url":".lake/packages/batteries/Batteries/Data/HashMap/WF.lean","commit":"","full_name":"Batteries.HashMap.Imp.erase_WF","start":[274,0],"end":[280,11],"file_path":".lake/packages/batteries/Batteries/Data/HashMap/WF.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.buckets.WF\n⊢ (m.erase k).buckets.WF","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.buckets.WF\n⊢ (match AssocList.contains k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat] with\n      | true =>\n        { size := m.size - 1,\n          buckets :=\n            m.buckets.update (mkIdx ⋯ (hash k).toUSize).val\n              (AssocList.erase k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat]) ⋯ }\n      | false => { size := m.size, buckets := m.buckets }).buckets.WF","tactic":"dsimp [erase, cond]","premises":[{"full_name":"Batteries.HashMap.Imp.erase","def_path":".lake/packages/batteries/Batteries/Data/HashMap/Basic.lean","def_pos":[178,4],"def_end_pos":[178,9]},{"full_name":"cond","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1006,20],"def_end_pos":[1006,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.buckets.WF\n⊢ (match AssocList.contains k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat] with\n      | true =>\n        { size := m.size - 1,\n          buckets :=\n            m.buckets.update (mkIdx ⋯ (hash k).toUSize).val\n              (AssocList.erase k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat]) ⋯ }\n      | false => { size := m.size, buckets := m.buckets }).buckets.WF","state_after":"case h_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.buckets.WF\nc✝ : Bool\nheq✝ : AssocList.contains k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat] = true\n⊢ { size := m.size - 1,\n        buckets :=\n          m.buckets.update (mkIdx ⋯ (hash k).toUSize).val\n            (AssocList.erase k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat]) ⋯ }.buckets.WF\n\ncase h_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.buckets.WF\nc✝ : Bool\nheq✝ : AssocList.contains k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat] = false\n⊢ { size := m.size, buckets := m.buckets }.buckets.WF","tactic":"split","premises":[]}]}
{"url":"Mathlib/NumberTheory/ZetaValues.lean","commit":"","full_name":"integral_bernoulliFun_eq_zero","start":[72,0],"end":[79,8],"file_path":"Mathlib/NumberTheory/ZetaValues.lean","tactics":[{"state_before":"k : ℕ\nhk : k ≠ 0\n⊢ ∫ (x : ℝ) in 0 ..1, bernoulliFun k x = 0","state_after":"k : ℕ\nhk : k ≠ 0\n⊢ bernoulliFun (k + 1) 1 / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0","tactic":"rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)\n      ((Polynomial.continuous _).intervalIntegrable _ _)]","premises":[{"full_name":"Continuous.intervalIntegrable","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[337,8],"def_end_pos":[337,37]},{"full_name":"Polynomial.continuous","def_path":"Mathlib/Topology/Algebra/Polynomial.lean","def_pos":[53,18],"def_end_pos":[53,28]},{"full_name":"antideriv_bernoulliFun","def_path":"Mathlib/NumberTheory/ZetaValues.lean","def_pos":[67,8],"def_end_pos":[67,30]},{"full_name":"intervalIntegral.integral_eq_sub_of_hasDerivAt","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[1141,8],"def_end_pos":[1141,37]}]},{"state_before":"k : ℕ\nhk : k ≠ 0\n⊢ bernoulliFun (k + 1) 1 / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0","state_after":"k : ℕ\nhk : k ≠ 0\n⊢ (bernoulliFun (k + 1) 0 + if k + 1 = 1 then 1 else 0) / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0","tactic":"rw [bernoulliFun_eval_one]","premises":[{"full_name":"bernoulliFun_eval_one","def_path":"Mathlib/NumberTheory/ZetaValues.lean","def_pos":[54,8],"def_end_pos":[54,29]}]},{"state_before":"k : ℕ\nhk : k ≠ 0\n⊢ (bernoulliFun (k + 1) 0 + if k + 1 = 1 then 1 else 0) / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0","state_after":"case pos\nk : ℕ\nhk : k ≠ 0\nh : k + 1 = 1\n⊢ (bernoulliFun (k + 1) 0 + 1) / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0\n\ncase neg\nk : ℕ\nhk : k ≠ 0\nh : ¬k + 1 = 1\n⊢ (bernoulliFun (k + 1) 0 + 0) / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Nat/Size.lean","commit":"","full_name":"Nat.size_pow","start":[118,0],"end":[120,25],"file_path":"Mathlib/Data/Nat/Size.lean","tactics":[{"state_before":"n : ℕ\n⊢ 1 < 2","state_after":"no goals","tactic":"decide","premises":[]}]}
{"url":"Mathlib/Algebra/Ring/Subsemiring/Basic.lean","commit":"","full_name":"Subsemiring.centralizer_centralizer_centralizer","start":[623,0],"end":[627,70],"file_path":"Mathlib/Algebra/Ring/Subsemiring/Basic.lean","tactics":[{"state_before":"R✝ : Type u\nS : Type v\nT : Type w\ninst✝³ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝² : NonAssocSemiring S\ninst✝¹ : NonAssocSemiring T\nR : Type u_1\ninst✝ : Semiring R\ns : Set R\n⊢ centralizer s.centralizer.centralizer = centralizer s","state_after":"case a\nR✝ : Type u\nS : Type v\nT : Type w\ninst✝³ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝² : NonAssocSemiring S\ninst✝¹ : NonAssocSemiring T\nR : Type u_1\ninst✝ : Semiring R\ns : Set R\n⊢ ↑(centralizer s.centralizer.centralizer) = ↑(centralizer s)","tactic":"apply SetLike.coe_injective","premises":[{"full_name":"SetLike.coe_injective","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[145,8],"def_end_pos":[145,21]}]},{"state_before":"case a\nR✝ : Type u\nS : Type v\nT : Type w\ninst✝³ : NonAssocSemiring R✝\nM : Submonoid R✝\ninst✝² : NonAssocSemiring S\ninst✝¹ : NonAssocSemiring T\nR : Type u_1\ninst✝ : Semiring R\ns : Set R\n⊢ ↑(centralizer s.centralizer.centralizer) = ↑(centralizer s)","state_after":"no goals","tactic":"simp only [coe_centralizer, Set.centralizer_centralizer_centralizer]","premises":[{"full_name":"Set.centralizer_centralizer_centralizer","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[151,6],"def_end_pos":[151,41]},{"full_name":"Subsemiring.coe_centralizer","def_path":"Mathlib/Algebra/Ring/Subsemiring/Basic.lean","def_pos":[593,8],"def_end_pos":[593,23]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","commit":"","full_name":"CategoryTheory.leftAdjointMate_comp_evaluation","start":[454,0],"end":[458,6],"file_path":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\n⊢ (X ◁ ᘁf) ≫ ε_ (ᘁX) X = f ▷ ᘁY ≫ ε_ (ᘁY) Y","state_after":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\n⊢ (tensorLeftHomEquiv (ᘁY) (ᘁX) X (𝟙_ C)) ((X ◁ ᘁf) ≫ ε_ (ᘁX) X) =\n    (tensorLeftHomEquiv (ᘁY) (ᘁX) X (𝟙_ C)) (f ▷ ᘁY ≫ ε_ (ᘁY) Y)","tactic":"apply_fun tensorLeftHomEquiv _ (ᘁX) X _","premises":[{"full_name":"CategoryTheory.HasLeftDual.leftDual","def_path":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","def_pos":[143,2],"def_end_pos":[143,10]},{"full_name":"CategoryTheory.tensorLeftHomEquiv","def_path":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","def_pos":[266,4],"def_end_pos":[266,22]},{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]}]},{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\n⊢ (tensorLeftHomEquiv (ᘁY) (ᘁX) X (𝟙_ C)) ((X ◁ ᘁf) ≫ ε_ (ᘁX) X) =\n    (tensorLeftHomEquiv (ᘁY) (ᘁX) X (𝟙_ C)) (f ▷ ᘁY ≫ ε_ (ᘁY) Y)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/GroupTheory/Congruence/Opposite.lean","commit":"","full_name":"Con.orderIsoOp_symm_apply","start":[46,0],"end":[56,93],"file_path":"Mathlib/GroupTheory/Congruence/Opposite.lean","tactics":[{"state_before":"M : Type u_1\ninst✝ : Mul M\nc d : Con M\n⊢ { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } c ≤\n      { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } d ↔\n    c ≤ d","state_after":"M : Type u_1\ninst✝ : Mul M\nc d : Con M\n⊢ (∀ {x y : Mᵐᵒᵖ},\n      ({ toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } c) x y →\n        ({ toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } d) x y) ↔\n    ∀ {x y : M}, c x y → d x y","tactic":"rw [le_def, le_def]","premises":[{"full_name":"Con.le_def","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[351,8],"def_end_pos":[351,14]}]},{"state_before":"M : Type u_1\ninst✝ : Mul M\nc d : Con M\n⊢ (∀ {x y : Mᵐᵒᵖ},\n      ({ toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } c) x y →\n        ({ toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } d) x y) ↔\n    ∀ {x y : M}, c x y → d x y","state_after":"no goals","tactic":"constructor <;> intro h _ _ h' <;> exact h h'","premises":[]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.sub_isLimit","start":[520,0],"end":[522,52],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_4}\nl : a.IsLimit\nh : b < a\n⊢ b + 0 < a","state_after":"no goals","tactic":"rwa [add_zero]","premises":[{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_4}\nl : a.IsLimit\nh✝ : b < a\nc : Ordinal.{u_4}\nh : c < a - b\n⊢ succ c < a - b","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_4}\nl : a.IsLimit\nh✝ : b < a\nc : Ordinal.{u_4}\nh : c < a - b\n⊢ succ (b + c) < a","tactic":"rw [lt_sub, add_succ]","premises":[{"full_name":"Ordinal.add_succ","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[934,8],"def_end_pos":[934,16]},{"full_name":"Ordinal.lt_sub","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[470,8],"def_end_pos":[470,14]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_4}\nl : a.IsLimit\nh✝ : b < a\nc : Ordinal.{u_4}\nh : c < a - b\n⊢ succ (b + c) < a","state_after":"no goals","tactic":"exact l.2 _ (lt_sub.1 h)","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ordinal.lt_sub","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[470,8],"def_end_pos":[470,14]}]}]}
{"url":"Mathlib/Algebra/Algebra/Spectrum.lean","commit":"","full_name":"spectrum.algebraMap_mem_iff","start":[128,0],"end":[132,84],"file_path":"Mathlib/Algebra/Algebra/Spectrum.lean","tactics":[{"state_before":"R✝ : Type u\nA✝ : Type v\ninst✝⁹ : CommSemiring R✝\ninst✝⁸ : Ring A✝\ninst✝⁷ : Algebra R✝ A✝\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ (algebraMap R S) r ∈ spectrum S a ↔ r ∈ σ a","state_after":"no goals","tactic":"simp only [spectrum.mem_iff, Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]","premises":[{"full_name":"Algebra.algebraMap_eq_smul_one","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[273,8],"def_end_pos":[273,30]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"smul_assoc","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[219,6],"def_end_pos":[219,16]},{"full_name":"spectrum.mem_iff","def_path":"Mathlib/Algebra/Algebra/Spectrum.lean","def_pos":[97,8],"def_end_pos":[97,15]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","commit":"","full_name":"CategoryTheory.Limits.cokernelIsoOfEq_inv_comp_desc","start":[813,0],"end":[817,15],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\ne : Y ⟶ Z\nhe : f ≫ e = 0\n⊢ g ≫ e = 0","state_after":"no goals","tactic":"simp [← h, he]","premises":[]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\ne : Y ⟶ Z\nhe : f ≫ e = 0\n⊢ (cokernelIsoOfEq h).inv ≫ cokernel.desc f e he = cokernel.desc g e ⋯","state_after":"case refl\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf : X ⟶ Y\ninst✝¹ : HasCokernel f\ne : Y ⟶ Z\nhe : f ≫ e = 0\ninst✝ : HasCokernel f\n⊢ (cokernelIsoOfEq ⋯).inv ≫ cokernel.desc f e he = cokernel.desc f e ⋯","tactic":"cases h","premises":[]},{"state_before":"case refl\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf : X ⟶ Y\ninst✝¹ : HasCokernel f\ne : Y ⟶ Z\nhe : f ≫ e = 0\ninst✝ : HasCokernel f\n⊢ (cokernelIsoOfEq ⋯).inv ≫ cokernel.desc f e he = cokernel.desc f e ⋯","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Sign.lean","commit":"","full_name":"exists_signed_sum'","start":[507,0],"end":[520,7],"file_path":"Mathlib/Data/Sign.lean","tactics":[{"state_before":"α✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\n⊢ ∃ β x sgn g,\n    (∀ (b : β), g b ∉ s → sgn b = 0) ∧ Fintype.card β = n ∧ ∀ a ∈ s, (∑ i : β, if g i = a then ↑(sgn i) else 0) = f a","state_after":"case intro.intro.intro.intro.intro.intro\nα✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u_1\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b : β, if g b = a then ↑(sgn b) else 0) = f a\n⊢ ∃ β x sgn g,\n    (∀ (b : β), g b ∉ s → sgn b = 0) ∧ Fintype.card β = n ∧ ∀ a ∈ s, (∑ i : β, if g i = a then ↑(sgn i) else 0) = f a","tactic":"obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f","premises":[{"full_name":"exists_signed_sum","def_path":"Mathlib/Data/Sign.lean","def_pos":[498,8],"def_end_pos":[498,25]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nα✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u_1\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b : β, if g b = a then ↑(sgn b) else 0) = f a\n⊢ ∃ β x sgn g,\n    (∀ (b : β), g b ∉ s → sgn b = 0) ∧ Fintype.card β = n ∧ ∀ a ∈ s, (∑ i : β, if g i = a then ↑(sgn i) else 0) = f a","state_after":"case intro.intro.intro.intro.intro.intro\nα✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u_1\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b : β, if g b = a then ↑(sgn b) else 0) = f a\n⊢ ∀ (b : β ⊕ Fin (n - ∑ i ∈ s, (f i).natAbs)),\n    Sum.elim g (Classical.arbitrary (Fin (n - ∑ i ∈ s, (f i).natAbs) → α)) b ∉ s → Sum.elim sgn 0 b = 0","tactic":"refine\n    ⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,\n      Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),\n        ?_, by simp [hβ, h], fun a ha => by simp [hf _ ha]⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Classical.arbitrary","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[85,31],"def_end_pos":[85,50]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Int.natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[262,4],"def_end_pos":[262,10]},{"full_name":"Sum","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[130,10],"def_end_pos":[130,13]},{"full_name":"Sum.elim","def_path":".lake/packages/batteries/Batteries/Data/Sum/Basic.lean","def_pos":[82,14],"def_end_pos":[82,18]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nα✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u_1\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b : β, if g b = a then ↑(sgn b) else 0) = f a\n⊢ ∀ (b : β ⊕ Fin (n - ∑ i ∈ s, (f i).natAbs)),\n    Sum.elim g (Classical.arbitrary (Fin (n - ∑ i ∈ s, (f i).natAbs) → α)) b ∉ s → Sum.elim sgn 0 b = 0","state_after":"case intro.intro.intro.intro.intro.intro.inl\nα✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u_1\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b : β, if g b = a then ↑(sgn b) else 0) = f a\nb : β\nhb : Sum.elim g (Classical.arbitrary (Fin (n - ∑ i ∈ s, (f i).natAbs) → α)) (Sum.inl b) ∉ s\n⊢ Sum.elim sgn 0 (Sum.inl b) = 0\n\ncase intro.intro.intro.intro.intro.intro.inr\nα✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u_1\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b : β, if g b = a then ↑(sgn b) else 0) = f a\nb : Fin (n - ∑ i ∈ s, (f i).natAbs)\nhb : Sum.elim g (Classical.arbitrary (Fin (n - ∑ i ∈ s, (f i).natAbs) → α)) (Sum.inr b) ∉ s\n⊢ Sum.elim sgn 0 (Sum.inr b) = 0","tactic":"rintro (b | b) hb","premises":[]}]}
{"url":"Mathlib/Analysis/Fourier/PoissonSummation.lean","commit":"","full_name":"isBigO_norm_Icc_restrict_atBot","start":[154,0],"end":[172,50],"file_path":"Mathlib/Analysis/Fourier/PoissonSummation.lean","tactics":[{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","state_after":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","tactic":"have h1 : (f.comp (ContinuousMap.mk _ continuous_neg)) =O[atTop] fun x : ℝ => |x| ^ (-b) := by\n    convert hf.comp_tendsto tendsto_neg_atTop_atBot using 1\n    ext1 x; simp only [Function.comp_apply, abs_neg]","premises":[{"full_name":"Asymptotics.IsBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[93,16],"def_end_pos":[93,22]},{"full_name":"Asymptotics.IsBigO.comp_tendsto","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[390,8],"def_end_pos":[390,27]},{"full_name":"ContinuousMap.comp","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[205,4],"def_end_pos":[205,8]},{"full_name":"ContinuousNeg.continuous_neg","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[141,2],"def_end_pos":[141,16]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Filter.tendsto_neg_atTop_atBot","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[761,8],"def_end_pos":[761,31]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"abs_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[70,2],"def_end_pos":[70,13]}]},{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","state_after":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    ((fun x => |x| ^ (-b)) ∘ Neg.neg)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","tactic":"have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_atBot_atTop","premises":[{"full_name":"Asymptotics.IsBigO.comp_tendsto","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[390,8],"def_end_pos":[390,27]},{"full_name":"Filter.tendsto_neg_atBot_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[764,8],"def_end_pos":[764,31]},{"full_name":"isBigO_norm_Icc_restrict_atTop","def_path":"Mathlib/Analysis/Fourier/PoissonSummation.lean","def_pos":[126,8],"def_end_pos":[126,38]}]},{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    ((fun x => |x| ^ (-b)) ∘ Neg.neg)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","state_after":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    ((fun x => |x| ^ (-b)) ∘ Neg.neg)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","tactic":"have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) := by\n    ext1 x; simp only [Function.comp_apply, abs_neg]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Neg.neg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1329,2],"def_end_pos":[1329,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"abs_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[70,2],"def_end_pos":[70,13]}]},{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    ((fun x => |x| ^ (-b)) ∘ Neg.neg)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","state_after":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","tactic":"rw [this] at h2","premises":[]},{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)","state_after":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx : ℝ\n⊢ ‖‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖‖ ≤\n    ‖((fun x =>\n            ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n          Neg.neg)\n        x‖","tactic":"refine (isBigO_of_le _ fun x => ?_).trans h2","premises":[{"full_name":"Asymptotics.IsBigO.trans","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[431,8],"def_end_pos":[431,20]},{"full_name":"Asymptotics.isBigO_of_le","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[507,8],"def_end_pos":[507,20]}]},{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx : ℝ\n⊢ ‖‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖‖ ≤\n    ‖((fun x =>\n            ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n          Neg.neg)\n        x‖","state_after":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx : ℝ\n⊢ ∀ (x_1 : ↑(Icc (x + R) (x + S))),\n    ‖(ContinuousMap.restrict (Icc (x + R) (x + S)) f) x_1‖ ≤\n      ‖ContinuousMap.restrict (Icc (-x + -S) (-x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖","tactic":"rw [norm_norm, Function.comp_apply, norm_norm, ContinuousMap.norm_le _ (norm_nonneg _)]","premises":[{"full_name":"ContinuousMap.norm_le","def_path":"Mathlib/Topology/ContinuousFunction/Compact.lean","def_pos":[187,8],"def_end_pos":[187,15]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]},{"full_name":"norm_norm","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[1028,8],"def_end_pos":[1028,17]}]},{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx : ℝ\n⊢ ∀ (x_1 : ↑(Icc (x + R) (x + S))),\n    ‖(ContinuousMap.restrict (Icc (x + R) (x + S)) f) x_1‖ ≤\n      ‖ContinuousMap.restrict (Icc (-x + -S) (-x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖","state_after":"case mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖(ContinuousMap.restrict (Icc (x✝ + R) (x✝ + S)) f) ⟨x, hx⟩‖ ≤\n    ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖","tactic":"rintro ⟨x, hx⟩","premises":[]},{"state_before":"case mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖(ContinuousMap.restrict (Icc (x✝ + R) (x✝ + S)) f) ⟨x, hx⟩‖ ≤\n    ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖","state_after":"case mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖f x‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖","tactic":"rw [ContinuousMap.restrict_apply_mk]","premises":[{"full_name":"ContinuousMap.restrict_apply_mk","def_path":"Mathlib/Topology/ContinuousFunction/Basic.lean","def_pos":[377,8],"def_end_pos":[377,25]}]},{"state_before":"case mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖f x‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖","state_after":"case mk.refine_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖f x‖ =\n    ‖(ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }))\n        ⟨-x, ?mk.refine_2⟩‖\n\ncase mk.refine_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ -x ∈ Icc (-x✝ + -S) (-x✝ + -R)","tactic":"refine (le_of_eq ?_).trans (ContinuousMap.norm_coe_le_norm _ ⟨-x, ?_⟩)","premises":[{"full_name":"ContinuousMap.norm_coe_le_norm","def_path":"Mathlib/Topology/ContinuousFunction/Compact.lean","def_pos":[179,8],"def_end_pos":[179,24]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]}]}
{"url":"Mathlib/Data/List/InsertNth.lean","commit":"","full_name":"List.length_le_length_insertNth","start":[111,0],"end":[116,40],"file_path":"Mathlib/Data/List/InsertNth.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\n⊢ l.length ≤ (insertNth n x l).length","state_after":"case inl\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : n ≤ l.length\n⊢ l.length ≤ (insertNth n x l).length\n\ncase inr\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : l.length < n\n⊢ l.length ≤ (insertNth n x l).length","tactic":"rcases le_or_lt n l.length with hn | hn","premises":[{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]}]}
{"url":"Mathlib/NumberTheory/FLT/Three.lean","commit":"","full_name":"FermatLastTheoremForThreeGen.Solution.Solution'_descent_multiplicity","start":[693,0],"end":[700,11],"file_path":"Mathlib/NumberTheory/FLT/Three.lean","tactics":[{"state_before":"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\n⊢ S.Solution'_descent.multiplicity = S.multiplicity - 1","state_after":"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nh : λ ^ (S.multiplicity - 1 + 1) ∣ S.Solution'_descent.c\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","tactic":"refine (multiplicity.unique' (by simp [Solution'_descent]) (fun h ↦ S.lambda_not_dvd_X ?_)).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FermatLastTheoremForThreeGen.Solution.Solution'_descent","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[678,18],"def_end_pos":[678,35]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_X","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[587,14],"def_end_pos":[587,30]},{"full_name":"multiplicity.unique'","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[120,8],"def_end_pos":[120,15]}]},{"state_before":"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nh : λ ^ (S.multiplicity - 1 + 1) ∣ S.Solution'_descent.c\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","state_after":"case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1 + 1) * k\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","tactic":"obtain ⟨k, hk : λ^(S.multiplicity-1)*S.X=λ^(S.multiplicity-1+1)*k⟩ := h","premises":[{"full_name":"FermatLastTheoremForThreeGen.Solution.multiplicity","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[218,4],"def_end_pos":[218,25]},{"full_name":"IsPrimitiveRoot.toInteger","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[170,7],"def_end_pos":[170,16]},{"full_name":"_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.X","def_path":"Mathlib/NumberTheory/FLT/Three.lean","def_pos":[565,26],"def_end_pos":[565,27]}]},{"state_before":"case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1 + 1) * k\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","state_after":"case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1) * (λ * k)\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","tactic":"rw [pow_succ, mul_assoc] at hk","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]},{"state_before":"case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1) * (λ * k)\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","state_after":"case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : FermatLastTheoremForThreeGen.Solution.X S = λ * k\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","tactic":"simp only [mul_eq_mul_left_iff, pow_eq_zero_iff', hζ.zeta_sub_one_prime'.ne_zero, ne_eq,\n    false_and, or_false] at hk","premises":[{"full_name":"IsPrimitiveRoot.zeta_sub_one_prime'","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[333,8],"def_end_pos":[333,27]},{"full_name":"Prime.ne_zero","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[47,8],"def_end_pos":[47,15]},{"full_name":"false_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[110,16],"def_end_pos":[110,25]},{"full_name":"mul_eq_mul_left_iff","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[197,8],"def_end_pos":[197,27]},{"full_name":"ne_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[89,16],"def_end_pos":[89,21]},{"full_name":"or_false","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[121,16],"def_end_pos":[121,24]},{"full_name":"pow_eq_zero_iff'","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[178,14],"def_end_pos":[178,30]}]},{"state_before":"case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : FermatLastTheoremForThreeGen.Solution.X S = λ * k\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S","state_after":"no goals","tactic":"simp [hk]","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Subobject/Limits.lean","commit":"","full_name":"CategoryTheory.Limits.cokernelOrderHom_coe","start":[210,0],"end":[230,57],"file_path":"Mathlib/CategoryTheory/Subobject/Limits.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX : C\n⊢ ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [inst : Mono f] [inst_1 : Mono g] (i : A ≅ B),\n    i.hom ≫ g = f →\n      (fun A f x => Subobject.mk (cokernel.π f).op) A f inst = (fun A f x => Subobject.mk (cokernel.π f).op) B g inst_1","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX A B : C\ng : B ⟶ X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ (fun A f x => Subobject.mk (cokernel.π f).op) A (i.hom ≫ g) hf = (fun A f x => Subobject.mk (cokernel.π f).op) B g hg","tactic":"rintro A B f g hf hg i rfl","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX A B : C\ng : B ⟶ X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ (fun A f x => Subobject.mk (cokernel.π f).op) A (i.hom ≫ g) hf = (fun A f x => Subobject.mk (cokernel.π f).op) B g hg","state_after":"case refine_1\nC : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX A B : C\ng : B ⟶ X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ cokernel g ≅ cokernel (i.hom ≫ g)\n\ncase refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX A B : C\ng : B ⟶ X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ (?refine_1.op.hom ≫ (cokernel.π g).op).unop = (cokernel.π (i.hom ≫ g)).op.unop","tactic":"refine Subobject.mk_eq_mk_of_comm _ _ (Iso.op ?_) (Quiver.Hom.unop_inj ?_)","premises":[{"full_name":"CategoryTheory.Iso.op","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[402,14],"def_end_pos":[402,16]},{"full_name":"CategoryTheory.Subobject.mk_eq_mk_of_comm","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[275,8],"def_end_pos":[275,24]},{"full_name":"Quiver.Hom.unop_inj","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[39,8],"def_end_pos":[39,27]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX : C\n⊢ ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [inst : Mono f] [inst_1 : Mono g],\n    Subobject.mk f ≤ Subobject.mk g →\n      Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤\n        Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤\n    Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)","tactic":"intro A B f g hf hg h","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤\n    Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ Subobject.mk (cokernel.π f).op ≤ Subobject.mk (cokernel.π g).op","tactic":"dsimp only [Subobject.lift_mk]","premises":[{"full_name":"CategoryTheory.Subobject.lift_mk","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[141,18],"def_end_pos":[141,25]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ Subobject.mk (cokernel.π f).op ≤ Subobject.mk (cokernel.π g).op","state_after":"case refine_1\nC : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ f ≫ cokernel.π g = 0\n\ncase refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ (cokernel.desc f (cokernel.π g) ?refine_1).op ≫ (cokernel.π f).op = (cokernel.π g).op","tactic":"refine Subobject.mk_le_mk_of_comm (cokernel.desc f (cokernel.π g) ?_).op ?_","premises":[{"full_name":"CategoryTheory.Limits.cokernel.desc","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[696,7],"def_end_pos":[696,20]},{"full_name":"CategoryTheory.Limits.cokernel.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[678,7],"def_end_pos":[678,17]},{"full_name":"CategoryTheory.Subobject.mk_le_mk_of_comm","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[229,8],"def_end_pos":[229,24]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]}]}
{"url":"Mathlib/Tactic/NormNum/DivMod.lean","commit":"","full_name":"Mathlib.Meta.NormNum.isInt_emod_zero","start":[85,0],"end":[86,36],"file_path":"Mathlib/Tactic/NormNum/DivMod.lean","tactics":[{"state_before":"x✝¹ x✝ : ℤ\ne : IsInt x✝¹ x✝\n⊢ IsInt (x✝¹ % ↑0) x✝","state_after":"no goals","tactic":"simp [e]","premises":[]}]}
{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"iInf_sUnion","start":[1922,0],"end":[1924,54],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : CompleteLattice β\nS : Set (Set α)\nf : α → β\n⊢ ⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x","state_after":"no goals","tactic":"rw [sUnion_eq_iUnion, iInf_iUnion, ← iInf_subtype'']","premises":[{"full_name":"Set.sUnion_eq_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1096,8],"def_end_pos":[1096,24]},{"full_name":"iInf_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1906,8],"def_end_pos":[1906,19]},{"full_name":"iInf_subtype''","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[990,8],"def_end_pos":[990,22]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","commit":"","full_name":"Real.BohrMollerup.ge_logGammaSeq","start":[213,0],"end":[222,9],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","tactics":[{"state_before":"f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + logGammaSeq x n ≤ f x","state_after":"f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + ↑m)) ≤ f x","tactic":"dsimp [logGammaSeq]","premises":[{"full_name":"Real.BohrMollerup.logGammaSeq","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","def_pos":[137,4],"def_end_pos":[137,15]}]},{"state_before":"f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + ↑m)) ≤ f x","state_after":"f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n !) ≤ f (↑(n + 1) + x)","tactic":"rw [← add_sub_assoc, sub_le_iff_le_add, ← f_add_nat_eq (@hf_feq) hx, add_comm x _]","premises":[{"full_name":"Real.BohrMollerup.f_add_nat_eq","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","def_pos":[153,8],"def_end_pos":[153,20]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"add_sub_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[329,2],"def_end_pos":[329,13]},{"full_name":"sub_le_iff_le_add","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[498,2],"def_end_pos":[498,13]}]},{"state_before":"f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n !) ≤ f (↑(n + 1) + x)","state_after":"case refine_1\nf : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n !) = f ↑(n + 1) + x * log (↑(n + 1) - 1)\n\ncase refine_2\nf : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ 2 ≤ n + 1","tactic":"refine le_trans (le_of_eq ?_) (f_add_nat_ge hf_conv @hf_feq ?_ hx)","premises":[{"full_name":"Real.BohrMollerup.f_add_nat_ge","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","def_pos":[174,8],"def_end_pos":[174,20]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]}]}
{"url":"Mathlib/CategoryTheory/Category/Cat/Limit.lean","commit":"","full_name":"CategoryTheory.Cat.HasLimits.homDiagram_obj","start":[41,0],"end":[57,60],"file_path":"Mathlib/CategoryTheory/Category/Cat/Limit.lean","tactics":[{"state_before":"J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\ng : (fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y) X✝\n⊢ (fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y) Y✝","state_after":"case refine_1\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\ng : (fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y) X✝\n⊢ limit.π (F ⋙ objects) Y✝ X = (F.map f).obj (limit.π (F ⋙ objects) X✝ X)\n\ncase refine_2\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\ng : (fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y) X✝\n⊢ (F.map f).obj (limit.π (F ⋙ objects) X✝ Y) = limit.π (F ⋙ objects) Y✝ Y","tactic":"refine eqToHom ?_ ≫ (F.map f).map g ≫ eqToHom ?_","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.eqToHom","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[41,4],"def_end_pos":[41,11]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)","state_after":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\nf :\n  { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    X\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) f =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)\n      f","tactic":"funext f","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\nf :\n  { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    X\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) f =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)\n      f","state_after":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\nf :\n  { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    X\nthis : Category.{v, v} (objects.obj (F.obj X)) := inferInstance\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) f =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)\n      f","tactic":"letI : Category (objects.obj (F.obj X)) := (inferInstance : Category (F.obj X))","premises":[{"full_name":"CategoryTheory.Cat.objects","def_path":"Mathlib/CategoryTheory/Category/Cat.lean","def_pos":[122,4],"def_end_pos":[122,11]},{"full_name":"CategoryTheory.Category","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[149,6],"def_end_pos":[149,14]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\nf :\n  { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    X\nthis : Category.{v, v} (objects.obj (F.obj X)) := inferInstance\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) f =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)\n      f","state_after":"no goals","tactic":"simp [Functor.congr_hom (F.map_id X) f]","premises":[{"full_name":"CategoryTheory.Functor.congr_hom","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[215,8],"def_end_pos":[215,17]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]}]},{"state_before":"J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nx✝¹ x✝ Z : J\nf : x✝¹ ⟶ x✝\ng : x✝ ⟶ Z\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (f ≫ g) =\n    { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n        f ≫\n      { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n        g","state_after":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nx✝¹ x✝ Z : J\nf : x✝¹ ⟶ x✝\ng : x✝ ⟶ Z\nh :\n  { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    x✝¹\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (f ≫ g) h =\n    ({ obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          f ≫\n        { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          g)\n      h","tactic":"funext h","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nx✝¹ x✝ Z : J\nf : x✝¹ ⟶ x✝\ng : x✝ ⟶ Z\nh :\n  { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    x✝¹\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (f ≫ g) h =\n    ({ obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          f ≫\n        { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          g)\n      h","state_after":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nx✝¹ x✝ Z : J\nf : x✝¹ ⟶ x✝\ng : x✝ ⟶ Z\nh :\n  { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    x✝¹\nthis : Category.{v, v} (objects.obj (F.obj Z)) := inferInstance\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (f ≫ g) h =\n    ({ obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          f ≫\n        { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          g)\n      h","tactic":"letI : Category (objects.obj (F.obj Z)) := (inferInstance : Category (F.obj Z))","premises":[{"full_name":"CategoryTheory.Cat.objects","def_path":"Mathlib/CategoryTheory/Category/Cat.lean","def_pos":[122,4],"def_end_pos":[122,11]},{"full_name":"CategoryTheory.Category","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[149,6],"def_end_pos":[149,14]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nx✝¹ x✝ Z : J\nf : x✝¹ ⟶ x✝\ng : x✝ ⟶ Z\nh :\n  { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    x✝¹\nthis : Category.{v, v} (objects.obj (F.obj Z)) := inferInstance\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (f ≫ g) h =\n    ({ obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          f ≫\n        { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n              map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n          g)\n      h","state_after":"no goals","tactic":"simp [Functor.congr_hom (F.map_comp f g) h, eqToHom_map]","premises":[{"full_name":"CategoryTheory.Functor.congr_hom","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[215,8],"def_end_pos":[215,17]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.eqToHom_map","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[268,8],"def_end_pos":[268,19]}]}]}
{"url":"Mathlib/Data/Finsupp/Basic.lean","commit":"","full_name":"Finsupp.graph_injective","start":[89,0],"end":[94,36],"file_path":"Mathlib/Data/Finsupp/Basic.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM✝ : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M✝\nα : Type u_13\nM : Type u_14\ninst✝ : Zero M\n⊢ Injective graph","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM✝ : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M✝\nα : Type u_13\nM : Type u_14\ninst✝ : Zero M\nf g : α →₀ M\nh : f.graph = g.graph\n⊢ f = g","tactic":"intro f g h","premises":[]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM✝ : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M✝\nα : Type u_13\nM : Type u_14\ninst✝ : Zero M\nf g : α →₀ M\nh : f.graph = g.graph\n⊢ f = g","state_after":"no goals","tactic":"classical\n    have hsup : f.support = g.support := by rw [← image_fst_graph, h, image_fst_graph]\n    refine ext_iff'.2 ⟨hsup, fun x hx => apply_eq_of_mem_graph <| h.symm ▸ ?_⟩\n    exact mk_mem_graph _ (hsup ▸ hx)","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finsupp.apply_eq_of_mem_graph","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[78,8],"def_end_pos":[78,29]},{"full_name":"Finsupp.ext_iff'","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[163,8],"def_end_pos":[163,16]},{"full_name":"Finsupp.image_fst_graph","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[86,8],"def_end_pos":[86,23]},{"full_name":"Finsupp.mk_mem_graph","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[75,8],"def_end_pos":[75,20]},{"full_name":"Finsupp.support","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[93,2],"def_end_pos":[93,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","commit":"","full_name":"CategoryTheory.ShortComplex.HomotopyEquiv.trans_hom","start":[710,0],"end":[721,60],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.239909, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\ne : S₁.HomotopyEquiv S₂\ne' : S₂.HomotopyEquiv S₃\n⊢ (e.hom ≫ e'.hom) ≫ e'.inv ≫ e.inv = e.hom ≫ (e'.hom ≫ e'.inv) ≫ e.inv","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.239909, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\ne : S₁.HomotopyEquiv S₂\ne' : S₂.HomotopyEquiv S₃\n⊢ e.hom ≫ 𝟙 S₂ ≫ e.inv = e.hom ≫ e.inv","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.239909, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\ne : S₁.HomotopyEquiv S₂\ne' : S₂.HomotopyEquiv S₃\n⊢ (e'.inv ≫ e.inv) ≫ e.hom ≫ e'.hom = e'.inv ≫ (e.inv ≫ e.hom) ≫ e'.hom","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.239909, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\ne : S₁.HomotopyEquiv S₂\ne' : S₂.HomotopyEquiv S₃\n⊢ e'.inv ≫ 𝟙 S₂ ≫ e'.hom = e'.inv ≫ e'.hom","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","commit":"","full_name":"AffineSubspace.nonempty_of_affineSpan_eq_top","start":[700,0],"end":[704,26],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : affineSpan k s = ⊤\n⊢ s.Nonempty","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : affineSpan k s = ⊤\n⊢ s ≠ ∅","tactic":"rw [Set.nonempty_iff_ne_empty]","premises":[{"full_name":"Set.nonempty_iff_ne_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[485,8],"def_end_pos":[485,29]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : affineSpan k s = ⊤\n⊢ s ≠ ∅","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : affineSpan k ∅ = ⊤\n⊢ False","tactic":"rintro rfl","premises":[]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : affineSpan k ∅ = ⊤\n⊢ False","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : ⊥ = ⊤\n⊢ False","tactic":"rw [AffineSubspace.span_empty] at h","premises":[{"full_name":"AffineSubspace.span_empty","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[618,8],"def_end_pos":[618,18]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : ⊥ = ⊤\n⊢ False","state_after":"no goals","tactic":"exact bot_ne_top k V P h","premises":[{"full_name":"AffineSubspace.bot_ne_top","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[692,8],"def_end_pos":[692,18]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Basic.lean","commit":"","full_name":"nhds_basis_uniformity","start":[662,0],"end":[666,32],"file_path":"Mathlib/Topology/UniformSpace/Basic.lean","tactics":[{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nh : (𝓤 α).HasBasis p s\nx : α\n⊢ (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}","state_after":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nx : α\nh : (comap Prod.swap (𝓤 α)).HasBasis p fun i => Prod.swap ⁻¹' s i\n⊢ (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}","tactic":"replace h := h.comap Prod.swap","premises":[{"full_name":"Filter.HasBasis.comap","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[697,8],"def_end_pos":[697,22]},{"full_name":"Prod.swap","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[133,4],"def_end_pos":[133,8]}]},{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nx : α\nh : (comap Prod.swap (𝓤 α)).HasBasis p fun i => Prod.swap ⁻¹' s i\n⊢ (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}","state_after":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nx : α\nh : (𝓤 α).HasBasis p fun i => Prod.swap ⁻¹' s i\n⊢ (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}","tactic":"rw [comap_swap_uniformity] at h","premises":[{"full_name":"comap_swap_uniformity","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[494,8],"def_end_pos":[494,29]}]},{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nx : α\nh : (𝓤 α).HasBasis p fun i => Prod.swap ⁻¹' s i\n⊢ (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}","state_after":"no goals","tactic":"exact nhds_basis_uniformity' h","premises":[{"full_name":"nhds_basis_uniformity'","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[657,8],"def_end_pos":[657,30]}]}]}
{"url":"Mathlib/Data/Finite/Card.lean","commit":"","full_name":"Finite.card_eq","start":[64,0],"end":[67,55],"file_path":"Mathlib/Data/Finite/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Finite α\ninst✝ : Finite β\n⊢ Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis : Fintype α\n⊢ Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)","tactic":"haveI := Fintype.ofFinite α","premises":[{"full_name":"Fintype.ofFinite","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[395,18],"def_end_pos":[395,34]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis : Fintype α\n⊢ Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis✝ : Fintype α\nthis : Fintype β\n⊢ Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)","tactic":"haveI := Fintype.ofFinite β","premises":[{"full_name":"Fintype.ofFinite","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[395,18],"def_end_pos":[395,34]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis✝ : Fintype α\nthis : Fintype β\n⊢ Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)","state_after":"no goals","tactic":"simp only [Nat.card_eq_fintype_card, Fintype.card_eq]","premises":[{"full_name":"Fintype.card_eq","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[190,8],"def_end_pos":[190,15]},{"full_name":"Nat.card_eq_fintype_card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[37,8],"def_end_pos":[37,28]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean","commit":"","full_name":"CochainComplex.mappingCone.inl_v_descShortComplex_f","start":[56,0],"end":[59,25],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\ni j : ℤ\nh : i + -1 = j\n⊢ (inl S.f).v i j h ≫ (descShortComplex S).f j = 0","state_after":"no goals","tactic":"simp [descShortComplex]","premises":[{"full_name":"CochainComplex.mappingCone.descShortComplex","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean","def_pos":[46,18],"def_end_pos":[46,34]}]}]}
{"url":"Mathlib/MeasureTheory/Function/AEEqFun.lean","commit":"","full_name":"MeasureTheory.AEEqFun.coeFn_compQuasiMeasurePreserving","start":[214,0],"end":[217,16],"file_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : MeasurableSpace β\nν : Measure β\nf : α → β\ng : β →ₘ[ν] γ\nhf : QuasiMeasurePreserving f μ ν\n⊢ ↑(g.compQuasiMeasurePreserving f hf) =ᶠ[ae μ] ↑g ∘ f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : MeasurableSpace β\nν : Measure β\nf : α → β\ng : β →ₘ[ν] γ\nhf : QuasiMeasurePreserving f μ ν\n⊢ ↑(mk (↑g ∘ f) ⋯) =ᶠ[ae μ] ↑g ∘ f","tactic":"rw [compQuasiMeasurePreserving_eq_mk]","premises":[{"full_name":"MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk","def_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","def_pos":[209,8],"def_end_pos":[209,40]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : MeasurableSpace β\nν : Measure β\nf : α → β\ng : β →ₘ[ν] γ\nhf : QuasiMeasurePreserving f μ ν\n⊢ ↑(mk (↑g ∘ f) ⋯) =ᶠ[ae μ] ↑g ∘ f","state_after":"no goals","tactic":"apply coeFn_mk","premises":[{"full_name":"MeasureTheory.AEEqFun.coeFn_mk","def_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","def_pos":[163,8],"def_end_pos":[163,16]}]}]}
{"url":"Mathlib/GroupTheory/CoprodI.lean","commit":"","full_name":"Monoid.CoprodI.of_leftInverse","start":[182,0],"end":[184,63],"file_path":"Mathlib/GroupTheory/CoprodI.lean","tactics":[{"state_before":"ι : Type u_1\nM : ι → Type u_2\ninst✝² : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝¹ : Monoid N\ninst✝ : DecidableEq ι\ni : ι\nx : M i\n⊢ (lift (Pi.mulSingle i (MonoidHom.id (M i)))) (of x) = x","state_after":"no goals","tactic":"simp only [lift_of, Pi.mulSingle_eq_same, MonoidHom.id_apply]","premises":[{"full_name":"Monoid.CoprodI.lift_of","def_path":"Mathlib/GroupTheory/CoprodI.lean","def_pos":[170,8],"def_end_pos":[170,15]},{"full_name":"MonoidHom.id_apply","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[671,23],"def_end_pos":[671,28]},{"full_name":"Pi.mulSingle_eq_same","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[268,8],"def_end_pos":[268,25]}]}]}
{"url":"Mathlib/Data/Matrix/Rank.lean","commit":"","full_name":"Matrix.rank_one","start":[46,0],"end":[49,71],"file_path":"Mathlib/Data/Matrix/Rank.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\ninst✝⁴ : Fintype n\ninst✝³ : Fintype o\ninst✝² : CommRing R\ninst✝¹ : StrongRankCondition R\ninst✝ : DecidableEq n\n⊢ rank 1 = Fintype.card n","state_after":"no goals","tactic":"rw [rank, mulVecLin_one, LinearMap.range_id, finrank_top, finrank_pi]","premises":[{"full_name":"FiniteDimensional.finrank_pi","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[264,8],"def_end_pos":[264,36]},{"full_name":"LinearMap.range_id","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[78,8],"def_end_pos":[78,16]},{"full_name":"Matrix.mulVecLin_one","def_path":"Mathlib/LinearAlgebra/Matrix/ToLin.lean","def_pos":[244,8],"def_end_pos":[244,28]},{"full_name":"Matrix.rank","def_path":"Mathlib/Data/Matrix/Rank.lean","def_pos":[43,18],"def_end_pos":[43,22]},{"full_name":"finrank_top","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[128,8],"def_end_pos":[128,19]}]}]}
{"url":"Mathlib/GroupTheory/FreeGroup/Basic.lean","commit":"","full_name":"FreeAddGroup.norm_add_le","start":[1153,0],"end":[1158,49],"file_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","tactics":[{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nx y : FreeGroup α\n⊢ (x * y).norm = (mk (x.toWord ++ y.toWord)).norm","state_after":"no goals","tactic":"rw [← mul_mk, mk_toWord, mk_toWord]","premises":[{"full_name":"FreeGroup.mk_toWord","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[1044,8],"def_end_pos":[1044,17]},{"full_name":"FreeGroup.mul_mk","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[451,8],"def_end_pos":[451,14]}]}]}
{"url":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","commit":"","full_name":"tendsto_comp_of_locally_uniform_limit","start":[839,0],"end":[846,63],"file_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousAt f x\nhg : Tendsto g p (𝓝 x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝 x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))","tactic":"rw [← continuousWithinAt_univ] at h","premises":[{"full_name":"continuousWithinAt_univ","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[445,8],"def_end_pos":[445,31]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝 x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝[univ] x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))","tactic":"rw [← nhdsWithin_univ] at hunif hg","premises":[{"full_name":"nhdsWithin_univ","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[66,14],"def_end_pos":[66,29]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝[univ] x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))","state_after":"no goals","tactic":"exact tendsto_comp_of_locally_uniform_limit_within h hg hunif","premises":[{"full_name":"tendsto_comp_of_locally_uniform_limit_within","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[828,8],"def_end_pos":[828,52]}]}]}
{"url":"Mathlib/Combinatorics/Additive/Energy.lean","commit":"","full_name":"Finset.addEnergy_pos_iff","start":[105,0],"end":[109,32],"file_path":"Mathlib/Combinatorics/Additive/Energy.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns s₁ s₂ t t₁ t₂ : Finset α\nh : 0 < Eₘ[s, t]\nH : ¬(s.Nonempty ∧ t.Nonempty)\n⊢ False","state_after":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns s₁ s₂ t t₁ t₂ : Finset α\nh : 0 < Eₘ[s, t]\nH : s = ∅ ∨ t = ∅\n⊢ False","tactic":"simp_rw [not_and_or, not_nonempty_iff_eq_empty] at H","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.not_nonempty_iff_eq_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[525,8],"def_end_pos":[525,33]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]}]},{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns s₁ s₂ t t₁ t₂ : Finset α\nh : 0 < Eₘ[s, t]\nH : s = ∅ ∨ t = ∅\n⊢ False","state_after":"no goals","tactic":"obtain rfl | rfl := H <;> simp [Nat.not_lt_zero] at h","premises":[{"full_name":"Nat.not_lt_zero","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1660,8],"def_end_pos":[1660,23]}]}]}
{"url":"Mathlib/Algebra/Group/Submonoid/Membership.lean","commit":"","full_name":"AddSubmonoid.mem_iSup_of_directed","start":[172,0],"end":[182,46],"file_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","tactics":[{"state_before":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\n⊢ x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i","state_after":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i","tactic":"refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"le_iSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[617,8],"def_end_pos":[617,15]}]},{"state_before":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i","state_after":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\n⊢ x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i","tactic":"suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by\n    simpa only [closure_iUnion, closure_eq (S _)] using this","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Submonoid.closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[329,4],"def_end_pos":[329,11]},{"full_name":"Submonoid.closure_eq","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[446,8],"def_end_pos":[446,18]},{"full_name":"Submonoid.closure_iUnion","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[466,8],"def_end_pos":[466,22]}]},{"state_before":"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\n⊢ x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i","state_after":"case refine_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ closure (⋃ i, ↑(S i))\n⊢ ∃ i, 1 ∈ S i\n\ncase refine_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : MulOneClass M\nι : Sort u_4\nhι : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ closure (⋃ i, ↑(S i))\n⊢ ∀ (x y : M), (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i","tactic":"refine fun hx ↦ closure_induction hx (fun _ ↦ mem_iUnion.1) ?_ ?_","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"Submonoid.closure_induction","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[375,8],"def_end_pos":[375,25]}]}]}
{"url":"Mathlib/Data/List/Cycle.lean","commit":"","full_name":"List.nextOr_concat","start":[78,0],"end":[82,58],"file_path":"Mathlib/Data/List/Cycle.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nh : x ∉ xs\n⊢ (xs ++ [x]).nextOr x d = d","state_after":"case nil\nα : Type u_1\ninst✝ : DecidableEq α\nx d : α\nh : x ∉ []\n⊢ ([] ++ [x]).nextOr x d = d\n\ncase cons\nα : Type u_1\ninst✝ : DecidableEq α\nx d z : α\nzs : List α\nIH : x ∉ zs → (zs ++ [x]).nextOr x d = d\nh : x ∉ z :: zs\n⊢ (z :: zs ++ [x]).nextOr x d = d","tactic":"induction' xs with z zs IH","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Types.lean","commit":"","full_name":"CategoryTheory.Functor.sectionsFunctor_obj","start":[124,0],"end":[129,70],"file_path":"Mathlib/CategoryTheory/Types.lean","tactics":[{"state_before":"J : Type u\ninst✝ : Category.{v, u} J\nF G : J ⥤ Type w\nφ : F ⟶ G\nx : (fun F => ↑F.sections) F\nj j' : J\nf : j ⟶ j'\n⊢ (F.map f ≫ φ.app j') (↑x j) = (fun j => φ.app j (↑x j)) j'","state_after":"no goals","tactic":"simp [x.2 f]","premises":[{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
{"url":"Mathlib/AlgebraicGeometry/AffineScheme.lean","commit":"","full_name":"AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage","start":[403,0],"end":[409,22],"file_path":"Mathlib/AlgebraicGeometry/AffineScheme.lean","tactics":[{"state_before":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen ((X.basicOpen r).ι ⁻¹ᵁ U)","state_after":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen ((X.basicOpen r).ι ''ᵁ (X.basicOpen r).ι ⁻¹ᵁ U)","tactic":"apply (X.basicOpen r).ι.isAffineOpen_iff_of_isOpenImmersion.mp","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion","def_path":"Mathlib/AlgebraicGeometry/AffineScheme.lean","def_pos":[306,8],"def_end_pos":[306,79]},{"full_name":"AlgebraicGeometry.Scheme.Opens.ι","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[50,4],"def_end_pos":[50,5]},{"full_name":"AlgebraicGeometry.Scheme.basicOpen","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[446,4],"def_end_pos":[446,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen ((X.basicOpen r).ι ''ᵁ (X.basicOpen r).ι ⁻¹ᵁ U)","state_after":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen (⋯.functor.obj ((X.basicOpen r).ι ⁻¹ᵁ U))","tactic":"dsimp [Scheme.Hom.opensFunctor, LocallyRingedSpace.IsOpenImmersion.opensFunctor]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.opensFunctor","def_path":"Mathlib/Geometry/RingedSpace/OpenImmersion.lean","def_pos":[1145,7],"def_end_pos":[1145,19]},{"full_name":"AlgebraicGeometry.Scheme.Hom.opensFunctor","def_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","def_pos":[83,7],"def_end_pos":[83,19]}]},{"state_before":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen (⋯.functor.obj ((X.basicOpen r).ι ⁻¹ᵁ U))","state_after":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen (X.basicOpen ((X.presheaf.map (homOfLE ⋯).op) r))","tactic":"rw [Opens.functor_obj_map_obj, Opens.openEmbedding_obj_top, inf_comm,\n    ← Scheme.basicOpen_res _ _ (homOfLE le_top).op]","premises":[{"full_name":"AlgebraicGeometry.Scheme.basicOpen_res","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[467,8],"def_end_pos":[467,21]},{"full_name":"CategoryTheory.homOfLE","def_path":"Mathlib/CategoryTheory/Category/Preorder.lean","def_pos":[65,4],"def_end_pos":[65,11]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"TopologicalSpace.Opens.functor_obj_map_obj","def_path":"Mathlib/Topology/Category/TopCat/Opens.lean","def_pos":[329,8],"def_end_pos":[329,27]},{"full_name":"TopologicalSpace.Opens.openEmbedding_obj_top","def_path":"Mathlib/Topology/Category/TopCat/Opens.lean","def_pos":[307,8],"def_end_pos":[307,29]},{"full_name":"inf_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[385,8],"def_end_pos":[385,16]},{"full_name":"le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[62,8],"def_end_pos":[62,14]}]},{"state_before":"X Y : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nr : ↑Γ(X, ⊤)\n⊢ IsAffineOpen (X.basicOpen ((X.presheaf.map (homOfLE ⋯).op) r))","state_after":"no goals","tactic":"exact hU.basicOpen _","premises":[{"full_name":"AlgebraicGeometry.IsAffineOpen.basicOpen","def_path":"Mathlib/AlgebraicGeometry/AffineScheme.lean","def_pos":[393,8],"def_end_pos":[393,17]}]}]}
{"url":"Mathlib/Order/Filter/Basic.lean","commit":"","full_name":"Filter.inf_principal_eq_bot","start":[903,0],"end":[905,55],"file_path":"Mathlib/Order/Filter/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nf✝ g : Filter α\ns✝ t : Set α\nf : Filter α\ns : Set α\n⊢ f ⊓ 𝓟 s = ⊥ ↔ sᶜ ∈ f","state_after":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nf✝ g : Filter α\ns✝ t : Set α\nf : Filter α\ns : Set α\n⊢ {x | x ∈ s → x ∈ ∅} ∈ f ↔ sᶜ ∈ f","tactic":"rw [← empty_mem_iff_bot, mem_inf_principal]","premises":[{"full_name":"Filter.empty_mem_iff_bot","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[620,8],"def_end_pos":[620,25]},{"full_name":"Filter.mem_inf_principal","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[896,6],"def_end_pos":[896,23]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nf✝ g : Filter α\ns✝ t : Set α\nf : Filter α\ns : Set α\n⊢ {x | x ∈ s → x ∈ ∅} ∈ f ↔ sᶜ ∈ f","state_after":"no goals","tactic":"simp only [mem_empty_iff_false, imp_false, compl_def]","premises":[{"full_name":"Set.compl_def","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1290,8],"def_end_pos":[1290,17]},{"full_name":"Set.mem_empty_iff_false","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[446,8],"def_end_pos":[446,27]},{"full_name":"imp_false","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1420,16],"def_end_pos":[1420,25]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","commit":"","full_name":"integral_cos_mul_complex","start":[497,0],"end":[511,33],"file_path":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","tactics":[{"state_before":"a✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∫ (x : ℝ) in a..b, Complex.cos (z * ↑x) = Complex.sin (z * ↑b) / z - Complex.sin (z * ↑a) / z","state_after":"case hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ x ∈ [[a, b]], HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x\n\ncase hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b","tactic":"apply integral_eq_sub_of_hasDerivAt","premises":[{"full_name":"intervalIntegral.integral_eq_sub_of_hasDerivAt","def_path":"Mathlib/MeasureTheory/Integral/FundThmCalculus.lean","def_pos":[1141,8],"def_end_pos":[1141,37]}]},{"state_before":"case hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ x ∈ [[a, b]], HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x\n\ncase hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b","state_after":"case hint\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ IntervalIntegrable (fun y => Complex.cos (z * ↑y)) MeasureTheory.volume a b\n\ncase hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ x ∈ [[a, b]], HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"swap","premises":[]},{"state_before":"case hderiv\na✝ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b : ℝ\n⊢ ∀ x ∈ [[a, b]], HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case hderiv\na✝¹ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b x : ℝ\na✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"intro x _","premises":[]},{"state_before":"case hderiv\na✝¹ b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na b x : ℝ\na✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case hderiv\na✝² b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b x : ℝ\na✝ : x ∈ [[a✝¹, b]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"have a := Complex.hasDerivAt_sin (↑x * z)","premises":[{"full_name":"Complex.hasDerivAt_sin","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean","def_pos":[43,8],"def_end_pos":[43,22]}]},{"state_before":"case hderiv\na✝² b✝ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b x : ℝ\na✝ : x ∈ [[a✝¹, b]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"have b : HasDerivAt (fun y => y * z : ℂ → ℂ) z ↑x := hasDerivAt_mul_const _","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"hasDerivAt_mul_const","def_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","def_pos":[229,8],"def_end_pos":[229,28]}]},{"state_before":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"have c : HasDerivAt (fun y : ℂ => Complex.sin (y * z)) _ ↑x := HasDerivAt.comp (𝕜 := ℂ) x a b","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.sin","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[55,4],"def_end_pos":[55,7]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"HasDerivAt.comp","def_path":"Mathlib/Analysis/Calculus/Deriv/Comp.lean","def_pos":[236,15],"def_end_pos":[236,30]}]},{"state_before":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (↑y * z) / z) (Complex.cos (↑x * z) * z / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"have d := HasDerivAt.comp_ofReal (c.div_const z)","premises":[{"full_name":"HasDerivAt.comp_ofReal","def_path":"Mathlib/Analysis/Complex/RealDeriv.lean","def_pos":[123,8],"def_end_pos":[123,30]},{"full_name":"HasDerivAt.div_const","def_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","def_pos":[351,8],"def_end_pos":[351,28]}]},{"state_before":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (↑y * z) / z) (Complex.cos (↑x * z) * z / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","tactic":"simp only [mul_comm] at d","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case hderiv\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ HasDerivAt (fun b => Complex.sin (z * ↑b) / z) (Complex.cos (z * ↑x)) x","state_after":"case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = z * Complex.cos (z * ↑x) / z","tactic":"convert d using 1","premises":[]},{"state_before":"case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = z * Complex.cos (z * ↑x) / z","state_after":"case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = Complex.cos (z * ↑x) * z / z","tactic":"conv_rhs => arg 1; rw [mul_comm]","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case h.e'_7\na✝² b✝¹ : ℝ\nn : ℕ\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y => y * z) z ↑x\nc : HasDerivAt (fun y => Complex.sin (y * z)) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y => Complex.sin (z * ↑y) / z) (z * Complex.cos (z * ↑x) / z) x\n⊢ Complex.cos (z * ↑x) = Complex.cos (z * ↑x) * z / z","state_after":"no goals","tactic":"rw [mul_div_cancel_right₀ _ hz]","premises":[{"full_name":"mul_div_cancel_right₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[164,14],"def_end_pos":[164,35]}]}]}
{"url":"Mathlib/Order/SupClosed.lean","commit":"","full_name":"infClosure_prod","start":[388,0],"end":[395,31],"file_path":"Mathlib/Order/SupClosed.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Set β\n⊢ infClosure s ×ˢ infClosure t ≤ infClosure (s ×ˢ t)","state_after":"case mk.intro.intro.intro.intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ (u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t)","tactic":"rintro ⟨_, _⟩ ⟨⟨u, hu, hus, rfl⟩, v, hv, hvt, rfl⟩","premises":[]},{"state_before":"case mk.intro.intro.intro.intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ (u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t)","state_after":"case mk.intro.intro.intro.intro.intro.intro.intro.refine_1\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t\n\ncase mk.intro.intro.intro.intro.intro.intro.intro.refine_2\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ (u ×ˢ v).inf' ⋯ id = (u.inf' hu id, v.inf' hv id)","tactic":"refine ⟨u ×ˢ v, hu.product hv, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.Nonempty.product","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[171,8],"def_end_pos":[171,24]}]}]}
{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.sdiff_union_erase_cancel","start":[1978,0],"end":[1979,78],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\nhts : t ⊆ s\nha : a ∈ t\n⊢ s \\ t ∪ t.erase a = s.erase a","state_after":"no goals","tactic":"simp_rw [erase_eq, sdiff_union_sdiff_cancel hts (singleton_subset_iff.2 ha)]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.erase_eq","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1911,8],"def_end_pos":[1911,16]},{"full_name":"Finset.sdiff_union_sdiff_cancel","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1975,8],"def_end_pos":[1975,32]},{"full_name":"Finset.singleton_subset_iff","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[667,8],"def_end_pos":[667,28]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","commit":"","full_name":"SimpleGraph.Walk.reverse_copy","start":[287,0],"end":[291,5],"file_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u' v' : V\np : G.Walk u v\nhu : u = u'\nhv : v = v'\n⊢ (p.copy hu hv).reverse = p.reverse.copy hv hu","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' v' : V\np : G.Walk u' v'\n⊢ (p.copy ⋯ ⋯).reverse = p.reverse.copy ⋯ ⋯","tactic":"subst_vars","premises":[]},{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' v' : V\np : G.Walk u' v'\n⊢ (p.copy ⋯ ⋯).reverse = p.reverse.copy ⋯ ⋯","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/ENNReal/Inv.lean","commit":"","full_name":"ENNReal.zpow_lt_top","start":[523,0],"end":[526,67],"file_path":"Mathlib/Data/ENNReal/Inv.lean","tactics":[{"state_before":"a b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\nn : ℤ\n⊢ a ^ n < ⊤","state_after":"case ofNat\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\na✝ : ℕ\n⊢ a ^ Int.ofNat a✝ < ⊤\n\ncase negSucc\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nh'a : a ≠ ⊤\na✝ : ℕ\n⊢ a ^ Int.negSucc a✝ < ⊤","tactic":"cases n","premises":[]}]}
{"url":"Mathlib/Data/Finset/Image.lean","commit":"","full_name":"Finset.fiber_nonempty_iff_mem_image","start":[350,0],"end":[351,85],"file_path":"Mathlib/Data/Finset/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na : α\nb c : β\nf : α → β\ns : Finset α\ny : β\n⊢ (filter (fun x => f x = y) s).Nonempty ↔ y ∈ image f s","state_after":"no goals","tactic":"simp [Finset.Nonempty]","premises":[{"full_name":"Finset.Nonempty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[417,14],"def_end_pos":[417,22]}]}]}
{"url":"Mathlib/Order/Cover.lean","commit":"","full_name":"WCovBy.eq_or_eq","start":[135,0],"end":[138,24],"file_path":"Mathlib/Order/Cover.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2 : a ≤ c\nh3 : c ≤ b\n⊢ c = a ∨ c = b","state_after":"case inl\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3 : c ≤ b\nh2 : a = c\n⊢ c = a ∨ c = b\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3 : c ≤ b\nh2 : a < c\n⊢ c = a ∨ c = b","tactic":"rcases h2.eq_or_lt with (h2 | h2)","premises":[]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3 : c ≤ b\nh2 : a < c\n⊢ c = a ∨ c = b","state_after":"case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3✝ : c ≤ b\nh2 : a < c\nh3 : c = b\n⊢ c = a ∨ c = b\n\ncase inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3✝ : c ≤ b\nh2 : a < c\nh3 : c < b\n⊢ c = a ∨ c = b","tactic":"rcases h3.eq_or_lt with (h3 | h3)","premises":[]},{"state_before":"case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3✝ : c ≤ b\nh2 : a < c\nh3 : c < b\n⊢ c = a ∨ c = b","state_after":"no goals","tactic":"exact (h.2 h2 h3).elim","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"False.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[236,20],"def_end_pos":[236,30]}]}]}
{"url":"Mathlib/Topology/Algebra/Valued/ValuationTopology.lean","commit":"","full_name":"Valuation.subgroups_basis","start":[34,0],"end":[79,49],"file_path":"Mathlib/Topology/Algebra/Valued/ValuationTopology.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\n⊢ ∀ (i j : Γ₀ˣ), ∃ k, v.ltAddSubgroup k ≤ v.ltAddSubgroup i ⊓ v.ltAddSubgroup j","state_after":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ ∃ k, v.ltAddSubgroup k ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁","tactic":"rintro γ₀ γ₁","premises":[]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ ∃ k, v.ltAddSubgroup k ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁","state_after":"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ v.ltAddSubgroup (min γ₀ γ₁) ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁","tactic":"use min γ₀ γ₁","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Min.min","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1142,2],"def_end_pos":[1142,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ v.ltAddSubgroup (min γ₀ γ₁) ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁","state_after":"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ ∀ (a : R), v a < ↑γ₀ → v a < ↑γ₁ → v a < ↑γ₀","tactic":"simp only [ltAddSubgroup, Units.min_val, Units.val_le_val, lt_min_iff,\n        AddSubgroup.mk_le_mk, setOf_subset_setOf, le_inf_iff, and_imp, imp_self, implies_true,\n        forall_const, and_true]","premises":[{"full_name":"AddSubgroup.mk_le_mk","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[345,2],"def_end_pos":[345,13]},{"full_name":"Set.setOf_subset_setOf","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[237,8],"def_end_pos":[237,26]},{"full_name":"Units.min_val","def_path":"Mathlib/Algebra/Order/Monoid/Units.lean","def_pos":[49,8],"def_end_pos":[49,15]},{"full_name":"Units.val_le_val","def_path":"Mathlib/Algebra/Order/Monoid/Units.lean","def_pos":[23,8],"def_end_pos":[23,18]},{"full_name":"Valuation.ltAddSubgroup","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[357,4],"def_end_pos":[357,17]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"and_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[104,16],"def_end_pos":[104,24]},{"full_name":"forall_const","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[265,16],"def_end_pos":[265,28]},{"full_name":"imp_self","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1418,21],"def_end_pos":[1418,29]},{"full_name":"implies_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[132,16],"def_end_pos":[132,28]},{"full_name":"le_inf_iff","def_path":"Mathlib/Order/Lattice.lean","def_pos":[333,8],"def_end_pos":[333,18]},{"full_name":"lt_min_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[47,8],"def_end_pos":[47,18]}]},{"state_before":"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ ∀ (a : R), v a < ↑γ₀ → v a < ↑γ₁ → v a < ↑γ₀","state_after":"no goals","tactic":"tauto","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\n⊢ ∀ (i : Γ₀ˣ), ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup i)","state_after":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)","tactic":"rintro γ","premises":[]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)","state_after":"case intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)","tactic":"cases' exists_square_le γ with γ₀ h","premises":[{"full_name":"exists_square_le","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[1005,8],"def_end_pos":[1005,24]}]},{"state_before":"case intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)","state_after":"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ↑(v.ltAddSubgroup γ₀) * ↑(v.ltAddSubgroup γ₀) ⊆ ↑(v.ltAddSubgroup γ)","tactic":"use γ₀","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ↑(v.ltAddSubgroup γ₀) * ↑(v.ltAddSubgroup γ₀) ⊆ ↑(v.ltAddSubgroup γ)","state_after":"case h.intro.intro.intro.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\nr : R\nr_in : r ∈ ↑(v.ltAddSubgroup γ₀)\ns : R\ns_in : s ∈ ↑(v.ltAddSubgroup γ₀)\n⊢ (fun x x_1 => x * x_1) r s ∈ ↑(v.ltAddSubgroup γ)","tactic":"rintro - ⟨r, r_in, s, s_in, rfl⟩","premises":[]},{"state_before":"case h.intro.intro.intro.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\nr : R\nr_in : r ∈ ↑(v.ltAddSubgroup γ₀)\ns : R\ns_in : s ∈ ↑(v.ltAddSubgroup γ₀)\n⊢ (fun x x_1 => x * x_1) r s ∈ ↑(v.ltAddSubgroup γ)","state_after":"no goals","tactic":"calc\n        (v (r * s) : Γ₀) = v r * v s := Valuation.map_mul _ _ _\n        _ < γ₀ * γ₀ := mul_lt_mul₀ r_in s_in\n        _ ≤ γ := mod_cast h","premises":[{"full_name":"Valuation.map_mul","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[155,8],"def_end_pos":[155,15]},{"full_name":"mul_lt_mul₀","def_path":"Mathlib/Algebra/Order/GroupWithZero/Canonical.lean","def_pos":[155,8],"def_end_pos":[155,19]}]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\n⊢ ∀ (x : R) (i : Γ₀ˣ), ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup i)","state_after":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup γ)","tactic":"rintro x γ","premises":[]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup γ)","state_after":"case inl\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\nHx : v x = 0\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup γ)\n\ncase inr.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ γx : Γ₀ˣ\nHx : v x = ↑γx\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup γ)","tactic":"rcases GroupWithZero.eq_zero_or_unit (v x) with (Hx | ⟨γx, Hx⟩)","premises":[{"full_name":"GroupWithZero.eq_zero_or_unit","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[205,8],"def_end_pos":[205,44]}]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\n⊢ ∀ (x : R) (i : Γ₀ˣ), ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(v.ltAddSubgroup i)","state_after":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(v.ltAddSubgroup γ)","tactic":"rintro x γ","premises":[]},{"state_before":"R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(v.ltAddSubgroup γ)","state_after":"case inl\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\nHx : v x = 0\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(v.ltAddSubgroup γ)\n\ncase inr.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ γx : Γ₀ˣ\nHx : v x = ↑γx\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(v.ltAddSubgroup γ)","tactic":"rcases GroupWithZero.eq_zero_or_unit (v x) with (Hx | ⟨γx, Hx⟩)","premises":[{"full_name":"GroupWithZero.eq_zero_or_unit","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[205,8],"def_end_pos":[205,44]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","commit":"","full_name":"SimpleGraph.incidenceSet_inter_incidenceSet_of_adj","start":[648,0],"end":[652,83],"file_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","tactics":[{"state_before":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : G.Adj a b\n⊢ G.incidenceSet a ∩ G.incidenceSet b = {s(a, b)}","state_after":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : G.Adj a b\n⊢ {s(a, b)} ⊆ G.incidenceSet a ∩ G.incidenceSet b","tactic":"refine (G.incidenceSet_inter_incidenceSet_subset <| h.ne).antisymm ?_","premises":[{"full_name":"SimpleGraph.Adj.ne","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[180,18],"def_end_pos":[180,24]},{"full_name":"SimpleGraph.incidenceSet_inter_incidenceSet_subset","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[644,8],"def_end_pos":[644,46]}]},{"state_before":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : G.Adj a b\n⊢ {s(a, b)} ⊆ G.incidenceSet a ∩ G.incidenceSet b","state_after":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : G.Adj a b\n⊢ s(a, b) ∈ G.incidenceSet a ∩ G.incidenceSet b","tactic":"rintro _ (rfl : _ = s(a, b))","premises":[{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Sym2.mk","def_path":"Mathlib/Data/Sym/Sym2.lean","def_pos":[97,17],"def_end_pos":[97,24]}]},{"state_before":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : G.Adj a b\n⊢ s(a, b) ∈ G.incidenceSet a ∩ G.incidenceSet b","state_after":"no goals","tactic":"exact ⟨G.mk'_mem_incidenceSet_left_iff.2 h, G.mk'_mem_incidenceSet_right_iff.2 h⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"SimpleGraph.mk'_mem_incidenceSet_left_iff","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[635,8],"def_end_pos":[635,37]},{"full_name":"SimpleGraph.mk'_mem_incidenceSet_right_iff","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[638,8],"def_end_pos":[638,38]}]}]}
{"url":"Mathlib/Data/Seq/Computation.lean","commit":"","full_name":"Computation.LiftRelRec.lem","start":[1089,0],"end":[1101,28],"file_path":"Mathlib/Data/Seq/Computation.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nR✝ : α → β → Prop\nC✝ : Computation α → Computation β → Prop\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C ca.destruct cb.destruct\nca : Computation α\ncb : Computation β\nHc : C ca cb\na : α\nha : a ∈ ca\n⊢ LiftRel R ca cb","state_after":"α : Type u\nβ : Type v\nγ : Type w\nR✝ : α → β → Prop\nC✝ : Computation α → Computation β → Prop\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C ca.destruct cb.destruct\nca : Computation α\na : α\nha : a ∈ ca\n⊢ ∀ (cb : Computation β), C ca cb → LiftRel R ca cb","tactic":"revert cb","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nR✝ : α → β → Prop\nC✝ : Computation α → Computation β → Prop\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C ca.destruct cb.destruct\nca : Computation α\na : α\nha : a ∈ ca\n⊢ ∀ (cb : Computation β), C ca cb → LiftRel R ca cb","state_after":"case refine_1\nα : Type u\nβ : Type v\nγ : Type w\nR✝ : α → β → Prop\nC✝ : Computation α → Computation β → Prop\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C ca.destruct cb.destruct\nca : Computation α\na : α\nha : a ∈ ca\ncb : Computation β\nHc : C (pure a) cb\nh : LiftRelAux R C (pure a).destruct cb.destruct\n⊢ LiftRel R (pure a) cb\n\ncase refine_2\nα : Type u\nβ : Type v\nγ : Type w\nR✝ : α → β → Prop\nC✝ : Computation α → Computation β → Prop\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C ca.destruct cb.destruct\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C ca'.think cb\nh : LiftRelAux R C ca'.think.destruct cb.destruct\n⊢ LiftRel R ca'.think cb","tactic":"refine memRecOn (C := (fun ca ↦ ∀ (cb : Computation β), C ca cb → LiftRel R ca cb))\n    ha ?_ (fun ca' IH => ?_) <;> intro cb Hc <;> have h := H Hc","premises":[{"full_name":"Computation","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[31,4],"def_end_pos":[31,15]},{"full_name":"Computation.LiftRel","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[872,4],"def_end_pos":[872,11]},{"full_name":"Computation.memRecOn","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[523,4],"def_end_pos":[523,12]}]}]}
{"url":"Mathlib/Data/NNReal/Basic.lean","commit":"","full_name":"Real.natCastle_toNNReal'","start":[664,0],"end":[666,74],"file_path":"Mathlib/Data/NNReal/Basic.lean","tactics":[{"state_before":"n : ℕ\nr : ℝ\n⊢ ↑n ≤ r.toNNReal ↔ ↑n ≤ r ∨ n = 0","state_after":"no goals","tactic":"simpa [n.cast_nonneg.le_iff_eq] using toNNReal_le_toNNReal_iff' (r := n)","premises":[{"full_name":"LE.le.le_iff_eq","def_path":"Mathlib/Order/Basic.lean","def_pos":[238,8],"def_end_pos":[238,17]},{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]},{"full_name":"Real.toNNReal_le_toNNReal_iff'","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[651,6],"def_end_pos":[651,31]}]}]}
{"url":"Mathlib/RingTheory/PowerSeries/Basic.lean","commit":"","full_name":"Polynomial.coe_injective","start":[822,0],"end":[825,7],"file_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","tactics":[{"state_before":"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : R[X]\nh : Coe.coe x = Coe.coe y\n⊢ x = y","state_after":"case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : R[X]\nh : Coe.coe x = Coe.coe y\nn✝ : ℕ\n⊢ x.coeff n✝ = y.coeff n✝","tactic":"ext","premises":[]},{"state_before":"case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : R[X]\nh : Coe.coe x = Coe.coe y\nn✝ : ℕ\n⊢ x.coeff n✝ = y.coeff n✝","state_after":"case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : R[X]\nh : Coe.coe x = Coe.coe y\nn✝ : ℕ\n⊢ (PowerSeries.coeff R n✝) ↑x = (PowerSeries.coeff R n✝) ↑y","tactic":"simp_rw [← coeff_coe]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Polynomial.coeff_coe","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[780,8],"def_end_pos":[780,17]}]},{"state_before":"case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : R[X]\nh : Coe.coe x = Coe.coe y\nn✝ : ℕ\n⊢ (PowerSeries.coeff R n✝) ↑x = (PowerSeries.coeff R n✝) ↑y","state_after":"no goals","tactic":"congr","premises":[]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","commit":"","full_name":"op_vadd_coe_set","start":[44,0],"end":[47,73],"file_path":"Mathlib/Algebra/Group/Subgroup/Pointwise.lean","tactics":[{"state_before":"α : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝² : Group G\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\ns : S\na : G\nha : a ∈ s\n⊢ MulOpposite.op a • ↑s = ↑s","state_after":"case h\nα : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝² : Group G\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\ns : S\na : G\nha : a ∈ s\nx✝ : G\n⊢ x✝ ∈ MulOpposite.op a • ↑s ↔ x✝ ∈ ↑s","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝² : Group G\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\ns : S\na : G\nha : a ∈ s\nx✝ : G\n⊢ x✝ ∈ MulOpposite.op a • ↑s ↔ x✝ ∈ ↑s","state_after":"no goals","tactic":"simp [Set.mem_smul_set_iff_inv_smul_mem, mul_mem_cancel_right, ha]","premises":[{"full_name":"Set.mem_smul_set_iff_inv_smul_mem","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[705,8],"def_end_pos":[705,37]},{"full_name":"mul_mem_cancel_right","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[157,8],"def_end_pos":[157,28]}]}]}
{"url":"Mathlib/RingTheory/OreLocalization/Basic.lean","commit":"","full_name":"OreLocalization.zero_oreDiv","start":[782,0],"end":[785,23],"file_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\ns : ↥S\n⊢ 0 /ₒ s = 0","state_after":"R : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\ns : ↥S\n⊢ ∃ u v, u • 0 = v • 0 ∧ ↑u * ↑1 = v * ↑s","tactic":"rw [OreLocalization.zero_def, oreDiv_eq_iff]","premises":[{"full_name":"OreLocalization.oreDiv_eq_iff","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[110,8],"def_end_pos":[110,21]},{"full_name":"OreLocalization.zero_def","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[634,18],"def_end_pos":[634,26]}]},{"state_before":"R : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\ns : ↥S\n⊢ ∃ u v, u • 0 = v • 0 ∧ ↑u * ↑1 = v * ↑s","state_after":"no goals","tactic":"exact ⟨s, 1, by simp⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Laurent.lean","commit":"","full_name":"LaurentPolynomial.isLocalization","start":[505,0],"end":[519,22],"file_path":"Mathlib/Algebra/Polynomial/Laurent.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\nt : ↥(Submonoid.closure {X})\n⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑t)","state_after":"case mk\nR : Type u_1\ninst✝ : CommSemiring R\nt : R[X]\nht : t ∈ Submonoid.closure {X}\n⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨t, ht⟩)","tactic":"cases' t with t ht","premises":[]},{"state_before":"case mk\nR : Type u_1\ninst✝ : CommSemiring R\nt : R[X]\nht : t ∈ Submonoid.closure {X}\n⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨t, ht⟩)","state_after":"case mk.intro\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nht : X ^ n ∈ Submonoid.closure {X}\n⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨X ^ n, ht⟩)","tactic":"rcases Submonoid.mem_closure_singleton.mp ht with ⟨n, rfl⟩","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Submonoid.mem_closure_singleton","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[282,8],"def_end_pos":[282,29]}]},{"state_before":"case mk.intro\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nht : X ^ n ∈ Submonoid.closure {X}\n⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨X ^ n, ht⟩)","state_after":"no goals","tactic":"simp only [isUnit_T n, algebraMap_eq_toLaurent, Polynomial.toLaurent_X_pow]","premises":[{"full_name":"LaurentPolynomial.algebraMap_eq_toLaurent","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[502,8],"def_end_pos":[502,31]},{"full_name":"LaurentPolynomial.isUnit_T","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[236,8],"def_end_pos":[236,16]},{"full_name":"Polynomial.toLaurent_X_pow","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[219,8],"def_end_pos":[219,41]}]},{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\nf : R[T;T⁻¹]\n⊢ ∃ x, f * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1","state_after":"case Qf\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nn : ℕ\n⊢ ∃ x, toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1","tactic":"induction' f using LaurentPolynomial.induction_on_mul_T with f n","premises":[{"full_name":"LaurentPolynomial.induction_on_mul_T","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[354,8],"def_end_pos":[354,26]}]},{"state_before":"case Qf\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nn : ℕ\n⊢ ∃ x, toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1","state_after":"case Qf\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nn : ℕ\nthis : X ^ n ∈ Submonoid.closure {X}\n⊢ ∃ x, toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1","tactic":"have := (Submonoid.closure ({X} : Set R[X])).pow_mem Submonoid.mem_closure_singleton_self n","premises":[{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Submonoid.closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[329,4],"def_end_pos":[329,11]},{"full_name":"Submonoid.mem_closure_singleton_self","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[285,8],"def_end_pos":[285,34]},{"full_name":"Submonoid.pow_mem","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[526,18],"def_end_pos":[526,25]}]},{"state_before":"case Qf\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nn : ℕ\nthis : X ^ n ∈ Submonoid.closure {X}\n⊢ ∃ x, toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1","state_after":"case Qf\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nn : ℕ\nthis : X ^ n ∈ Submonoid.closure {X}\n⊢ toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑(f, ⟨X ^ n, this⟩).2 =\n    (algebraMap R[X] R[T;T⁻¹]) (f, ⟨X ^ n, this⟩).1","tactic":"refine ⟨(f, ⟨_, this⟩), ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]}]},{"state_before":"case Qf\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nn : ℕ\nthis : X ^ n ∈ Submonoid.closure {X}\n⊢ toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑(f, ⟨X ^ n, this⟩).2 =\n    (algebraMap R[X] R[T;T⁻¹]) (f, ⟨X ^ n, this⟩).1","state_after":"no goals","tactic":"simp only [algebraMap_eq_toLaurent, Polynomial.toLaurent_X_pow, mul_T_assoc,\n        add_left_neg, T_zero, mul_one]","premises":[{"full_name":"LaurentPolynomial.T_zero","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[163,8],"def_end_pos":[163,14]},{"full_name":"LaurentPolynomial.algebraMap_eq_toLaurent","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[502,8],"def_end_pos":[502,31]},{"full_name":"LaurentPolynomial.mul_T_assoc","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[178,8],"def_end_pos":[178,19]},{"full_name":"Polynomial.toLaurent_X_pow","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[219,8],"def_end_pos":[219,41]},{"full_name":"add_left_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1038,2],"def_end_pos":[1038,13]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\nf g : R[X]\n⊢ (algebraMap R[X] R[T;T⁻¹]) f = (algebraMap R[X] R[T;T⁻¹]) g → ∃ c, ↑c * f = ↑c * g","state_after":"R : Type u_1\ninst✝ : CommSemiring R\nf g : R[X]\n⊢ f = g → ∃ c, ↑c * f = ↑c * g","tactic":"rw [algebraMap_eq_toLaurent, algebraMap_eq_toLaurent, Polynomial.toLaurent_inj]","premises":[{"full_name":"LaurentPolynomial.algebraMap_eq_toLaurent","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[502,8],"def_end_pos":[502,31]},{"full_name":"Polynomial.toLaurent_inj","def_path":"Mathlib/Algebra/Polynomial/Laurent.lean","def_pos":[334,8],"def_end_pos":[334,39]}]},{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\nf g : R[X]\n⊢ f = g → ∃ c, ↑c * f = ↑c * g","state_after":"R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\n⊢ ∃ c, ↑c * f = ↑c * f","tactic":"rintro rfl","premises":[]},{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\n⊢ ∃ c, ↑c * f = ↑c * f","state_after":"no goals","tactic":"exact ⟨1, rfl⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Deprecated/Subgroup.lean","commit":"","full_name":"IsAddGroupHom.neg_iff_ker'","start":[314,0],"end":[316,45],"file_path":"Mathlib/Deprecated/Subgroup.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\n⊢ f a = f b ↔ a⁻¹ * b ∈ ker f","state_after":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\n⊢ f a = f b ↔ f (a⁻¹ * b) = 1","tactic":"rw [mem_ker]","premises":[{"full_name":"IsGroupHom.mem_ker","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[272,8],"def_end_pos":[272,15]}]},{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\n⊢ f a = f b ↔ f (a⁻¹ * b) = 1","state_after":"no goals","tactic":"exact one_iff_ker_inv' hf _ _","premises":[{"full_name":"IsGroupHom.one_iff_ker_inv'","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[307,8],"def_end_pos":[307,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Types.lean","commit":"","full_name":"CategoryTheory.Limits.Types.colimit_sound","start":[540,0],"end":[542,27],"file_path":"Mathlib/CategoryTheory/Limits/Types.lean","tactics":[{"state_before":"J : Type v\ninst✝¹ : Category.{w, v} J\nF : J ⥤ Type u\ninst✝ : HasColimit F\nj j' : J\nx : F.obj j\nx' : F.obj j'\nf : j ⟶ j'\nw : F.map f x = x'\n⊢ colimit.ι F j x = colimit.ι F j' x'","state_after":"no goals","tactic":"rw [← w, Colimit.w_apply]","premises":[{"full_name":"CategoryTheory.Limits.Types.Colimit.w_apply","def_path":"Mathlib/CategoryTheory/Limits/Types.lean","def_pos":[510,8],"def_end_pos":[510,23]}]}]}
{"url":"Mathlib/Data/FinEnum.lean","commit":"","full_name":"FinEnum.mem_toList","start":[61,0],"end":[63,37],"file_path":"Mathlib/Data/FinEnum.lean","tactics":[{"state_before":"α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ x ∈ toList α","state_after":"α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ ∃ a, equiv.symm a = x","tactic":"simp [toList]","premises":[{"full_name":"FinEnum.toList","def_path":"Mathlib/Data/FinEnum.lean","def_pos":[56,4],"def_end_pos":[56,10]}]},{"state_before":"α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ ∃ a, equiv.symm a = x","state_after":"α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ equiv.symm (equiv x) = x","tactic":"exists equiv x","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"FinEnum.equiv","def_path":"Mathlib/Data/FinEnum.lean","def_pos":[28,2],"def_end_pos":[28,7]}]},{"state_before":"α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ equiv.symm (equiv x) = x","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Logic/Basic.lean","commit":"","full_name":"forall₂_true_iff","start":[505,0],"end":[505,78],"file_path":"Mathlib/Logic/Basic.lean","tactics":[{"state_before":"α : Sort u_1\nβ✝ : Sort u_2\np q : α → Prop\nβ : α → Sort u_3\n⊢ (∀ (a : α), β a → True) ↔ True","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","commit":"","full_name":"ContinuousMultilinearMap.hasFiniteFPowerSeriesOnBall","start":[277,0],"end":[282,72],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Analytic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁶ : NormedAddCommGroup E✝\ninst✝⁵ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\ny : (i : ι) → E i\nx✝ : y ∈ EMetric.ball 0 ⊤\n⊢ (∑ i ∈ Finset.range (Fintype.card ι + 1), (f.toFormalMultilinearSeries i) fun x => y) = f (0 + y)","state_after":"𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁶ : NormedAddCommGroup E✝\ninst✝⁵ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\ny : (i : ι) → E i\nx✝ : y ∈ EMetric.ball 0 ⊤\n⊢ ((f.toFormalMultilinearSeries (Fintype.card ι)) fun x => y) = f y\n\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE✝ : Type u\ninst✝⁶ : NormedAddCommGroup E✝\ninst✝⁵ : NormedSpace 𝕜 E✝\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\ny : (i : ι) → E i\nx✝ : y ∈ EMetric.ball 0 ⊤\n⊢ ∀ b ∈ Finset.range (Fintype.card ι + 1), b ≠ Fintype.card ι → ((f.toFormalMultilinearSeries b) fun x => y) = 0","tactic":"rw [Finset.sum_eq_single_of_mem _ (Finset.self_mem_range_succ _), zero_add]","premises":[{"full_name":"Finset.self_mem_range_succ","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2476,8],"def_end_pos":[2476,27]},{"full_name":"Finset.sum_eq_single_of_mem","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[854,2],"def_end_pos":[854,13]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
{"url":"Mathlib/Order/Filter/Ultrafilter.lean","commit":"","full_name":"Ultrafilter.comap_id","start":[219,0],"end":[223,24],"file_path":"Mathlib/Order/Filter/Ultrafilter.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type u_1\nf✝ g : Ultrafilter α\ns t : Set α\np q : α → Prop\nf : Ultrafilter α\nh₀ : optParam (Injective id) ⋯\n⊢ range id ∈ f","state_after":"α : Type u\nβ : Type v\nγ : Type u_1\nf✝ g : Ultrafilter α\ns t : Set α\np q : α → Prop\nf : Ultrafilter α\nh₀ : optParam (Injective id) ⋯\n⊢ univ ∈ f","tactic":"rw [range_id]","premises":[{"full_name":"Set.range_id","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[731,8],"def_end_pos":[731,16]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type u_1\nf✝ g : Ultrafilter α\ns t : Set α\np q : α → Prop\nf : Ultrafilter α\nh₀ : optParam (Injective id) ⋯\n⊢ univ ∈ f","state_after":"no goals","tactic":"exact univ_mem","premises":[{"full_name":"Filter.univ_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[136,8],"def_end_pos":[136,16]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.prod_apply_dite","start":[1016,0],"end":[1034,99],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter p s }\n⊢ p ↑x","state_after":"no goals","tactic":"simpa using (mem_filter.mp x.2).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n⊢ ¬p ↑x","state_after":"no goals","tactic":"simpa using (mem_filter.mp x.2).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter p s }\n⊢ p ↑x","state_after":"no goals","tactic":"simpa using (mem_filter.mp x.2).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n⊢ ¬p ↑x","state_after":"no goals","tactic":"simpa using (mem_filter.mp x.2).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter p s }\n_hx : x ∈ (filter p s).attach\n⊢ p ↑x","state_after":"no goals","tactic":"simpa using (mem_filter.mp x.2).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n_hx : x ∈ (filter (fun x => ¬p x) s).attach\n⊢ ¬p ↑x","state_after":"no goals","tactic":"simpa using (mem_filter.mp x.2).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
{"url":"Mathlib/Algebra/Field/Subfield.lean","commit":"","full_name":"Subfield.map_comap_eq_self","start":[849,0],"end":[851,54],"file_path":"Mathlib/Algebra/Field/Subfield.lean","tactics":[{"state_before":"K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\nf : K →+* L\ns : Subfield L\nh : s ≤ f.fieldRange\n⊢ map f (comap f s) = s","state_after":"no goals","tactic":"simpa only [inf_of_le_left h] using map_comap_eq f s","premises":[{"full_name":"Subfield.map_comap_eq","def_path":"Mathlib/Algebra/Field/Subfield.lean","def_pos":[846,8],"def_end_pos":[846,20]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Projection.lean","commit":"","full_name":"norm_eq_iInf_iff_inner_eq_zero","start":[321,0],"end":[348,60],"file_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","tactic":"letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E","premises":[{"full_name":"InnerProductSpace","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[103,6],"def_end_pos":[103,23]},{"full_name":"InnerProductSpace.rclikeToReal","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1939,4],"def_end_pos":[1939,34]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis✝ : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\nthis : Module ℝ E := RestrictScalars.module ℝ 𝕜 E\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","tactic":"letI : Module ℝ E := RestrictScalars.module ℝ 𝕜 E","premises":[{"full_name":"Module","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[57,6],"def_end_pos":[57,12]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RestrictScalars.module","def_path":"Mathlib/Algebra/Algebra/RestrictScalars.lean","def_pos":[104,9],"def_end_pos":[104,31]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis✝ : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\nthis : Module ℝ E := RestrictScalars.module ℝ 𝕜 E\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis✝ : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\nthis : Module ℝ E := RestrictScalars.module ℝ 𝕜 E\nK' : Submodule ℝ E := Submodule.restrictScalars ℝ K\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","tactic":"let K' : Submodule ℝ E := K.restrictScalars ℝ","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Submodule","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[36,10],"def_end_pos":[36,19]},{"full_name":"Submodule.restrictScalars","def_path":"Mathlib/Algebra/Module/Submodule/RestrictScalars.lean","def_pos":[31,4],"def_end_pos":[31,19]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis✝ : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\nthis : Module ℝ E := RestrictScalars.module ℝ 𝕜 E\nK' : Submodule ℝ E := Submodule.restrictScalars ℝ K\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0","state_after":"case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis✝ : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\nthis : Module ℝ E := RestrictScalars.module ℝ 𝕜 E\nK' : Submodule ℝ E := Submodule.restrictScalars ℝ K\n⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ → ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0\n\ncase mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : _root_.RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis✝ : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\nthis : Module ℝ E := RestrictScalars.module ℝ 𝕜 E\nK' : Submodule ℝ E := Submodule.restrictScalars ℝ K\n⊢ (∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0) → ‖u - v‖ = ⨅ w, ‖u - ↑w‖","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.leftUnshift_v","start":[90,0],"end":[95,5],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn' a : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\np q : ℤ\nhpq : p + n = q\np' : ℤ\nhp' : p' + n' = q\n⊢ p = p' + a","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn' a : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\np q : ℤ\nhpq : p + n = q\np' : ℤ\nhp' : p' + n' = q\n⊢ p' + n' = q","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn' a : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\np q : ℤ\nhpq : p + n = q\np' : ℤ\nhp' : p' + n' = q\n⊢ (γ.leftUnshift n hn).v p q hpq =\n    (a * n' + a * (a - 1) / 2).negOnePow • (K.shiftFunctorObjXIso a p' p ⋯).inv ≫ γ.v p' q ⋯","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn' a : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\np q : ℤ\nhpq : p + n = q\nhp' : p - a + n' = q\n⊢ (γ.leftUnshift n hn).v p q hpq =\n    (a * n' + a * (a - 1) / 2).negOnePow • (K.shiftFunctorObjXIso a (p - a) p ⋯).inv ≫ γ.v (p - a) q ⋯","tactic":"obtain rfl : p' = p - a := by omega","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn' a : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\np q : ℤ\nhpq : p + n = q\nhp' : p - a + n' = q\n⊢ (γ.leftUnshift n hn).v p q hpq =\n    (a * n' + a * (a - 1) / 2).negOnePow • (K.shiftFunctorObjXIso a (p - a) p ⋯).inv ≫ γ.v (p - a) q ⋯","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Function/LpSpace.lean","commit":"","full_name":"MeasureTheory.Lp.edist_toLp_zero","start":[293,0],"end":[296,6],"file_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhf : Memℒp f p μ\n⊢ edist (Memℒp.toLp f hf) 0 = eLpNorm f p μ","state_after":"case h.e'_3.h.e'_5\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhf : Memℒp f p μ\n⊢ f = f - 0","tactic":"convert edist_toLp_toLp f 0 hf zero_memℒp","premises":[{"full_name":"MeasureTheory.Lp.edist_toLp_toLp","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[288,8],"def_end_pos":[288,23]},{"full_name":"MeasureTheory.zero_memℒp","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[255,8],"def_end_pos":[255,18]}]},{"state_before":"case h.e'_3.h.e'_5\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhf : Memℒp f p μ\n⊢ f = f - 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Int/Cast/Basic.lean","commit":"","full_name":"Int.cast_add","start":[93,0],"end":[103,62],"file_path":"Mathlib/Data/Int/Cast/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝ : AddGroupWithOne R\nm n : ℕ\n⊢ ↑(↑m + ↑n) = ↑↑m + ↑↑n","state_after":"no goals","tactic":"simp [-Int.natCast_add, ← Int.ofNat_add]","premises":[{"full_name":"Int.natCast_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[522,16],"def_end_pos":[522,27]},{"full_name":"Int.ofNat_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[25,21],"def_end_pos":[25,30]}]},{"state_before":"R : Type u\ninst✝ : AddGroupWithOne R\nm n : ℕ\n⊢ ↑(↑m + -[n+1]) = ↑↑m + ↑-[n+1]","state_after":"no goals","tactic":"erw [cast_subNatNat, cast_natCast, cast_negSucc, sub_eq_add_neg]","premises":[{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.cast_negSucc","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[48,8],"def_end_pos":[48,20]},{"full_name":"Int.cast_subNatNat","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[82,8],"def_end_pos":[82,22]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"R : Type u\ninst✝ : AddGroupWithOne R\nm n : ℕ\n⊢ ↑(-[m+1] + ↑n) = ↑-[m+1] + ↑↑n","state_after":"no goals","tactic":"erw [cast_subNatNat, cast_natCast, cast_negSucc, sub_eq_iff_eq_add, add_assoc,\n      eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]","premises":[{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.cast_negSucc","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[48,8],"def_end_pos":[48,20]},{"full_name":"Int.cast_subNatNat","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[82,8],"def_end_pos":[82,22]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Nat.add_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[160,18],"def_end_pos":[160,26]},{"full_name":"Nat.cast_add","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"eq_neg_add_iff_add_eq","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[657,2],"def_end_pos":[657,13]},{"full_name":"sub_eq_iff_eq_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[756,2],"def_end_pos":[756,13]}]},{"state_before":"R : Type u\ninst✝ : AddGroupWithOne R\nm n : ℕ\n⊢ ↑-[m + n + 1+1] = ↑-[m+1] + ↑-[n+1]","state_after":"no goals","tactic":"rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,\n        Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]","premises":[{"full_name":"Int.cast_negSucc","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[48,8],"def_end_pos":[48,20]},{"full_name":"Nat.add_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[168,18],"def_end_pos":[168,27]},{"full_name":"Nat.add_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[160,18],"def_end_pos":[160,26]},{"full_name":"Nat.add_right_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[176,18],"def_end_pos":[176,32]},{"full_name":"Nat.cast_add","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"neg_add_rev","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[981,29],"def_end_pos":[981,40]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/NaturalOps.lean","commit":"","full_name":"Ordinal.le_self_nadd","start":[421,0],"end":[421,96],"file_path":"Mathlib/SetTheory/Ordinal/NaturalOps.lean","tactics":[{"state_before":"a b : Ordinal.{u_1}\n⊢ a ≤ a ♯ b","state_after":"no goals","tactic":"simpa using nadd_le_nadd_left (Ordinal.zero_le b) a","premises":[{"full_name":"Ordinal.nadd_le_nadd_left","def_path":"Mathlib/SetTheory/Ordinal/NaturalOps.lean","def_pos":[221,8],"def_end_pos":[221,25]},{"full_name":"Ordinal.zero_le","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[338,18],"def_end_pos":[338,25]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Fintype.prod_subset","start":[1989,0],"end":[1991,73],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι✝ : Type u_1\nκ✝ : Type u_2\nα✝ : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α✝\na : α✝\nf✝ g : α✝ → β\nι : Type u_6\nκ : Type u_7\nα : Type u_8\ninst✝² : Fintype ι\ninst✝¹ : Fintype κ\ninst✝ : CommMonoid α\ns : Finset ι\nf : ι → α\nh : ∀ (i : ι), f i ≠ 1 → i ∈ s\n⊢ ∀ x ∈ univ, x ∉ s → f x = 1","state_after":"no goals","tactic":"simpa [not_imp_comm (a := _ ∈ s)]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"not_imp_comm","def_path":"Mathlib/Logic/Basic.lean","def_pos":[213,8],"def_end_pos":[213,20]}]}]}
{"url":"Mathlib/MeasureTheory/PiSystem.lean","commit":"","full_name":"mem_generatePiSystem_iUnion_elim","start":[242,0],"end":[273,41],"file_path":"Mathlib/MeasureTheory/PiSystem.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nh_t : t ∈ generatePiSystem (⋃ b, g b)\n⊢ ∃ T f, t = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b","state_after":"case base\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nh_s : s ∈ ⋃ b, g b\n⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b\n\ncase inter\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s t' : Set α\nh_gen_s : generatePiSystem (⋃ b, g b) s\nh_gen_t' : generatePiSystem (⋃ b, g b) t'\nh_nonempty : (s ∩ t').Nonempty\nh_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b\nh_t' : ∃ T f, t' = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b\n⊢ ∃ T f, s ∩ t' = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b","tactic":"induction' h_t with s h_s s t' h_gen_s h_gen_t' h_nonempty h_s h_t'","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","commit":"","full_name":"CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.of_equivalence","start":[162,0],"end":[175,73],"file_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","tactics":[{"state_before":"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\nD₃ : Type u₆\ninst✝⁸ : Category.{v₁, u₁} C₁\ninst✝⁷ : Category.{v₂, u₂} C₂\ninst✝⁶ : Category.{v₃, u₃} C₃\ninst✝⁵ : Category.{v₄, u₄} D₁\ninst✝⁴ : Category.{v₅, u₅} D₂\ninst✝³ : Category.{v₆, u₆} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nG : D₁ ⥤ D₂\ninst✝ : Φ.functor.IsEquivalence\nh : W₂ ≤ W₁.map Φ.functor\n⊢ Φ.IsLocalizedEquivalence","state_after":"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\nD₃ : Type u₆\ninst✝⁸ : Category.{v₁, u₁} C₁\ninst✝⁷ : Category.{v₂, u₂} C₂\ninst✝⁶ : Category.{v₃, u₃} C₃\ninst✝⁵ : Category.{v₄, u₄} D₁\ninst✝⁴ : Category.{v₅, u₅} D₂\ninst✝³ : Category.{v₆, u₆} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nG : D₁ ⥤ D₂\ninst✝ : Φ.functor.IsEquivalence\nh : W₂ ≤ W₁.map Φ.functor\nthis : (Φ.functor ⋙ W₂.Q).IsLocalization W₁\n⊢ Φ.IsLocalizedEquivalence","tactic":"haveI : Functor.IsLocalization (Φ.functor ⋙ MorphismProperty.Q W₂) W₁ := by\n    refine Functor.IsLocalization.of_equivalence_source W₂.Q W₂ (Φ.functor ⋙ W₂.Q) W₁\n      (Functor.asEquivalence Φ.functor).symm ?_ (Φ.inverts W₂.Q)\n      ((Functor.associator _ _ _).symm ≪≫ isoWhiskerRight ((Equivalence.unitIso _).symm) _ ≪≫\n        Functor.leftUnitor _)\n    erw [W₁.isoClosure.inverseImage_equivalence_functor_eq_map_inverse]\n    rw [MorphismProperty.map_isoClosure]\n    exact h","premises":[{"full_name":"CategoryTheory.Equivalence.symm","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[262,4],"def_end_pos":[262,8]},{"full_name":"CategoryTheory.Equivalence.unitIso","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[85,2],"def_end_pos":[85,9]},{"full_name":"CategoryTheory.Functor.IsLocalization","def_path":"Mathlib/CategoryTheory/Localization/Predicate.lean","def_pos":[48,6],"def_end_pos":[48,20]},{"full_name":"CategoryTheory.Functor.IsLocalization.of_equivalence_source","def_path":"Mathlib/CategoryTheory/Localization/Equivalence.lean","def_pos":[67,6],"def_end_pos":[67,27]},{"full_name":"CategoryTheory.Functor.asEquivalence","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[528,18],"def_end_pos":[528,31]},{"full_name":"CategoryTheory.Functor.associator","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[242,4],"def_end_pos":[242,14]},{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Functor.leftUnitor","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[220,4],"def_end_pos":[220,14]},{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.Iso.trans","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[129,4],"def_end_pos":[129,9]},{"full_name":"CategoryTheory.LocalizerMorphism.functor","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[43,2],"def_end_pos":[43,9]},{"full_name":"CategoryTheory.LocalizerMorphism.inverts","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[75,6],"def_end_pos":[75,13]},{"full_name":"CategoryTheory.MorphismProperty.Q","def_path":"Mathlib/CategoryTheory/Localization/Construction.lean","def_pos":[104,4],"def_end_pos":[104,5]},{"full_name":"CategoryTheory.MorphismProperty.inverseImage_equivalence_functor_eq_map_inverse","def_path":"Mathlib/CategoryTheory/MorphismProperty/Basic.lean","def_pos":[261,6],"def_end_pos":[261,53]},{"full_name":"CategoryTheory.MorphismProperty.isoClosure","def_path":"Mathlib/CategoryTheory/MorphismProperty/Basic.lean","def_pos":[114,4],"def_end_pos":[114,14]},{"full_name":"CategoryTheory.MorphismProperty.map_isoClosure","def_path":"Mathlib/CategoryTheory/MorphismProperty/Basic.lean","def_pos":[197,6],"def_end_pos":[197,20]},{"full_name":"CategoryTheory.isoWhiskerRight","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[167,4],"def_end_pos":[167,19]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\nD₃ : Type u₆\ninst✝⁸ : Category.{v₁, u₁} C₁\ninst✝⁷ : Category.{v₂, u₂} C₂\ninst✝⁶ : Category.{v₃, u₃} C₃\ninst✝⁵ : Category.{v₄, u₄} D₁\ninst✝⁴ : Category.{v₅, u₅} D₂\ninst✝³ : Category.{v₆, u₆} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nG : D₁ ⥤ D₂\ninst✝ : Φ.functor.IsEquivalence\nh : W₂ ≤ W₁.map Φ.functor\nthis : (Φ.functor ⋙ W₂.Q).IsLocalization W₁\n⊢ Φ.IsLocalizedEquivalence","state_after":"no goals","tactic":"exact IsLocalizedEquivalence.of_isLocalization_of_isLocalization Φ W₂.Q","premises":[{"full_name":"CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.of_isLocalization_of_isLocalization","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[155,6],"def_end_pos":[155,64]},{"full_name":"CategoryTheory.MorphismProperty.Q","def_path":"Mathlib/CategoryTheory/Localization/Construction.lean","def_pos":[104,4],"def_end_pos":[104,5]}]}]}
{"url":"Mathlib/Algebra/ContinuedFractions/Translations.lean","commit":"","full_name":"GenContFract.convs'Aux_succ_some","start":[151,0],"end":[153,21],"file_path":"Mathlib/Algebra/ContinuedFractions/Translations.lean","tactics":[{"state_before":"K : Type u_1\ng : GenContFract K\nn✝ : ℕ\ninst✝ : DivisionRing K\ns : Stream'.Seq (Pair K)\np : Pair K\nh : s.head = some p\nn : ℕ\n⊢ convs'Aux s (n + 1) = p.a / (p.b + convs'Aux s.tail n)","state_after":"no goals","tactic":"simp [convs'Aux, h]","premises":[{"full_name":"GenContFract.convs'Aux","def_path":"Mathlib/Algebra/ContinuedFractions/Basic.lean","def_pos":[353,4],"def_end_pos":[353,13]}]}]}
{"url":"Mathlib/RingTheory/DedekindDomain/Different.lean","commit":"","full_name":"coeSubmodule_differentIdeal_fractionRing","start":[388,0],"end":[405,21],"file_path":"Mathlib/RingTheory/DedekindDomain/Different.lean","tactics":[{"state_before":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\n⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / Submodule.traceDual A (FractionRing A) 1","state_after":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis : IsIntegralClosure B A (FractionRing B)\n⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / Submodule.traceDual A (FractionRing A) 1","tactic":"have : IsIntegralClosure B A (FractionRing B) :=\n    IsIntegralClosure.of_isIntegrallyClosed _ _ _","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"IsIntegralClosure","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Defs.lean","def_pos":[27,6],"def_end_pos":[27,23]},{"full_name":"IsIntegralClosure.of_isIntegrallyClosed","def_path":"Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean","def_pos":[215,9],"def_end_pos":[215,55]}]},{"state_before":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis : IsIntegralClosure B A (FractionRing B)\n⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / Submodule.traceDual A (FractionRing A) 1","state_after":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis : IsIntegralClosure B A (FractionRing B)\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","tactic":"rw [coeSubmodule, differentIdeal, Submodule.map_comap_eq, inf_eq_right]","premises":[{"full_name":"IsLocalization.coeSubmodule","def_path":"Mathlib/RingTheory/Localization/Submodule.lean","def_pos":[32,4],"def_end_pos":[32,16]},{"full_name":"Submodule.map_comap_eq","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[154,8],"def_end_pos":[154,37]},{"full_name":"differentIdeal","def_path":"Mathlib/RingTheory/DedekindDomain/Different.lean","def_pos":[384,4],"def_end_pos":[384,18]},{"full_name":"inf_eq_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[341,8],"def_end_pos":[341,20]}]},{"state_before":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis : IsIntegralClosure B A (FractionRing B)\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","state_after":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝ : IsIntegralClosure B A (FractionRing B)\nthis : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","tactic":"have := FractionalIdeal.dual_inv_le (A := A) (K := FractionRing A)\n    (1 : FractionalIdeal B⁰ (FractionRing B))","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.dual_inv_le","def_path":"Mathlib/RingTheory/DedekindDomain/Different.lean","def_pos":[314,6],"def_end_pos":[314,17]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝ : IsIntegralClosure B A (FractionRing B)\nthis : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","state_after":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : (fun a => ↑a) (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ ↑1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","tactic":"have : _ ≤ ((1 : FractionalIdeal B⁰ (FractionRing B)) : Submodule B (FractionRing B)) := this","premises":[{"full_name":"FractionRing","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[266,7],"def_end_pos":[266,19]},{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"Submodule","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[36,10],"def_end_pos":[36,19]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : (fun a => ↑a) (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ ↑1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","state_after":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","tactic":"simp only [← one_div, FractionalIdeal.val_eq_coe] at this","premises":[{"full_name":"FractionalIdeal.val_eq_coe","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[189,8],"def_end_pos":[189,18]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]}]},{"state_before":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))","state_after":"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : ↑1 / Submodule.traceDual A (FractionRing A) ↑1 ≤ ↑1\n⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))\n\ncase hI\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : ↑1 / ↑(FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1\n⊢ 1 ≠ 0\n\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²² : CommRing A\ninst✝²¹ : Field K\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Field L\ninst✝¹⁸ : Algebra A K\ninst✝¹⁷ : Algebra B L\ninst✝¹⁶ : Algebra A B\ninst✝¹⁵ : Algebra K L\ninst✝¹⁴ : Algebra A L\ninst✝¹³ : IsScalarTower A K L\ninst✝¹² : IsScalarTower A B L\ninst✝¹¹ : IsDomain A\ninst✝¹⁰ : IsFractionRing A K\ninst✝⁹ : IsIntegralClosure B A L\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : FiniteDimensional K L\ninst✝⁶ : Algebra.IsSeparable K L\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : NoZeroSMulDivisors A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝ : FiniteDimensional (FractionRing A) (FractionRing B)\nthis✝¹ : IsIntegralClosure B A (FractionRing B)\nthis✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1\nthis : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1\n⊢ 1 ≠ 0","tactic":"rw [FractionalIdeal.coe_div (FractionalIdeal.dual_ne_zero _ _ _),\n    FractionalIdeal.coe_dual] at this","premises":[{"full_name":"FractionalIdeal.coe_div","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[393,8],"def_end_pos":[393,15]},{"full_name":"FractionalIdeal.coe_dual","def_path":"Mathlib/RingTheory/DedekindDomain/Different.lean","def_pos":[234,6],"def_end_pos":[234,14]},{"full_name":"FractionalIdeal.dual_ne_zero","def_path":"Mathlib/RingTheory/DedekindDomain/Different.lean","def_pos":[257,6],"def_end_pos":[257,18]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","commit":"","full_name":"ENNReal.top_rpow_of_pos","start":[378,0],"end":[379,93],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","tactics":[{"state_before":"y : ℝ\nh : 0 < y\n⊢ ⊤ ^ y = ⊤","state_after":"no goals","tactic":"simp [top_rpow_def, h]","premises":[{"full_name":"ENNReal.top_rpow_def","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[375,8],"def_end_pos":[375,20]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","commit":"","full_name":"ProbabilityTheory.StieltjesFunction.measurable_measure","start":[478,0],"end":[504,44],"file_path":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\n⊢ Measurable fun a => (f a).measure","state_after":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\n⊢ Measurable fun b => (f b).measure s","tactic":"refine Measure.measurable_measure.mpr fun s hs ↦ ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.Measure.measurable_measure","def_path":"Mathlib/MeasureTheory/Measure/GiryMonad.lean","def_pos":[67,8],"def_end_pos":[67,26]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\n⊢ Measurable fun b => (f b).measure s","state_after":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\nthis : ∀ (a : α), IsProbabilityMeasure (f a).measure\n⊢ Measurable fun b => (f b).measure s","tactic":"have : ∀ a, IsProbabilityMeasure (f a).measure :=\n    fun a ↦ (f a).isProbabilityMeasure (hf_bot a) (hf_top a)","premises":[{"full_name":"MeasureTheory.IsProbabilityMeasure","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[207,6],"def_end_pos":[207,26]},{"full_name":"StieltjesFunction.isProbabilityMeasure","def_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","def_pos":[453,6],"def_end_pos":[453,26]},{"full_name":"StieltjesFunction.measure","def_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","def_pos":[355,26],"def_end_pos":[355,33]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\nthis : ∀ (a : α), IsProbabilityMeasure (f a).measure\n⊢ Measurable fun b => (f b).measure s","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\nthis : ∀ (a : α), IsProbabilityMeasure (f a).measure\n⊢ (fun s => Measurable fun b => (f b).measure s) ∅\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\nthis : ∀ (a : α), IsProbabilityMeasure (f a).measure\n⊢ ∀ t ∈ range Iic, (fun s => Measurable fun b => (f b).measure s) t\n\ncase refine_3\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\nthis : ∀ (a : α), IsProbabilityMeasure (f a).measure\n⊢ ∀ (t : Set ℝ),\n    MeasurableSet t →\n      (fun s => Measurable fun b => (f b).measure s) t → (fun s => Measurable fun b => (f b).measure s) tᶜ\n\ncase refine_4\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α (β × ℝ)\nν✝ : Kernel α β\nx✝ : MeasurableSpace β\nf✝ : α × β → StieltjesFunction\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α → StieltjesFunction\nhf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q\nhf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)\nhf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)\ns : Set ℝ\nhs : MeasurableSet s\nthis : ∀ (a : α), IsProbabilityMeasure (f a).measure\n⊢ ∀ (f_1 : ℕ → Set ℝ),\n    Pairwise (Disjoint on f_1) →\n      (∀ (i : ℕ), MeasurableSet (f_1 i)) →\n        (∀ (i : ℕ), (fun s => Measurable fun b => (f b).measure s) (f_1 i)) →\n          (fun s => Measurable fun b => (f b).measure s) (⋃ i, f_1 i)","tactic":"refine MeasurableSpace.induction_on_inter\n    (C := fun s ↦ Measurable fun b ↦ StieltjesFunction.measure (f b) s)\n    (borel_eq_generateFrom_Iic ℝ) isPiSystem_Iic ?_ ?_ ?_ ?_ hs","premises":[{"full_name":"Measurable","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[479,4],"def_end_pos":[479,14]},{"full_name":"MeasurableSpace.induction_on_inter","def_path":"Mathlib/MeasureTheory/PiSystem.lean","def_pos":[658,8],"def_end_pos":[658,26]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"StieltjesFunction.measure","def_path":"Mathlib/MeasureTheory/Measure/Stieltjes.lean","def_pos":[355,26],"def_end_pos":[355,33]},{"full_name":"borel_eq_generateFrom_Iic","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[77,8],"def_end_pos":[77,33]},{"full_name":"isPiSystem_Iic","def_path":"Mathlib/MeasureTheory/PiSystem.lean","def_pos":[139,8],"def_end_pos":[139,22]}]}]}
{"url":"Mathlib/Data/Int/Cast/Basic.lean","commit":"","full_name":"Int.cast_ofNat","start":[64,0],"end":[67,73],"file_path":"Mathlib/Data/Int/Cast/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : AddGroupWithOne R\nn : ℕ\ninst✝ : n.AtLeastTwo\n⊢ ↑(OfNat.ofNat n) = OfNat.ofNat n","state_after":"no goals","tactic":"simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n","premises":[{"full_name":"AddGroupWithOne.intCast_ofNat","def_path":"Mathlib/Data/Int/Cast/Defs.lean","def_pos":[40,2],"def_end_pos":[40,15]},{"full_name":"OfNat.ofNat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1105,2],"def_end_pos":[1105,7]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","commit":"","full_name":"Batteries.BinomialHeap.Imp.Heap.realSize_deleteMin","start":[247,0],"end":[257,95],"file_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","tactics":[{"state_before":"α : Type u_1\nle : α → α → Bool\na : α\ns' s : Heap α\neq : deleteMin le s = some (a, s')\n⊢ s.realSize = s'.realSize + 1","state_after":"case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\n⊢ (cons r a c s).realSize =\n    (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n          ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n      1","tactic":"cases s with cases eq | cons r a c s => ?_","premises":[{"full_name":"Batteries.BinomialHeap.Imp.Heap.cons","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[76,4],"def_end_pos":[76,8]}]},{"state_before":"case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\n⊢ (cons r a c s).realSize =\n    (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n          ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n      1","state_after":"case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\nthis :\n  Batteries.BinomialHeap.Imp.FindMin.HasSize\n    (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1)\n⊢ (cons r a c s).realSize =\n    (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n          ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n      1","tactic":"have : (s.findMin le (cons r a c) ⟨id, a, c, s⟩).HasSize (c.realSize + s.realSize + 1) :=\n    Heap.realSize_findMin (c.realSize + 1) (by simp) (Nat.add_right_comm ..) ⟨0, by simp⟩","premises":[{"full_name":"Batteries.BinomialHeap.Imp.Heap.cons","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[76,4],"def_end_pos":[76,8]},{"full_name":"Batteries.BinomialHeap.Imp.Heap.findMin","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[182,18],"def_end_pos":[182,30]},{"full_name":"Batteries.BinomialHeap.Imp.Heap.realSize","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[84,12],"def_end_pos":[84,25]},{"full_name":"Batteries.BinomialHeap.Imp.HeapNode.realSize","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[38,12],"def_end_pos":[38,29]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Nat.add_right_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[176,18],"def_end_pos":[176,32]},{"full_name":"_private.«.lake».packages.batteries.Batteries.Data.BinomialHeap.Basic.0.Batteries.BinomialHeap.Imp.FindMin.HasSize","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[223,12],"def_end_pos":[223,27]},{"full_name":"_private.«.lake».packages.batteries.Batteries.Data.BinomialHeap.Basic.0.Batteries.BinomialHeap.Imp.Heap.realSize_findMin","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[228,16],"def_end_pos":[228,37]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\nthis :\n  Batteries.BinomialHeap.Imp.FindMin.HasSize\n    (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1)\n⊢ (cons r a c s).realSize =\n    (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n          ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n            (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n      1","state_after":"case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\n⊢ Batteries.BinomialHeap.Imp.FindMin.HasSize\n      (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1) →\n    (cons r a c s).realSize =\n      (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n            ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n              (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n        1","tactic":"revert this","premises":[]},{"state_before":"case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\n⊢ Batteries.BinomialHeap.Imp.FindMin.HasSize\n      (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1) →\n    (cons r a c s).realSize =\n      (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n            ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n              (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n        1","state_after":"no goals","tactic":"match s.findMin le (cons r a c) ⟨id, a, c, s⟩ with\n  | { before, val, node, next } =>\n    intro ⟨m, ih₁, ih₂⟩; dsimp only at ih₁ ih₂\n    rw [realSize, Nat.add_right_comm, ih₂]\n    simp only [realSize_merge, HeapNode.realSize_toHeap, ih₁, Nat.add_assoc, Nat.add_left_comm]","premises":[{"full_name":"Batteries.BinomialHeap.Imp.Heap.cons","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[76,4],"def_end_pos":[76,8]},{"full_name":"Batteries.BinomialHeap.Imp.Heap.findMin","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[182,18],"def_end_pos":[182,30]},{"full_name":"Batteries.BinomialHeap.Imp.Heap.realSize","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[84,12],"def_end_pos":[84,25]},{"full_name":"Batteries.BinomialHeap.Imp.Heap.realSize_merge","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[205,8],"def_end_pos":[205,27]},{"full_name":"Batteries.BinomialHeap.Imp.HeapNode.realSize_toHeap","def_path":".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean","def_pos":[242,8],"def_end_pos":[242,32]},{"full_name":"Nat.add_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[168,18],"def_end_pos":[168,27]},{"full_name":"Nat.add_left_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[173,18],"def_end_pos":[173,31]},{"full_name":"Nat.add_right_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[176,18],"def_end_pos":[176,32]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
{"url":"Mathlib/Data/Matrix/Rank.lean","commit":"","full_name":"Matrix.rank_submatrix_le","start":[113,0],"end":[119,36],"file_path":"Mathlib/Data/Matrix/Rank.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\ninst✝⁴ : Fintype n\ninst✝³ : Fintype o\ninst✝² : CommRing R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype m\nf : n → m\ne : n ≃ m\nA : Matrix m m R\n⊢ (A.submatrix f ⇑e).rank ≤ A.rank","state_after":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\ninst✝⁴ : Fintype n\ninst✝³ : Fintype o\ninst✝² : CommRing R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype m\nf : n → m\ne : n ≃ m\nA : Matrix m m R\n⊢ finrank R ↥(Submodule.map (LinearMap.funLeft R R f) (LinearMap.range A.mulVecLin)) ≤\n    finrank R ↥(LinearMap.range A.mulVecLin)","tactic":"rw [rank, rank, mulVecLin_submatrix, LinearMap.range_comp, LinearMap.range_comp,\n    show LinearMap.funLeft R R e.symm = LinearEquiv.funCongrLeft R R e.symm from rfl,\n    LinearEquiv.range, Submodule.map_top]","premises":[{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"LinearEquiv.funCongrLeft","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[671,4],"def_end_pos":[671,16]},{"full_name":"LinearEquiv.range","def_path":"Mathlib/Algebra/Module/Submodule/Equiv.lean","def_pos":[114,18],"def_end_pos":[114,23]},{"full_name":"LinearMap.funLeft","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[627,4],"def_end_pos":[627,11]},{"full_name":"LinearMap.range_comp","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[81,8],"def_end_pos":[81,18]},{"full_name":"Matrix.mulVecLin_submatrix","def_path":"Mathlib/LinearAlgebra/Matrix/ToLin.lean","def_pos":[228,8],"def_end_pos":[228,34]},{"full_name":"Matrix.rank","def_path":"Mathlib/Data/Matrix/Rank.lean","def_pos":[43,18],"def_end_pos":[43,22]},{"full_name":"Submodule.map_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[259,8],"def_end_pos":[259,15]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\ninst✝⁴ : Fintype n\ninst✝³ : Fintype o\ninst✝² : CommRing R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype m\nf : n → m\ne : n ≃ m\nA : Matrix m m R\n⊢ finrank R ↥(Submodule.map (LinearMap.funLeft R R f) (LinearMap.range A.mulVecLin)) ≤\n    finrank R ↥(LinearMap.range A.mulVecLin)","state_after":"no goals","tactic":"exact Submodule.finrank_map_le _ _","premises":[{"full_name":"Submodule.finrank_map_le","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[385,8],"def_end_pos":[385,32]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/HasLimits.lean","commit":"","full_name":"CategoryTheory.Limits.hasLimitsOfShape_of_equivalence","start":[554,0],"end":[560,35],"file_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","tactics":[{"state_before":"J : Type u₁\ninst✝⁴ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝³ : Category.{v₂, u₂} K\nC : Type u\ninst✝² : Category.{v, u} C\nF : J ⥤ C\nJ' : Type u₂\ninst✝¹ : Category.{v₂, u₂} J'\ne : J ≌ J'\ninst✝ : HasLimitsOfShape J C\n⊢ HasLimitsOfShape J' C","state_after":"case has_limit\nJ : Type u₁\ninst✝⁴ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝³ : Category.{v₂, u₂} K\nC : Type u\ninst✝² : Category.{v, u} C\nF : J ⥤ C\nJ' : Type u₂\ninst✝¹ : Category.{v₂, u₂} J'\ne : J ≌ J'\ninst✝ : HasLimitsOfShape J C\n⊢ autoParam (∀ (F : J' ⥤ C), HasLimit F) _auto✝","tactic":"constructor","premises":[]},{"state_before":"case has_limit\nJ : Type u₁\ninst✝⁴ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝³ : Category.{v₂, u₂} K\nC : Type u\ninst✝² : Category.{v, u} C\nF : J ⥤ C\nJ' : Type u₂\ninst✝¹ : Category.{v₂, u₂} J'\ne : J ≌ J'\ninst✝ : HasLimitsOfShape J C\n⊢ autoParam (∀ (F : J' ⥤ C), HasLimit F) _auto✝","state_after":"case has_limit\nJ : Type u₁\ninst✝⁴ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝³ : Category.{v₂, u₂} K\nC : Type u\ninst✝² : Category.{v, u} C\nF✝ : J ⥤ C\nJ' : Type u₂\ninst✝¹ : Category.{v₂, u₂} J'\ne : J ≌ J'\ninst✝ : HasLimitsOfShape J C\nF : J' ⥤ C\n⊢ HasLimit F","tactic":"intro F","premises":[]},{"state_before":"case has_limit\nJ : Type u₁\ninst✝⁴ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝³ : Category.{v₂, u₂} K\nC : Type u\ninst✝² : Category.{v, u} C\nF✝ : J ⥤ C\nJ' : Type u₂\ninst✝¹ : Category.{v₂, u₂} J'\ne : J ≌ J'\ninst✝ : HasLimitsOfShape J C\nF : J' ⥤ C\n⊢ HasLimit F","state_after":"no goals","tactic":"apply hasLimitOfEquivalenceComp e","premises":[{"full_name":"CategoryTheory.Limits.hasLimitOfEquivalenceComp","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[436,8],"def_end_pos":[436,33]}]}]}
{"url":"Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean","commit":"","full_name":"Finset.truncatedSup_infs","start":[255,0],"end":[260,29],"file_path":"Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ (s ⊼ t).truncatedSup a = s.truncatedSup a ⊓ t.truncatedSup a","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ ((filter (fun x => a ≤ x) s ⊼ filter (fun x => a ≤ x) t).sup' ⋯ fun x => id x) =\n    (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2","tactic":"simp only [truncatedSup_of_mem, hs, ht, infs_aux.2 ⟨hs, ht⟩, sup'_inf_sup', filter_infs_le]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Finset.filter_infs_le","def_path":"Mathlib/Data/Finset/Sups.lean","def_pos":[307,6],"def_end_pos":[307,20]},{"full_name":"Finset.sup'_inf_sup'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[1101,8],"def_end_pos":[1101,21]},{"full_name":"Finset.truncatedSup_of_mem","def_path":"Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean","def_pos":[115,6],"def_end_pos":[115,25]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"_private.Mathlib.Combinatorics.SetFamily.AhlswedeZhang.0.Finset.infs_aux","def_path":"Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean","def_pos":[249,14],"def_end_pos":[249,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ ((filter (fun x => a ≤ x) s ⊼ filter (fun x => a ≤ x) t).sup' ⋯ fun x => id x) =\n    (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ ((image (Function.uncurry fun x x_1 => x ⊓ x_1) (filter (fun x => a ≤ x) s ×ˢ filter (fun x => a ≤ x) t)).sup' ⋯\n      fun x => id x) =\n    (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2","tactic":"simp_rw [← image_inf_product]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.image_inf_product","def_path":"Mathlib/Data/Finset/Sups.lean","def_pos":[318,8],"def_end_pos":[318,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ ((image (Function.uncurry fun x x_1 => x ⊓ x_1) (filter (fun x => a ≤ x) s ×ˢ filter (fun x => a ≤ x) t)).sup' ⋯\n      fun x => id x) =\n    (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ (filter (fun x => a ≤ x) s ×ˢ filter (fun x => a ≤ x) t).sup' ⋯\n      ((fun x => id x) ∘ Function.uncurry fun x x_1 => x ⊓ x_1) =\n    (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2","tactic":"rw [sup'_image]","premises":[{"full_name":"Finset.sup'_image","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[823,8],"def_end_pos":[823,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : DistribLattice α\ninst✝² : BoundedOrder α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : Finset α\na : α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ (filter (fun x => a ≤ x) s ×ˢ filter (fun x => a ≤ x) t).sup' ⋯\n      ((fun x => id x) ∘ Function.uncurry fun x x_1 => x ⊓ x_1) =\n    (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2","state_after":"no goals","tactic":"simp [Function.uncurry_def]","premises":[{"full_name":"Function.uncurry_def","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[685,8],"def_end_pos":[685,19]}]}]}
{"url":"Mathlib/Algebra/Group/Submonoid/Membership.lean","commit":"","full_name":"IsScalarTower.of_mclosure_eq_top","start":[512,0],"end":[521,41],"file_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","tactics":[{"state_before":"M : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\nα : Type u_5\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ x ∈ s, ∀ (y : N) (z : α), (x • y) • z = x • y • z\n⊢ IsScalarTower M N α","state_after":"case refine_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\nα : Type u_5\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ x ∈ s, ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\n⊢ ∀ (y : N) (z : α), (1 • y) • z = 1 • y • z\n\ncase refine_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\nα : Type u_5\ninst✝³ : Monoid M\ninst✝² : MulAction M N\ninst✝¹ : SMul N α\ninst✝ : MulAction M α\ns : Set M\nhtop : Submonoid.closure s = ⊤\nhs : ∀ x ∈ s, ∀ (y : N) (z : α), (x • y) • z = x • y • z\nx : M\n⊢ ∀ x ∈ s,\n    ∀ (y : M),\n      (∀ (y_1 : N) (z : α), (y • y_1) • z = y • y_1 • z) → ∀ (y_1 : N) (z : α), ((x * y) • y_1) • z = (x * y) • y_1 • z","tactic":"refine ⟨fun x => Submonoid.induction_of_closure_eq_top_left htop x ?_ ?_⟩","premises":[{"full_name":"Submonoid.induction_of_closure_eq_top_left","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[366,8],"def_end_pos":[366,40]}]}]}
{"url":"Mathlib/Data/DFinsupp/Basic.lean","commit":"","full_name":"DFinsupp.support_update","start":[1103,0],"end":[1109,38],"file_path":"Mathlib/Data/DFinsupp/Basic.lean","tactics":[{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\ninst✝ : Decidable (b = 0)\n⊢ (update i f b).support = if b = 0 then (erase i f).support else insert i f.support","state_after":"case a\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\ninst✝ : Decidable (b = 0)\nj : ι\n⊢ j ∈ (update i f b).support ↔ j ∈ if b = 0 then (erase i f).support else insert i f.support","tactic":"ext j","premises":[]},{"state_before":"case a\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\ninst✝ : Decidable (b = 0)\nj : ι\n⊢ j ∈ (update i f b).support ↔ j ∈ if b = 0 then (erase i f).support else insert i f.support","state_after":"case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\ninst✝ : Decidable (b = 0)\nj : ι\nhb : b = 0\n⊢ j ∈ (update i f b).support ↔ j ∈ (erase i f).support\n\ncase neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\ninst✝ : Decidable (b = 0)\nj : ι\nhb : ¬b = 0\n⊢ j ∈ (update i f b).support ↔ j ∈ insert i f.support","tactic":"split_ifs with hb","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/RingTheory/FiniteType.lean","commit":"","full_name":"Algebra.FiniteType.polynomial","start":[69,0],"end":[72,33],"file_path":"Mathlib/RingTheory/FiniteType.lean","tactics":[{"state_before":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nM : Type uM\nN : Type uN\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\n⊢ adjoin R ↑{X} = ⊤","state_after":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nM : Type uM\nN : Type uN\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\n⊢ adjoin R {X} = ⊤","tactic":"rw [Finset.coe_singleton]","premises":[{"full_name":"Finset.coe_singleton","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[621,8],"def_end_pos":[621,21]}]},{"state_before":"R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nM : Type uM\nN : Type uN\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\n⊢ adjoin R {X} = ⊤","state_after":"no goals","tactic":"exact Polynomial.adjoin_X","premises":[{"full_name":"Polynomial.adjoin_X","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[231,8],"def_end_pos":[231,16]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Lifts.lean","commit":"","full_name":"Polynomial.monomial_mem_lifts","start":[103,0],"end":[108,17],"file_path":"Mathlib/Algebra/Polynomial/Lifts.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\ns : S\nn : ℕ\nh : s ∈ Set.range ⇑f\n⊢ (monomial n) s ∈ lifts f","state_after":"case intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nr : R\nh : f r ∈ Set.range ⇑f\n⊢ (monomial n) (f r) ∈ lifts f","tactic":"obtain ⟨r, rfl⟩ := Set.mem_range.1 h","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[146,16],"def_end_pos":[146,25]}]},{"state_before":"case intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nr : R\nh : f r ∈ Set.range ⇑f\n⊢ (monomial n) (f r) ∈ lifts f","state_after":"case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nr : R\nh : f r ∈ Set.range ⇑f\n⊢ (mapRingHom f) ((monomial n) r) = (monomial n) (f r)","tactic":"use monomial n r","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Polynomial.monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[371,4],"def_end_pos":[371,12]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nr : R\nh : f r ∈ Set.range ⇑f\n⊢ (mapRingHom f) ((monomial n) r) = (monomial n) (f r)","state_after":"no goals","tactic":"simp only [coe_mapRingHom, Set.mem_univ, map_monomial, Subsemiring.coe_top, eq_self_iff_true,\n    and_self_iff]","premises":[{"full_name":"Polynomial.coe_mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[649,8],"def_end_pos":[649,22]},{"full_name":"Polynomial.map_monomial","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[609,8],"def_end_pos":[609,20]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"Subsemiring.coe_top","def_path":"Mathlib/Algebra/Ring/Subsemiring/Basic.lean","def_pos":[362,8],"def_end_pos":[362,15]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","commit":"","full_name":"Sum.elim_const_const","start":[243,0],"end":[246,17],"file_path":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","tactics":[{"state_before":"γ : Sort u_1\nα : Type u_2\nβ : Type u_3\nc : γ\n⊢ Sum.elim (const α c) (const β c) = const (α ⊕ β) c","state_after":"case h\nγ : Sort u_1\nα : Type u_2\nβ : Type u_3\nc : γ\nx : α ⊕ β\n⊢ Sum.elim (const α c) (const β c) x = const (α ⊕ β) c x","tactic":"ext x","premises":[]},{"state_before":"case h\nγ : Sort u_1\nα : Type u_2\nβ : Type u_3\nc : γ\nx : α ⊕ β\n⊢ Sum.elim (const α c) (const β c) x = const (α ⊕ β) c x","state_after":"no goals","tactic":"cases x <;> rfl","premises":[]}]}
{"url":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","commit":"","full_name":"Set.eq_of_mem_uIcc_of_mem_uIcc'","start":[142,0],"end":[143,59],"file_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : DistribLattice α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ b ∈ [[a, c]] → c ∈ [[a, b]] → b = c","state_after":"no goals","tactic":"simpa only [uIcc_comm a] using eq_of_mem_uIcc_of_mem_uIcc","premises":[{"full_name":"Set.eq_of_mem_uIcc_of_mem_uIcc","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[139,6],"def_end_pos":[139,32]},{"full_name":"Set.uIcc_comm","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[72,6],"def_end_pos":[72,15]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/Deriv.lean","commit":"","full_name":"LSeries.absicssaOfAbsConv_logPowMul","start":[119,0],"end":[126,87],"file_path":"Mathlib/NumberTheory/LSeries/Deriv.lean","tactics":[{"state_before":"f : ℕ → ℂ\nm : ℕ\n⊢ abscissaOfAbsConv (logMul^[m] f) = abscissaOfAbsConv f","state_after":"case zero\nf : ℕ → ℂ\n⊢ abscissaOfAbsConv (logMul^[0] f) = abscissaOfAbsConv f\n\ncase succ\nf : ℕ → ℂ\nn : ℕ\nih : abscissaOfAbsConv (logMul^[n] f) = abscissaOfAbsConv f\n⊢ abscissaOfAbsConv (logMul^[n + 1] f) = abscissaOfAbsConv f","tactic":"induction' m with n ih","premises":[]}]}
{"url":"Mathlib/ModelTheory/Basic.lean","commit":"","full_name":"FirstOrder.Language.Equiv.self_comp_symm_toHom","start":[865,0],"end":[868,47],"file_path":"Mathlib/ModelTheory/Basic.lean","tactics":[{"state_before":"L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nP : Type u_1\ninst✝¹ : L.Structure P\nQ : Type u_2\ninst✝ : L.Structure Q\nf : M ≃[L] N\n⊢ f.toHom.comp f.symm.toHom = Hom.id L N","state_after":"no goals","tactic":"rw [← comp_toHom, self_comp_symm, refl_toHom]","premises":[{"full_name":"FirstOrder.Language.Equiv.comp_toHom","def_path":"Mathlib/ModelTheory/Basic.lean","def_pos":[833,8],"def_end_pos":[833,18]},{"full_name":"FirstOrder.Language.Equiv.refl_toHom","def_path":"Mathlib/ModelTheory/Basic.lean","def_pos":[819,8],"def_end_pos":[819,18]},{"full_name":"FirstOrder.Language.Equiv.self_comp_symm","def_path":"Mathlib/ModelTheory/Basic.lean","def_pos":[843,8],"def_end_pos":[843,22]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","commit":"","full_name":"Real.Gamma_neg_nat_eq_zero","start":[518,0],"end":[522,65],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","tactics":[{"state_before":"n : ℕ\n⊢ Gamma (-↑n) = 0","state_after":"no goals","tactic":"simpa only [← Complex.ofReal_natCast, ← Complex.ofReal_neg, Complex.Gamma_ofReal,\n    Complex.ofReal_eq_zero] using Complex.Gamma_neg_nat_eq_zero n","premises":[{"full_name":"Complex.Gamma_neg_nat_eq_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[343,8],"def_end_pos":[343,29]},{"full_name":"Complex.Gamma_ofReal","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[500,8],"def_end_pos":[500,35]},{"full_name":"Complex.ofReal_eq_zero","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[136,8],"def_end_pos":[136,22]},{"full_name":"Complex.ofReal_natCast","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[424,25],"def_end_pos":[424,39]},{"full_name":"Complex.ofReal_neg","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[196,8],"def_end_pos":[196,18]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Principal.lean","commit":"","full_name":"Ordinal.mul_lt_omega_opow","start":[306,0],"end":[318,18],"file_path":"Mathlib/SetTheory/Ordinal/Principal.lean","tactics":[{"state_before":"a b c : Ordinal.{u_1}\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\n⊢ a * b < ω ^ c","state_after":"case inl\na b : Ordinal.{u_1}\nhb : b < ω\nc0 : 0 < 0\nha : a < ω ^ 0\n⊢ a * b < ω ^ 0\n\ncase inr.inl.intro\na b : Ordinal.{u_1}\nhb : b < ω\nc : Ordinal.{u_1}\nc0 : 0 < succ c\nha : a < ω ^ succ c\n⊢ a * b < ω ^ succ c\n\ncase inr.inr\na b c : Ordinal.{u_1}\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : c.IsLimit\n⊢ a * b < ω ^ c","tactic":"rcases zero_or_succ_or_limit c with (rfl | ⟨c, rfl⟩ | l)","premises":[{"full_name":"Ordinal.zero_or_succ_or_limit","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[267,8],"def_end_pos":[267,29]}]}]}
{"url":"Mathlib/Order/Filter/NAry.lean","commit":"","full_name":"Filter.map₂_left_comm","start":[176,0],"end":[180,49],"file_path":"Mathlib/Order/Filter/NAry.lean","tactics":[{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nm✝ : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nm : α → δ → ε\nn : β → γ → δ\nm' : α → γ → δ'\nn' : β → δ' → ε\nh_left_comm : ∀ (a : α) (b : β) (c : γ), m a (n b c) = n' b (m' a c)\n⊢ map₂ m f (map₂ n g h) = map₂ n' g (map₂ m' f h)","state_after":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nm✝ : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nm : α → δ → ε\nn : β → γ → δ\nm' : α → γ → δ'\nn' : β → δ' → ε\nh_left_comm : ∀ (a : α) (b : β) (c : γ), m a (n b c) = n' b (m' a c)\n⊢ map₂ (fun a b => m b a) (map₂ n g h) f = map₂ n' g (map₂ (fun a b => m' b a) h f)","tactic":"rw [map₂_swap m', map₂_swap m]","premises":[{"full_name":"Filter.map₂_swap","def_path":"Mathlib/Order/Filter/NAry.lean","def_pos":[123,8],"def_end_pos":[123,17]}]},{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nm✝ : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nm : α → δ → ε\nn : β → γ → δ\nm' : α → γ → δ'\nn' : β → δ' → ε\nh_left_comm : ∀ (a : α) (b : β) (c : γ), m a (n b c) = n' b (m' a c)\n⊢ map₂ (fun a b => m b a) (map₂ n g h) f = map₂ n' g (map₂ (fun a b => m' b a) h f)","state_after":"no goals","tactic":"exact map₂_assoc fun _ _ _ => h_left_comm _ _ _","premises":[{"full_name":"Filter.map₂_assoc","def_path":"Mathlib/Order/Filter/NAry.lean","def_pos":[166,8],"def_end_pos":[166,18]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"Specializes.zpow","start":[179,0],"end":[183,52],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Inv G✝\ninst✝⁴ : ContinuousInv G✝\nG : Type u_1\ninst✝³ : DivInvMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ContinuousMul G\ninst✝ : ContinuousInv G\nx y : G\nh : x ⤳ y\nn : ℕ\n⊢ (x ^ Int.ofNat n) ⤳ (y ^ Int.ofNat n)","state_after":"no goals","tactic":"simpa using h.pow n","premises":[{"full_name":"Specializes.pow","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[188,18],"def_end_pos":[188,33]}]},{"state_before":"G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Inv G✝\ninst✝⁴ : ContinuousInv G✝\nG : Type u_1\ninst✝³ : DivInvMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ContinuousMul G\ninst✝ : ContinuousInv G\nx y : G\nh : x ⤳ y\nn : ℕ\n⊢ (x ^ Int.negSucc n) ⤳ (y ^ Int.negSucc n)","state_after":"no goals","tactic":"simpa using (h.pow (n + 1)).inv","premises":[{"full_name":"Specializes.inv","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[172,18],"def_end_pos":[172,33]},{"full_name":"Specializes.pow","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[188,18],"def_end_pos":[188,33]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Lipschitz.lean","commit":"","full_name":"LocallyLipschitz.const_min","start":[309,0],"end":[310,41],"file_path":"Mathlib/Topology/MetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : PseudoEMetricSpace α\nf g : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x => min a (f x)","state_after":"no goals","tactic":"simpa [min_comm] using (hf.min_const a)","premises":[{"full_name":"LocallyLipschitz.min_const","def_path":"Mathlib/Topology/MetricSpace/Lipschitz.lean","def_pos":[306,8],"def_end_pos":[306,17]},{"full_name":"min_comm","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[76,8],"def_end_pos":[76,16]}]}]}
{"url":"Mathlib/Topology/Connected/Basic.lean","commit":"","full_name":"IsPreconnected.sUnion_directed","start":[136,0],"end":[144,22],"file_path":"Mathlib/Topology/Connected/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns t u v : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\n⊢ IsPreconnected (⋃₀ S)","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","tactic":"rintro u v hu hv Huv ⟨a, ⟨s, hsS, has⟩, hau⟩ ⟨b, ⟨t, htS, hbt⟩, hbv⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","tactic":"obtain ⟨r, hrS, hsr, htr⟩ : ∃ r ∈ S, s ⊆ r ∧ t ⊆ r := K s hsS t htS","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\nHnuv : (r ∩ (u ∩ v)).Nonempty\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","tactic":"have Hnuv : (r ∩ (u ∩ v)).Nonempty :=\n    H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv) ⟨a, hsr has, hau⟩ ⟨b, htr hbt, hbv⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]},{"full_name":"Set.subset_sUnion_of_mem","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[846,8],"def_end_pos":[846,28]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\nHnuv : (r ∩ (u ∩ v)).Nonempty\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\nHnuv : (r ∩ (u ∩ v)).Nonempty\nKruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v)\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","tactic":"have Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v) := inter_subset_inter_left _ (subset_sUnion_of_mem hrS)","premises":[{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.inter_subset_inter_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[810,8],"def_end_pos":[810,31]},{"full_name":"Set.sUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[163,4],"def_end_pos":[163,10]},{"full_name":"Set.subset_sUnion_of_mem","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[846,8],"def_end_pos":[846,28]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\nHnuv : (r ∩ (u ∩ v)).Nonempty\nKruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v)\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty","state_after":"no goals","tactic":"exact Hnuv.mono Kruv","premises":[{"full_name":"Set.Nonempty.mono","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[380,8],"def_end_pos":[380,21]}]}]}
{"url":"Mathlib/Order/RelSeries.lean","commit":"","full_name":"RelSeries.reverse_length","start":[365,0],"end":[381,11],"file_path":"Mathlib/Order/RelSeries.lean","tactics":[{"state_before":"α : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\n⊢ r ((p.toFun ∘ Fin.rev) i.succ) ((p.toFun ∘ Fin.rev) i.castSucc)","state_after":"α : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\n⊢ r (p.toFun i.succ.rev) (p.toFun i.castSucc.rev)","tactic":"rw [Function.comp_apply, Function.comp_apply]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]}]},{"state_before":"α : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\n⊢ r (p.toFun i.succ.rev) (p.toFun i.castSucc.rev)","state_after":"α : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\nhi : ↑i + 1 ≤ p.length\n⊢ r (p.toFun i.succ.rev) (p.toFun i.castSucc.rev)","tactic":"have hi : i.1 + 1 ≤ p.length := by omega","premises":[{"full_name":"Fin.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1836,2],"def_end_pos":[1836,5]},{"full_name":"RelSeries.length","def_path":"Mathlib/Order/RelSeries.lean","def_pos":[31,2],"def_end_pos":[31,8]}]},{"state_before":"α : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\nhi : ↑i + 1 ≤ p.length\n⊢ r (p.toFun i.succ.rev) (p.toFun i.castSucc.rev)","state_after":"case h.e'_1.h.e'_4\nα : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\nhi : ↑i + 1 ≤ p.length\n⊢ i.succ.rev = ⟨p.length - (↑i + 1), ⋯⟩.castSucc\n\ncase h.e'_2.h.e'_4\nα : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\nhi : ↑i + 1 ≤ p.length\n⊢ i.castSucc.rev = ⟨p.length - (↑i + 1), ⋯⟩.succ","tactic":"convert p.step ⟨p.length - (i.1 + 1), Nat.sub_lt_self (by omega) hi⟩","premises":[{"full_name":"Fin.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1833,2],"def_end_pos":[1833,4]},{"full_name":"Fin.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1836,2],"def_end_pos":[1836,5]},{"full_name":"Nat.sub_lt_self","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[174,17],"def_end_pos":[174,28]},{"full_name":"RelSeries.length","def_path":"Mathlib/Order/RelSeries.lean","def_pos":[31,2],"def_end_pos":[31,8]},{"full_name":"RelSeries.step","def_path":"Mathlib/Order/RelSeries.lean","def_pos":[35,2],"def_end_pos":[35,6]}]}]}
{"url":"Mathlib/SetTheory/Game/PGame.lean","commit":"","full_name":"SetTheory.PGame.zero_lf","start":[592,0],"end":[595,6],"file_path":"Mathlib/SetTheory/Game/PGame.lean","tactics":[{"state_before":"xl xr : Type u\nx : PGame\n⊢ 0 ⧏ x ↔ ∃ i, ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j","state_after":"xl xr : Type u\nx : PGame\n⊢ ((∃ i,\n        (∀ (i' : LeftMoves 0), moveLeft 0 i' ⧏ x.moveLeft i) ∧\n          ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j) ∨\n      ∃ j,\n        (∀ (i : (moveRight 0 j).LeftMoves), (moveRight 0 j).moveLeft i ⧏ x) ∧\n          ∀ (j' : x.RightMoves), moveRight 0 j ⧏ x.moveRight j') ↔\n    ∃ i, ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j","tactic":"rw [lf_def]","premises":[{"full_name":"SetTheory.PGame.lf_def","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[553,8],"def_end_pos":[553,14]}]},{"state_before":"xl xr : Type u\nx : PGame\n⊢ ((∃ i,\n        (∀ (i' : LeftMoves 0), moveLeft 0 i' ⧏ x.moveLeft i) ∧\n          ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j) ∨\n      ∃ j,\n        (∀ (i : (moveRight 0 j).LeftMoves), (moveRight 0 j).moveLeft i ⧏ x) ∧\n          ∀ (j' : x.RightMoves), moveRight 0 j ⧏ x.moveRight j') ↔\n    ∃ i, ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Order/Lattice.lean","commit":"","full_name":"sup_right_comm","start":[212,0],"end":[213,39],"file_path":"Mathlib/Order/Lattice.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na✝ b✝ c✝ d a b c : α\n⊢ a ⊔ b ⊔ c = a ⊔ c ⊔ b","state_after":"no goals","tactic":"rw [sup_assoc, sup_assoc, sup_comm b]","premises":[{"full_name":"sup_assoc","def_path":"Mathlib/Order/Lattice.lean","def_pos":[197,8],"def_end_pos":[197,17]},{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean","commit":"","full_name":"ProbabilityTheory.Kernel.compProd_fst_condKernelCountable","start":[433,0],"end":[436,91],"file_path":"Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝⁵ : CountablyGenerated γ\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : Countable α\nκ : Kernel α (β × Ω)\ninst✝ : IsFiniteKernel κ\nx y : α\nh : x ∈ measurableAtom y\n⊢ (fun a => (κ a).condKernel) x = (fun a => (κ a).condKernel) y","state_after":"no goals","tactic":"simp [apply_congr_of_mem_measurableAtom _ h]","premises":[{"full_name":"ProbabilityTheory.Kernel.apply_congr_of_mem_measurableAtom","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[211,6],"def_end_pos":[211,39]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Complex.cos_int_mul_two_pi_add_pi","start":[1067,0],"end":[1068,89],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"n : ℤ\n⊢ cos (↑n * (2 * ↑π) + ↑π) = -1","state_after":"no goals","tactic":"simpa only [cos_zero] using (cos_periodic.int_mul n).add_antiperiod_eq cos_antiperiodic","premises":[{"full_name":"Complex.cos_antiperiodic","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[1017,8],"def_end_pos":[1017,24]},{"full_name":"Complex.cos_periodic","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[1019,8],"def_end_pos":[1019,20]},{"full_name":"Complex.cos_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[452,8],"def_end_pos":[452,16]},{"full_name":"Function.Periodic.add_antiperiod_eq","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[505,8],"def_end_pos":[505,34]},{"full_name":"Function.Periodic.int_mul","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[189,18],"def_end_pos":[189,34]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean","commit":"","full_name":"HurwitzKernelBounds.F_nat_zero_zero_sub_le","start":[122,0],"end":[128,25],"file_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean","tactics":[{"state_before":"t : ℝ\nht : 0 < t\n⊢ ‖F_nat 0 0 t - 1‖ ≤ rexp (-π * t) / (1 - rexp (-π * t))","state_after":"case h.e'_3.h.e'_3\nt : ℝ\nht : 0 < t\n⊢ F_nat 0 0 t - 1 = F_nat 0 1 t\n\ncase h.e'_4.h.e'_5\nt : ℝ\nht : 0 < t\n⊢ rexp (-π * t) = rexp (-π * 1 ^ 2 * t)","tactic":"convert F_nat_zero_le zero_le_one ht using 2","premises":[{"full_name":"HurwitzKernelBounds.F_nat_zero_le","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean","def_pos":[113,6],"def_end_pos":[113,19]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]}]}
{"url":"Mathlib/CategoryTheory/Subobject/Limits.lean","commit":"","full_name":"CategoryTheory.Limits.kernelSubobject_factors","start":[101,0],"end":[103,30],"file_path":"Mathlib/CategoryTheory/Subobject/Limits.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nX Y Z : C\ninst✝¹ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝ : HasKernel f\nW : C\nh : W ⟶ X\nw : h ≫ f = 0\n⊢ kernel.lift f h w ≫ (MonoOver.mk' (kernel.ι f)).arrow = h","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/ENNReal/Operations.lean","commit":"","full_name":"ENNReal.lt_add_right","start":[148,0],"end":[149,75],"file_path":"Mathlib/Data/ENNReal/Operations.lean","tactics":[{"state_before":"a b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ ⊤\nhb : b ≠ 0\n⊢ a < a + b","state_after":"no goals","tactic":"rwa [← pos_iff_ne_zero, ← ENNReal.add_lt_add_iff_left ha, add_zero] at hb","premises":[{"full_name":"ENNReal.add_lt_add_iff_left","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[133,18],"def_end_pos":[133,37]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"pos_iff_ne_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[123,2],"def_end_pos":[123,13]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.Sized.size_eq_zero","start":[128,0],"end":[130,32],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\nt : Ordnode α\nht : t.Sized\n⊢ t.size = 0 ↔ t = nil","state_after":"no goals","tactic":"cases t <;> [simp;simp [ht.1]]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]}]}]}
{"url":"Mathlib/Analysis/Convex/Join.lean","commit":"","full_name":"convexJoin_empty_left","start":[48,0],"end":[49,88],"file_path":"Mathlib/Analysis/Convex/Join.lean","tactics":[{"state_before":"ι : Sort u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns t✝ s₁ s₂ t₁ t₂ u : Set E\nx y : E\nt : Set E\n⊢ convexJoin 𝕜 ∅ t = ∅","state_after":"no goals","tactic":"simp [convexJoin]","premises":[{"full_name":"convexJoin","def_path":"Mathlib/Analysis/Convex/Join.lean","def_pos":[28,4],"def_end_pos":[28,14]}]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","commit":"","full_name":"contDiffOn_top_iff_fderivWithin","start":[1097,0],"end":[1111,46],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : UniqueDiffOn 𝕜 s\n⊢ ContDiffOn 𝕜 ⊤ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ⊤ (fun y => fderivWithin 𝕜 f s y) s","state_after":"case mp\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : UniqueDiffOn 𝕜 s\n⊢ ContDiffOn 𝕜 ⊤ f s → DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ⊤ (fun y => fderivWithin 𝕜 f s y) s\n\ncase mpr\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : UniqueDiffOn 𝕜 s\n⊢ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ⊤ (fun y => fderivWithin 𝕜 f s y) s → ContDiffOn 𝕜 ⊤ f s","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","commit":"","full_name":"UniformOnFun.inf_eq","start":[830,0],"end":[836,17],"file_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\n⊢ uniformSpace α γ 𝔖 = uniformSpace α γ 𝔖 ⊓ uniformSpace α γ 𝔖","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\n⊢ ⨅ i, uniformSpace α γ 𝔖 = ⨅ b, bif b then uniformSpace α γ 𝔖 else uniformSpace α γ 𝔖","tactic":"rw [inf_eq_iInf, inf_eq_iInf, UniformOnFun.iInf_eq]","premises":[{"full_name":"UniformOnFun.iInf_eq","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[824,18],"def_end_pos":[824,25]},{"full_name":"inf_eq_iInf","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1258,8],"def_end_pos":[1258,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\n⊢ ⨅ i, uniformSpace α γ 𝔖 = ⨅ b, bif b then uniformSpace α γ 𝔖 else uniformSpace α γ 𝔖","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\ni : Bool\n⊢ uniformSpace α γ 𝔖 = bif i then uniformSpace α γ 𝔖 else uniformSpace α γ 𝔖","tactic":"refine iInf_congr fun i => ?_","premises":[{"full_name":"iInf_congr","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[567,8],"def_end_pos":[567,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\ni : Bool\n⊢ uniformSpace α γ 𝔖 = bif i then uniformSpace α γ 𝔖 else uniformSpace α γ 𝔖","state_after":"no goals","tactic":"cases i <;> rfl","premises":[]}]}
{"url":"Mathlib/Topology/MetricSpace/Polish.lean","commit":"","full_name":"IsOpen.polishSpace","start":[326,0],"end":[332,52],"file_path":"Mathlib/Topology/MetricSpace/Polish.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen s\n⊢ PolishSpace ↑s","state_after":"α✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen s\nthis : UpgradedPolishSpace α := upgradePolishSpace α\n⊢ PolishSpace ↑s","tactic":"letI := upgradePolishSpace α","premises":[{"full_name":"upgradePolishSpace","def_path":"Mathlib/Topology/MetricSpace/Polish.lean","def_pos":[92,4],"def_end_pos":[92,22]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen s\nthis : UpgradedPolishSpace α := upgradePolishSpace α\n⊢ PolishSpace ↑s","state_after":"case intro\nα✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nthis : UpgradedPolishSpace α := upgradePolishSpace α\ns : Opens α\n⊢ PolishSpace ↑↑s","tactic":"lift s to Opens α using hs","premises":[{"full_name":"TopologicalSpace.Opens","def_path":"Mathlib/Topology/Sets/Opens.lean","def_pos":[61,10],"def_end_pos":[61,15]}]},{"state_before":"case intro\nα✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nthis : UpgradedPolishSpace α := upgradePolishSpace α\ns : Opens α\n⊢ PolishSpace ↑↑s","state_after":"case intro\nα✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nthis✝ : UpgradedPolishSpace α := upgradePolishSpace α\ns : Opens α\nthis : SecondCountableTopology s.CompleteCopy\n⊢ PolishSpace ↑↑s","tactic":"have : SecondCountableTopology s.CompleteCopy := inferInstanceAs (SecondCountableTopology s)","premises":[{"full_name":"SecondCountableTopology","def_path":"Mathlib/Topology/Bases.lean","def_pos":[704,6],"def_end_pos":[704,36]},{"full_name":"TopologicalSpace.Opens.CompleteCopy","def_path":"Mathlib/Topology/MetricSpace/Polish.lean","def_pos":[239,4],"def_end_pos":[239,16]},{"full_name":"inferInstanceAs","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[113,7],"def_end_pos":[113,22]}]},{"state_before":"case intro\nα✝ : Type u_1\nβ : Type u_2\ninst✝² : MetricSpace α✝\ns✝ : Opens α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nthis✝ : UpgradedPolishSpace α := upgradePolishSpace α\ns : Opens α\nthis : SecondCountableTopology s.CompleteCopy\n⊢ PolishSpace ↑↑s","state_after":"no goals","tactic":"exact inferInstanceAs (PolishSpace s.CompleteCopy)","premises":[{"full_name":"PolishSpace","def_path":"Mathlib/Topology/MetricSpace/Polish.lean","def_pos":[64,6],"def_end_pos":[64,17]},{"full_name":"TopologicalSpace.Opens.CompleteCopy","def_path":"Mathlib/Topology/MetricSpace/Polish.lean","def_pos":[239,4],"def_end_pos":[239,16]},{"full_name":"inferInstanceAs","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[113,7],"def_end_pos":[113,22]}]}]}
{"url":"Mathlib/MeasureTheory/Group/Arithmetic.lean","commit":"","full_name":"nullMeasurableSet_eq_fun","start":[322,0],"end":[328,22],"file_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","tactics":[{"state_before":"α✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ NullMeasurableSet {x | f x = g x} μ","state_after":"α✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ {x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =ᶠ[ae μ] {x | f x = g x}","tactic":"apply (measurableSet_eq_fun hf.measurable_mk hg.measurable_mk).nullMeasurableSet.congr","premises":[{"full_name":"AEMeasurable.measurable_mk","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[392,8],"def_end_pos":[392,21]},{"full_name":"MeasurableSet.nullMeasurableSet","def_path":"Mathlib/MeasureTheory/Measure/NullMeasurable.lean","def_pos":[92,8],"def_end_pos":[92,46]},{"full_name":"MeasureTheory.NullMeasurableSet.congr","def_path":"Mathlib/MeasureTheory/Measure/NullMeasurable.lean","def_pos":[122,18],"def_end_pos":[122,23]},{"full_name":"measurableSet_eq_fun","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[301,8],"def_end_pos":[301,28]}]},{"state_before":"α✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ {x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =ᶠ[ae μ] {x | f x = g x}","state_after":"case h\nα✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nx : α\nhfx : f x = AEMeasurable.mk f hf x\nhgx : g x = AEMeasurable.mk g hg x\n⊢ {x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} x = {x | f x = g x} x","tactic":"filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx","premises":[{"full_name":"AEMeasurable.ae_eq_mk","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[395,8],"def_end_pos":[395,16]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nx : α\nhfx : f x = AEMeasurable.mk f hf x\nhgx : g x = AEMeasurable.mk g hg x\n⊢ {x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} x = {x | f x = g x} x","state_after":"case h\nα✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nx : α\nhfx : f x = AEMeasurable.mk f hf x\nhgx : g x = AEMeasurable.mk g hg x\n⊢ (AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) = (f x = g x)","tactic":"change (hf.mk f x = hg.mk g x) = (f x = g x)","premises":[{"full_name":"AEMeasurable.mk","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[388,4],"def_end_pos":[388,6]}]},{"state_before":"case h\nα✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nx : α\nhfx : f x = AEMeasurable.mk f hf x\nhgx : g x = AEMeasurable.mk g hg x\n⊢ (AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) = (f x = g x)","state_after":"no goals","tactic":"simp only [hfx, hgx]","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/QuotientPi.lean","commit":"","full_name":"Submodule.quotientPi_aux.right_inv","start":[92,0],"end":[101,93],"file_path":"Mathlib/LinearAlgebra/QuotientPi.lean","tactics":[{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ Function.RightInverse (invFun p) (toFun p)","state_after":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ Function.RightInverse ⇑(piQuotientLift p (pi Set.univ p) single ⋯) ⇑(quotientPiLift p (fun i => (p i).mkQ) ⋯)","tactic":"dsimp only [toFun, invFun]","premises":[{"full_name":"Submodule.quotientPi_aux.invFun","def_path":"Mathlib/LinearAlgebra/QuotientPi.lean","def_pos":[82,4],"def_end_pos":[82,10]},{"full_name":"Submodule.quotientPi_aux.toFun","def_path":"Mathlib/LinearAlgebra/QuotientPi.lean","def_pos":[78,4],"def_end_pos":[78,9]}]},{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ Function.RightInverse ⇑(piQuotientLift p (pi Set.univ p) single ⋯) ⇑(quotientPiLift p (fun i => (p i).mkQ) ⋯)","state_after":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ ⇑(quotientPiLift p (fun i => (p i).mkQ) ⋯ ∘ₗ piQuotientLift p (pi Set.univ p) single ⋯) = ⇑LinearMap.id","tactic":"rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]","premises":[{"full_name":"Function.rightInverse_iff_comp","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[287,8],"def_end_pos":[287,29]},{"full_name":"LinearMap.coe_comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[508,8],"def_end_pos":[508,16]},{"full_name":"LinearMap.id_coe","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[282,8],"def_end_pos":[282,14]}]},{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ ⇑(quotientPiLift p (fun i => (p i).mkQ) ⋯ ∘ₗ piQuotientLift p (pi Set.univ p) single ⋯) = ⇑LinearMap.id","state_after":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (quotientPiLift p (fun i => (p i).mkQ) ⋯ ∘ₗ piQuotientLift p (pi Set.univ p) single ⋯)\n      (Pi.single i (Quotient.mk'' x')) j =\n    LinearMap.id (Pi.single i (Quotient.mk'' x')) j","tactic":"refine congr_arg _ (pi_ext fun i x => Quotient.inductionOn' x fun x' => funext fun j => ?_)","premises":[{"full_name":"LinearMap.pi_ext","def_path":"Mathlib/LinearAlgebra/Pi.lean","def_pos":[162,8],"def_end_pos":[162,14]},{"full_name":"Quotient.inductionOn'","def_path":"Mathlib/Data/Quot.lean","def_pos":[597,18],"def_end_pos":[597,30]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (quotientPiLift p (fun i => (p i).mkQ) ⋯ ∘ₗ piQuotientLift p (pi Set.univ p) single ⋯)\n      (Pi.single i (Quotient.mk'' x')) j =\n    LinearMap.id (Pi.single i (Quotient.mk'' x')) j","state_after":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (fun i_1 => (p i_1).mkQ ((single i) x' i_1)) j = Pi.single i (Quotient.mk x') j","tactic":"rw [comp_apply, piQuotientLift_single, Quotient.mk''_eq_mk, mapQ_apply,\n    quotientPiLift_mk, id_apply]","premises":[{"full_name":"LinearMap.comp_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[504,8],"def_end_pos":[504,18]},{"full_name":"LinearMap.id_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[278,8],"def_end_pos":[278,16]},{"full_name":"Submodule.Quotient.mk''_eq_mk","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[62,8],"def_end_pos":[62,18]},{"full_name":"Submodule.mapQ_apply","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[356,8],"def_end_pos":[356,18]},{"full_name":"Submodule.piQuotientLift_single","def_path":"Mathlib/LinearAlgebra/QuotientPi.lean","def_pos":[46,8],"def_end_pos":[46,29]},{"full_name":"Submodule.quotientPiLift_mk","def_path":"Mathlib/LinearAlgebra/QuotientPi.lean","def_pos":[67,8],"def_end_pos":[67,25]}]},{"state_before":"ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (fun i_1 => (p i_1).mkQ ((single i) x' i_1)) j = Pi.single i (Quotient.mk x') j","state_after":"case pos\nι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\nhij : i = j\n⊢ Quotient.mk (Pi.single i x' j) = Pi.single i (Quotient.mk x') j\n\ncase neg\nι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\nhij : ¬i = j\n⊢ Quotient.mk (Pi.single i x' j) = Pi.single i (Quotient.mk x') j","tactic":"by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]","premises":[{"full_name":"LinearMap.coe_single","def_path":"Mathlib/LinearAlgebra/Pi.lean","def_pos":[123,8],"def_end_pos":[123,18]},{"full_name":"Submodule.mkQ_apply","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[284,8],"def_end_pos":[284,17]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/RingTheory/WittVector/Compare.lean","commit":"","full_name":"WittVector.toZModPow_compat","start":[137,0],"end":[145,64],"file_path":"Mathlib/RingTheory/WittVector/Compare.lean","tactics":[{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\n⊢ (ZMod.castHom ⋯ (ZMod (p ^ m))).comp ((zmodEquivTrunc p n).symm.toRingHom.comp (truncate n)) =\n    ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n)","state_after":"no goals","tactic":"rw [commutes_symm, RingHom.comp_assoc]","premises":[{"full_name":"RingHom.comp_assoc","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[567,8],"def_end_pos":[567,18]},{"full_name":"TruncatedWittVector.commutes_symm","def_path":"Mathlib/RingTheory/WittVector/Compare.lean","def_pos":[117,8],"def_end_pos":[117,21]}]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\n⊢ ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n) =\n    (zmodEquivTrunc p m).symm.toRingHom.comp (truncate m)","state_after":"no goals","tactic":"rw [RingHom.comp_assoc, truncate_comp_wittVector_truncate]","premises":[{"full_name":"RingHom.comp_assoc","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[567,8],"def_end_pos":[567,18]},{"full_name":"TruncatedWittVector.truncate_comp_wittVector_truncate","def_path":"Mathlib/RingTheory/WittVector/Truncated.lean","def_pos":[343,8],"def_end_pos":[343,41]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/Basic.lean","commit":"","full_name":"LSeries_zero","start":[225,0],"end":[228,81],"file_path":"Mathlib/NumberTheory/LSeries/Basic.lean","tactics":[{"state_before":"⊢ LSeries 0 = 0","state_after":"case h\nx✝ : ℂ\n⊢ LSeries 0 x✝ = 0 x✝","tactic":"ext","premises":[]},{"state_before":"case h\nx✝ : ℂ\n⊢ LSeries 0 x✝ = 0 x✝","state_after":"no goals","tactic":"simp only [LSeries, LSeries.term, Pi.zero_apply, zero_div, ite_self, tsum_zero]","premises":[{"full_name":"LSeries","def_path":"Mathlib/NumberTheory/LSeries/Basic.lean","def_pos":[120,4],"def_end_pos":[120,11]},{"full_name":"LSeries.term","def_path":"Mathlib/NumberTheory/LSeries/Basic.lean","def_pos":[64,4],"def_end_pos":[64,8]},{"full_name":"Pi.zero_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[59,2],"def_end_pos":[59,13]},{"full_name":"ite_self","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[102,16],"def_end_pos":[102,24]},{"full_name":"tsum_zero","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[373,2],"def_end_pos":[373,13]},{"full_name":"zero_div","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[298,8],"def_end_pos":[298,16]}]}]}
{"url":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean","commit":"","full_name":"CategoryTheory.Localization.Preadditive.comp_add'","start":[196,0],"end":[210,15],"file_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","state_after":"case intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","tactic":"obtain ⟨α, hα⟩ := exists_leftFraction L W f","premises":[{"full_name":"CategoryTheory.Localization.exists_leftFraction","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[725,6],"def_end_pos":[725,38]}]},{"state_before":"case intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","state_after":"case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","tactic":"obtain ⟨β, hβ₁, hβ₂⟩ := exists_leftFraction₂ L W g₁ g₂","premises":[{"full_name":"CategoryTheory.Localization.exists_leftFraction₂","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[261,6],"def_end_pos":[261,26]}]},{"state_before":"case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","state_after":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : (RightFraction₂.mk α.s ⋯ β.f β.f').f ≫ γ.s = (RightFraction₂.mk α.s ⋯ β.f β.f').s ≫ γ.f\nhγ₂ : (RightFraction₂.mk α.s ⋯ β.f β.f').f' ≫ γ.s = (RightFraction₂.mk α.s ⋯ β.f β.f').s ≫ γ.f'\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","tactic":"obtain ⟨γ, hγ₁, hγ₂⟩ := (RightFraction₂.mk _ α.hs β.f β.f').exists_leftFraction₂","premises":[{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.hs","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[47,2],"def_end_pos":[47,4]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.f","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[46,2],"def_end_pos":[46,3]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.f'","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[48,2],"def_end_pos":[48,4]},{"full_name":"CategoryTheory.MorphismProperty.RightFraction₂.exists_leftFraction₂","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[238,6],"def_end_pos":[238,26]}]},{"state_before":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : (RightFraction₂.mk α.s ⋯ β.f β.f').f ≫ γ.s = (RightFraction₂.mk α.s ⋯ β.f β.f').s ≫ γ.f\nhγ₂ : (RightFraction₂.mk α.s ⋯ β.f β.f').f' ≫ γ.s = (RightFraction₂.mk α.s ⋯ β.f β.f').s ≫ γ.f'\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","state_after":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","tactic":"dsimp at hγ₁ hγ₂","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)","state_after":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ (α.comp₀ β.add γ.add).map L ⋯ = (LeftFraction₂.mk (α.f ≫ γ.f) (α.f ≫ γ.f') (β.s ≫ γ.s) ⋯).add.map L ⋯","tactic":"rw [add'_eq W g₁ g₂ β hβ₁ hβ₂, add'_eq W (f ≫ g₁) (f ≫ g₂)\n    (LeftFraction₂.mk (α.f ≫ γ.f) (α.f ≫ γ.f') (β.s ≫ γ.s) (W.comp_mem _ _ β.hs γ.hs))\n    (by simpa only [hα, hβ₁] using LeftFraction.map_comp_map_eq_map α β.fst γ.fst hγ₁ L)\n    (by simpa only [hα, hβ₂] using LeftFraction.map_comp_map_eq_map α β.snd γ.snd hγ₂ L),\n    hα, LeftFraction.map_comp_map_eq_map α β.add γ.add\n      (by simp only [add_comp, hγ₁, hγ₂, comp_add])]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Localization.Preadditive.add'_eq","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean","def_pos":[123,6],"def_end_pos":[123,13]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.f","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[43,2],"def_end_pos":[43,3]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.map_comp_map_eq_map","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[696,6],"def_end_pos":[696,25]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.add","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean","def_pos":[56,7],"def_end_pos":[56,10]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.f","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[46,2],"def_end_pos":[46,3]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.f'","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[48,2],"def_end_pos":[48,4]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.fst","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[98,7],"def_end_pos":[98,10]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.hs","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[52,2],"def_end_pos":[52,4]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.s","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.snd","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean","def_pos":[105,7],"def_end_pos":[105,10]},{"full_name":"CategoryTheory.MorphismProperty.comp_mem","def_path":"Mathlib/CategoryTheory/MorphismProperty/Composition.lean","def_pos":[69,6],"def_end_pos":[69,14]},{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ (α.comp₀ β.add γ.add).map L ⋯ = (LeftFraction₂.mk (α.f ≫ γ.f) (α.f ≫ γ.f') (β.s ≫ γ.s) ⋯).add.map L ⋯","state_after":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ (LeftFraction.mk (α.f ≫ (γ.f + γ.f')) (β.s ≫ γ.s) ⋯).map L ⋯ =\n    (LeftFraction.mk (α.f ≫ γ.f + α.f ≫ γ.f') (β.s ≫ γ.s) ⋯).map L ⋯","tactic":"dsimp [LeftFraction₂.add]","premises":[{"full_name":"CategoryTheory.MorphismProperty.LeftFraction₂.add","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean","def_pos":[56,7],"def_end_pos":[56,10]}]},{"state_before":"case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ (LeftFraction.mk (α.f ≫ (γ.f + γ.f')) (β.s ≫ γ.s) ⋯).map L ⋯ =\n    (LeftFraction.mk (α.f ≫ γ.f + α.f ≫ γ.f') (β.s ≫ γ.s) ⋯).map L ⋯","state_after":"no goals","tactic":"rw [comp_add]","premises":[{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]}]}]}
{"url":"Mathlib/RingTheory/Algebraic.lean","commit":"","full_name":"isAlgebraic_rat","start":[126,0],"end":[129,43],"file_path":"Mathlib/RingTheory/Algebraic.lean","tactics":[{"state_before":"R✝ : Type u\nS : Type u_1\nA✝ : Type v\ninst✝⁹ : CommRing R✝\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A✝\ninst✝⁶ : Algebra R✝ A✝\ninst✝⁵ : Algebra R✝ S\ninst✝⁴ : Algebra S A✝\ninst✝³ : IsScalarTower R✝ S A✝\nR : Type u\nA : Type v\ninst✝² : DivisionRing A\ninst✝¹ : Field R\ninst✝ : Algebra R A\nn : ℚ\n⊢ IsAlgebraic R ↑n","state_after":"R✝ : Type u\nS : Type u_1\nA✝ : Type v\ninst✝⁹ : CommRing R✝\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A✝\ninst✝⁶ : Algebra R✝ A✝\ninst✝⁵ : Algebra R✝ S\ninst✝⁴ : Algebra S A✝\ninst✝³ : IsScalarTower R✝ S A✝\nR : Type u\nA : Type v\ninst✝² : DivisionRing A\ninst✝¹ : Field R\ninst✝ : Algebra R A\nn : ℚ\n⊢ IsAlgebraic R ((algebraMap R A) ↑n)","tactic":"rw [← map_ratCast (algebraMap R A)]","premises":[{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"map_ratCast","def_path":"Mathlib/Data/Rat/Cast/Defs.lean","def_pos":[219,8],"def_end_pos":[219,19]}]},{"state_before":"R✝ : Type u\nS : Type u_1\nA✝ : Type v\ninst✝⁹ : CommRing R✝\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A✝\ninst✝⁶ : Algebra R✝ A✝\ninst✝⁵ : Algebra R✝ S\ninst✝⁴ : Algebra S A✝\ninst✝³ : IsScalarTower R✝ S A✝\nR : Type u\nA : Type v\ninst✝² : DivisionRing A\ninst✝¹ : Field R\ninst✝ : Algebra R A\nn : ℚ\n⊢ IsAlgebraic R ((algebraMap R A) ↑n)","state_after":"no goals","tactic":"exact isAlgebraic_algebraMap (Rat.cast n)","premises":[{"full_name":"Rat.cast","def_path":".lake/packages/batteries/Batteries/Classes/RatCast.lean","def_pos":[17,47],"def_end_pos":[17,55]},{"full_name":"isAlgebraic_algebraMap","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[111,8],"def_end_pos":[111,30]}]}]}
{"url":"Mathlib/Analysis/Distribution/SchwartzSpace.lean","commit":"","full_name":"ContinuousLinearMap.hasTemperateGrowth","start":[569,0],"end":[574,51],"file_path":"Mathlib/Analysis/Distribution/SchwartzSpace.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕜' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nV : Type u_7\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\n⊢ Function.HasTemperateGrowth ⇑f","state_after":"𝕜 : Type u_1\n𝕜' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nV : Type u_7\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\n⊢ Function.HasTemperateGrowth (fderiv ℝ ⇑f)\n\n𝕜 : Type u_1\n𝕜' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nV : Type u_7\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nx : E\n⊢ ‖f x‖ ≤ ‖f‖ * (1 + ‖x‖) ^ 1","tactic":"apply Function.HasTemperateGrowth.of_fderiv ?_ f.differentiable (k := 1) (C := ‖f‖) (fun x ↦ ?_)","premises":[{"full_name":"ContinuousLinearMap.differentiable","def_path":"Mathlib/Analysis/Calculus/FDeriv/Linear.lean","def_pos":[84,18],"def_end_pos":[84,52]},{"full_name":"Function.HasTemperateGrowth.of_fderiv","def_path":"Mathlib/Analysis/Distribution/SchwartzSpace.lean","def_pos":[548,6],"def_end_pos":[548,50]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]}]}]}
{"url":"Mathlib/Topology/Constructions.lean","commit":"","full_name":"Subtype.dense_iff","start":[994,0],"end":[996,5],"file_path":"Mathlib/Topology/Constructions.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\ns : Set X\nt : Set ↑s\n⊢ Dense t ↔ s ⊆ closure (val '' t)","state_after":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\ns : Set X\nt : Set ↑s\n⊢ (∀ (x : X) (h : x ∈ s), ↑⟨x, h⟩ ∈ closure (val '' t)) ↔ s ⊆ closure (val '' t)","tactic":"rw [inducing_subtype_val.dense_iff, SetCoe.forall]","premises":[{"full_name":"Inducing.dense_iff","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[150,8],"def_end_pos":[150,17]},{"full_name":"SetCoe.forall","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[146,8],"def_end_pos":[146,21]},{"full_name":"inducing_subtype_val","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[938,8],"def_end_pos":[938,28]}]},{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\ns : Set X\nt : Set ↑s\n⊢ (∀ (x : X) (h : x ∈ s), ↑⟨x, h⟩ ∈ closure (val '' t)) ↔ s ⊆ closure (val '' t)","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean","commit":"","full_name":"Finsupp.toFreeAbelianGroup_toFinsupp","start":[64,0],"end":[67,98],"file_path":"Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean","tactics":[{"state_before":"X✝ : Type u_1\nX : Type u_2\nx : FreeAbelianGroup X\n⊢ toFreeAbelianGroup (toFinsupp x) = x","state_after":"no goals","tactic":"rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]","premises":[{"full_name":"AddMonoidHom.comp_apply","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[717,2],"def_end_pos":[717,13]},{"full_name":"AddMonoidHom.id_apply","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[671,23],"def_end_pos":[671,28]},{"full_name":"Finsupp.toFreeAbelianGroup_comp_toFinsupp","def_path":"Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean","def_pos":[57,8],"def_end_pos":[57,49]}]}]}
{"url":"Mathlib/Topology/Algebra/Monoid.lean","commit":"","full_name":"continuous_nsmul","start":[494,0],"end":[499,46],"file_path":"Mathlib/Topology/Algebra/Monoid.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\n⊢ Continuous fun a => a ^ 0","state_after":"no goals","tactic":"simpa using continuous_const","premises":[{"full_name":"continuous_const","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1436,8],"def_end_pos":[1436,24]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nk : ℕ\n⊢ Continuous fun a => a ^ (k + 1)","state_after":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nk : ℕ\n⊢ Continuous fun a => a * a ^ k","tactic":"simp only [pow_succ']","premises":[{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nk : ℕ\n⊢ Continuous fun a => a * a ^ k","state_after":"no goals","tactic":"exact continuous_id.mul (continuous_pow _)","premises":[{"full_name":"Continuous.mul","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[94,8],"def_end_pos":[94,22]},{"full_name":"continuous_id","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1382,8],"def_end_pos":[1382,21]}]}]}
{"url":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","commit":"","full_name":"hasFDerivAt_stereoInvFunAux_comp_coe","start":[153,0],"end":[157,77],"file_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","tactics":[{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv✝ v : E\n⊢ HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0","state_after":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv✝ v : E\nthis : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((Submodule.span ℝ {v})ᗮ.subtypeL 0)\n⊢ HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0","tactic":"have : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((ℝ ∙ v)ᗮ.subtypeL 0) :=\n    hasFDerivAt_stereoInvFunAux v","premises":[{"full_name":"ContinuousLinearMap.id","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[590,4],"def_end_pos":[590,6]},{"full_name":"HasFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[149,4],"def_end_pos":[149,15]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Submodule.orthogonal","def_path":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","def_pos":[37,4],"def_end_pos":[37,14]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]},{"full_name":"Submodule.subtypeL","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[929,4],"def_end_pos":[929,29]},{"full_name":"hasFDerivAt_stereoInvFunAux","def_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","def_pos":[136,8],"def_end_pos":[136,35]},{"full_name":"stereoInvFunAux","def_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","def_pos":[113,4],"def_end_pos":[113,19]}]},{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv✝ v : E\nthis : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((Submodule.span ℝ {v})ᗮ.subtypeL 0)\n⊢ HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0","state_after":"no goals","tactic":"convert this.comp (0 : (ℝ ∙ v)ᗮ) (by apply ContinuousLinearMap.hasFDerivAt)","premises":[{"full_name":"ContinuousLinearMap.hasFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Linear.lean","def_pos":[63,18],"def_end_pos":[63,49]},{"full_name":"HasFDerivAt.comp","def_path":"Mathlib/Analysis/Calculus/FDeriv/Comp.lean","def_pos":[94,8],"def_end_pos":[94,24]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Submodule.orthogonal","def_path":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","def_pos":[37,4],"def_end_pos":[37,14]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]}]}]}
{"url":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","commit":"","full_name":"CategoryTheory.LocalizerMorphism.homMap_apply","start":[69,0],"end":[84,13],"file_path":"Mathlib/CategoryTheory/Localization/HomEquiv.lean","tactics":[{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\n⊢ Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\n⊢ Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y","tactic":"let G' := Φ.localizedFunctor L₁ L₂","premises":[{"full_name":"CategoryTheory.LocalizerMorphism.localizedFunctor","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[81,18],"def_end_pos":[81,34]}]},{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\n⊢ Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\n⊢ Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y","tactic":"let e' := CatCommSq.iso Φ.functor L₁ L₂ G'","premises":[{"full_name":"CategoryTheory.CatCommSq.iso","def_path":"Mathlib/CategoryTheory/CatCommSq.lean","def_pos":[42,4],"def_end_pos":[42,7]},{"full_name":"CategoryTheory.LocalizerMorphism.functor","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[43,2],"def_end_pos":[43,9]}]},{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\n⊢ Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","tactic":"change e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = _","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","tactic":"letI : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := ⟨e.symm⟩","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.Localization.Lifting","def_path":"Mathlib/CategoryTheory/Localization/Predicate.lean","def_pos":[280,6],"def_end_pos":[280,13]},{"full_name":"CategoryTheory.LocalizerMorphism.functor","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[43,2],"def_end_pos":[43,9]}]},{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\nα : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","tactic":"let α : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) _ _ e'.symm","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Iso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[49,10],"def_end_pos":[49,13]},{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.Localization.liftNatIso","def_path":"Mathlib/CategoryTheory/Localization/Predicate.lean","def_pos":[348,4],"def_end_pos":[348,14]},{"full_name":"CategoryTheory.LocalizerMorphism.functor","def_path":"Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean","def_pos":[43,2],"def_end_pos":[43,9]}]},{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\nα : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis✝ : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\nα : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm\nthis : e = e' ≪≫ isoWhiskerLeft L₁ α\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","tactic":"have : e = e' ≪≫ isoWhiskerLeft _ α := by\n    ext X\n    dsimp [α]\n    rw [Localization.liftNatTrans_app]\n    erw [id_comp]\n    rw [Iso.hom_inv_id_app_assoc]\n    rfl","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Iso.trans","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[129,4],"def_end_pos":[129,9]},{"full_name":"CategoryTheory.Localization.liftNatTrans_app","def_path":"Mathlib/CategoryTheory/Localization/Predicate.lean","def_pos":[323,8],"def_end_pos":[323,24]},{"full_name":"CategoryTheory.isoWhiskerLeft","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[151,4],"def_end_pos":[151,18]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis✝ : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\nα : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm\nthis : e = e' ≪≫ isoWhiskerLeft L₁ α\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y","state_after":"no goals","tactic":"simp [this]","premises":[]}]}
{"url":"Mathlib/GroupTheory/Complement.lean","commit":"","full_name":"Subgroup.MemRightTransversals.toFun_mul_inv_mem","start":[540,0],"end":[543,92],"file_path":"Mathlib/GroupTheory/Complement.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhS : S ∈ rightTransversals ↑H\ng : G\n⊢ (g * (↑(toFun hS g))⁻¹)⁻¹ = ↑(toFun hS g) * g⁻¹","state_after":"no goals","tactic":"rw [mul_inv_rev, inv_inv]","premises":[{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]},{"full_name":"mul_inv_rev","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[982,8],"def_end_pos":[982,19]}]}]}
{"url":"Mathlib/Data/Real/Cardinality.lean","commit":"","full_name":"Cardinal.not_countable_real","start":[202,0],"end":[205,14],"file_path":"Mathlib/Data/Real/Cardinality.lean","tactics":[{"state_before":"c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ¬Set.univ.Countable","state_after":"c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ℵ₀ < 𝔠","tactic":"rw [← le_aleph0_iff_set_countable, not_le, mk_univ_real]","premises":[{"full_name":"Cardinal.le_aleph0_iff_set_countable","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1401,8],"def_end_pos":[1401,35]},{"full_name":"Cardinal.mk_univ_real","def_path":"Mathlib/Data/Real/Cardinality.lean","def_pos":[200,8],"def_end_pos":[200,20]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]},{"state_before":"c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ℵ₀ < 𝔠","state_after":"no goals","tactic":"apply cantor","premises":[{"full_name":"Cardinal.cantor","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[618,8],"def_end_pos":[618,14]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"_private.Mathlib.SetTheory.Ordinal.Arithmetic.0.Ordinal.add_lt_add_iff_left'","start":[118,0],"end":[119,46],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal.{u_4}\n⊢ a + b < a + c ↔ b < c","state_after":"no goals","tactic":"rw [← not_le, ← not_le, add_le_add_iff_left]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"add_le_add_iff_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[86,2],"def_end_pos":[86,13]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]}]}
{"url":"Mathlib/Algebra/Homology/ImageToKernel.lean","commit":"","full_name":"imageToKernel_zero_left","start":[81,0],"end":[85,6],"file_path":"Mathlib/Algebra/Homology/ImageToKernel.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝¹ : HasKernels V\ninst✝ : HasZeroObject V\nw : 0 ≫ g = 0\n⊢ imageToKernel 0 g w = 0","state_after":"case h\nι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝¹ : HasKernels V\ninst✝ : HasZeroObject V\nw : 0 ≫ g = 0\n⊢ imageToKernel 0 g w ≫ (kernelSubobject g).arrow = 0 ≫ (kernelSubobject g).arrow","tactic":"ext","premises":[]},{"state_before":"case h\nι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝¹ : HasKernels V\ninst✝ : HasZeroObject V\nw : 0 ≫ g = 0\n⊢ imageToKernel 0 g w ≫ (kernelSubobject g).arrow = 0 ≫ (kernelSubobject g).arrow","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Real.two_pi_pos","start":[160,0],"end":[160,54],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"⊢ 0 < 2 * π","state_after":"no goals","tactic":"linarith [pi_pos]","premises":[{"full_name":"Real.pi_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[148,8],"def_end_pos":[148,14]}]}]}
{"url":"Mathlib/Probability/CondCount.lean","commit":"","full_name":"ProbabilityTheory.condCount_add_compl_eq","start":[172,0],"end":[182,32],"file_path":"Mathlib/Probability/CondCount.lean","tactics":[{"state_before":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ = (condCount s) t","state_after":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ = (condCount s) t","tactic":"have : condCount s t = (condCount (s ∩ u) t * condCount (s ∩ u ∪ s ∩ uᶜ) (s ∩ u) +\n      condCount (s ∩ uᶜ) t * condCount (s ∩ u ∪ s ∩ uᶜ) (s ∩ uᶜ)) := by\n    rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]","premises":[{"full_name":"Disjoint.mono","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[63,8],"def_end_pos":[63,21]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"ProbabilityTheory.condCount","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[52,4],"def_end_pos":[52,13]},{"full_name":"ProbabilityTheory.condCount_disjoint_union","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[151,8],"def_end_pos":[151,32]},{"full_name":"Set.Finite.inter_of_left","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[651,8],"def_end_pos":[651,28]},{"full_name":"Set.inter_union_compl","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1477,8],"def_end_pos":[1477,25]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"disjoint_compl_right","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[645,8],"def_end_pos":[645,28]},{"full_name":"inf_le_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[312,8],"def_end_pos":[312,20]}]},{"state_before":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ = (condCount s) t","state_after":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)","tactic":"rw [this]","premises":[]},{"state_before":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)","state_after":"no goals","tactic":"simp [condCount_inter_self hs]","premises":[{"full_name":"ProbabilityTheory.condCount_inter_self","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[91,8],"def_end_pos":[91,28]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/CofilteredLimit.lean","commit":"","full_name":"Profinite.exists_locallyConstant_finite_nonempty","start":[157,0],"end":[186,39],"file_path":"Mathlib/Topology/Category/Profinite/CofilteredLimit.lean","tactics":[{"state_before":"J : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","state_after":"J : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","tactic":"inhabit α","premises":[]},{"state_before":"J : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","state_after":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","tactic":"obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f","premises":[{"full_name":"Profinite.exists_locallyConstant_finite_aux","def_path":"Mathlib/Topology/Category/Profinite/CofilteredLimit.lean","def_pos":[124,8],"def_end_pos":[124,41]}]},{"state_before":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","state_after":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","tactic":"let ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]},{"state_before":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","state_after":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","tactic":"let σ : (α → Fin 2) → α := fun f => if h : ∃ a : α, ι a = f then h.choose else default","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.choose","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[174,31],"def_end_pos":[174,44]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g","state_after":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\n⊢ f = LocallyConstant.comap (C.π.app j) (LocallyConstant.map σ gg)","tactic":"refine ⟨j, gg.map σ, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"LocallyConstant.map","def_path":"Mathlib/Topology/LocallyConstant/Basic.lean","def_pos":[342,4],"def_end_pos":[342,7]}]},{"state_before":"case intro.intro\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\n⊢ f = LocallyConstant.comap (C.π.app j) (LocallyConstant.map σ gg)","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = (LocallyConstant.comap (C.π.app j) (LocallyConstant.map σ gg)) x","tactic":"ext x","premises":[]},{"state_before":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = (LocallyConstant.comap (C.π.app j) (LocallyConstant.map σ gg)) x","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = σ (gg ((C.π.app j) x))","tactic":"simp only [Functor.const_obj_obj, LocallyConstant.coe_comap, LocallyConstant.map_apply,\n    Function.comp_apply]","premises":[{"full_name":"CategoryTheory.Functor.const_obj_obj","def_path":"Mathlib/CategoryTheory/Functor/Const.lean","def_pos":[32,2],"def_end_pos":[32,7]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"LocallyConstant.coe_comap","def_path":"Mathlib/Topology/LocallyConstant/Basic.lean","def_pos":[390,8],"def_end_pos":[390,17]},{"full_name":"LocallyConstant.map_apply","def_path":"Mathlib/Topology/LocallyConstant/Basic.lean","def_pos":[346,8],"def_end_pos":[346,17]}]},{"state_before":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = σ (gg ((C.π.app j) x))","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default","tactic":"dsimp [σ]","premises":[]},{"state_before":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default","tactic":"have h1 : ι (f x) = gg (C.π.app j x) := by\n    change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _\n    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n    erw [h]\n    rfl","premises":[{"full_name":"CategoryTheory.Limits.Cone.π","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[119,2],"def_end_pos":[119,3]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"LocallyConstant.map","def_path":"Mathlib/Topology/LocallyConstant/Basic.lean","def_pos":[342,4],"def_end_pos":[342,7]},{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]},{"state_before":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default","tactic":"have h2 : ∃ a : α, ι a = gg (C.π.app j x) := ⟨f x, h1⟩","premises":[{"full_name":"CategoryTheory.Limits.Cone.π","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[119,2],"def_end_pos":[119,3]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ f x = h2.choose","tactic":"erw [dif_pos h2]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]},{"state_before":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ f x = h2.choose","state_after":"case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ ι (f x) = ι h2.choose\n\ncase intro.intro.h.inj\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ Function.Injective ι","tactic":"apply_fun ι","premises":[{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]}]}]}
{"url":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","commit":"","full_name":"HahnModule.support_smul_subset_vadd_support","start":[315,0],"end":[322,41],"file_path":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","tactics":[{"state_before":"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : AddCommMonoid V\ninst✝¹ : MulZeroClass R\ninst✝ : SMulWithZero R V\nx : HahnSeries Γ R\ny : HahnModule Γ' R V\n⊢ ((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support","state_after":"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : AddCommMonoid V\ninst✝¹ : MulZeroClass R\ninst✝ : SMulWithZero R V\nx : HahnSeries Γ R\ny : HahnModule Γ' R V\nh : x.support +ᵥ ((of R).symm y).support = x.support +ᵥ ((of R).symm y).support\n⊢ ((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support","tactic":"have h : x.support +ᵥ ((of R).symm y).support =\n      x.support +ᵥ ((of R).symm y).support := by\n    exact rfl","premises":[{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"HVAdd.hVAdd","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[61,2],"def_end_pos":[61,7]},{"full_name":"HahnModule.of","def_path":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","def_pos":[96,4],"def_end_pos":[96,6]},{"full_name":"HahnSeries.support","def_path":"Mathlib/RingTheory/HahnSeries/Basic.lean","def_pos":[64,11],"def_end_pos":[64,18]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : AddCommMonoid V\ninst✝¹ : MulZeroClass R\ninst✝ : SMulWithZero R V\nx : HahnSeries Γ R\ny : HahnModule Γ' R V\nh : x.support +ᵥ ((of R).symm y).support = x.support +ᵥ ((of R).symm y).support\n⊢ ((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support","state_after":"no goals","tactic":"exact support_smul_subset_vadd_support'","premises":[{"full_name":"HahnModule.support_smul_subset_vadd_support'","def_path":"Mathlib/RingTheory/HahnSeries/Multiplication.lean","def_pos":[304,8],"def_end_pos":[304,41]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"inner_map_polarization","start":[1033,0],"end":[1044,6],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : _root_.RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\nV : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nT : V →ₗ[ℂ] V\nx y : V\n⊢ ⟪T y, x⟫_ℂ =\n    (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ + Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ -\n        Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /\n      4","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : _root_.RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\nV : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nT : V →ₗ[ℂ] V\nx y : V\n⊢ ⟪T y, x⟫_ℂ =\n    (⟪T x, x⟫_ℂ + ⟪T y, x⟫_ℂ + (⟪T x, y⟫_ℂ + ⟪T y, y⟫_ℂ) - (⟪T x, x⟫_ℂ - (⟪T y, x⟫_ℂ + (⟪T x, y⟫_ℂ - ⟪T y, y⟫_ℂ))) +\n          (Complex.I * ⟪T x, x⟫_ℂ + ⟪T y, x⟫_ℂ + (-⟪T x, y⟫_ℂ + Complex.I * ⟪T y, y⟫_ℂ)) -\n        (Complex.I * ⟪T x, x⟫_ℂ - (⟪T y, x⟫_ℂ + (-⟪T x, y⟫_ℂ - Complex.I * ⟪T y, y⟫_ℂ)))) /\n      4","tactic":"simp only [map_add, map_sub, inner_add_left, inner_add_right, LinearMap.map_smul, inner_smul_left,\n    inner_smul_right, Complex.conj_I, ← pow_two, Complex.I_sq, inner_sub_left, inner_sub_right,\n    mul_add, ← mul_assoc, mul_neg, neg_neg, sub_neg_eq_add, one_mul, neg_one_mul, mul_sub, sub_sub]","premises":[{"full_name":"Complex.I_sq","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[632,8],"def_end_pos":[632,12]},{"full_name":"Complex.conj_I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[476,8],"def_end_pos":[476,14]},{"full_name":"LinearMap.map_smul","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[348,18],"def_end_pos":[348,26]},{"full_name":"inner_add_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[402,8],"def_end_pos":[402,22]},{"full_name":"inner_add_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[405,8],"def_end_pos":[405,23]},{"full_name":"inner_smul_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[413,8],"def_end_pos":[413,23]},{"full_name":"inner_smul_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[422,8],"def_end_pos":[422,24]},{"full_name":"inner_sub_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[547,8],"def_end_pos":[547,22]},{"full_name":"inner_sub_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[550,8],"def_end_pos":[550,23]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"map_sub","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[414,2],"def_end_pos":[414,13]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_one_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[299,8],"def_end_pos":[299,19]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]},{"full_name":"sub_neg_eq_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[500,2],"def_end_pos":[500,13]},{"full_name":"sub_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[542,2],"def_end_pos":[542,13]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : _root_.RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\nV : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nT : V →ₗ[ℂ] V\nx y : V\n⊢ ⟪T y, x⟫_ℂ =\n    (⟪T x, x⟫_ℂ + ⟪T y, x⟫_ℂ + (⟪T x, y⟫_ℂ + ⟪T y, y⟫_ℂ) - (⟪T x, x⟫_ℂ - (⟪T y, x⟫_ℂ + (⟪T x, y⟫_ℂ - ⟪T y, y⟫_ℂ))) +\n          (Complex.I * ⟪T x, x⟫_ℂ + ⟪T y, x⟫_ℂ + (-⟪T x, y⟫_ℂ + Complex.I * ⟪T y, y⟫_ℂ)) -\n        (Complex.I * ⟪T x, x⟫_ℂ - (⟪T y, x⟫_ℂ + (-⟪T x, y⟫_ℂ - Complex.I * ⟪T y, y⟫_ℂ)))) /\n      4","state_after":"no goals","tactic":"ring","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Alternating/Basic.lean","commit":"","full_name":"AlternatingMap.map_congr_perm","start":[612,0],"end":[615,6],"file_path":"Mathlib/LinearAlgebra/Alternating/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁶ : Semiring R\nM : Type u_2\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type u_3\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type u_4\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type u_5\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_6\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_7\nι' : Type u_8\nι'' : Type u_9\nM₂ : Type u_10\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type u_11\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : M [⋀^ι]→ₗ[R] N\ng g₂ : M [⋀^ι]→ₗ[R] N'\ng' : M' [⋀^ι]→ₗ[R] N'\nv : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nσ : Equiv.Perm ι\n⊢ g v = Equiv.Perm.sign σ • g (v ∘ ⇑σ)","state_after":"R : Type u_1\ninst✝¹⁶ : Semiring R\nM : Type u_2\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type u_3\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type u_4\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type u_5\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_6\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_7\nι' : Type u_8\nι'' : Type u_9\nM₂ : Type u_10\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type u_11\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : M [⋀^ι]→ₗ[R] N\ng g₂ : M [⋀^ι]→ₗ[R] N'\ng' : M' [⋀^ι]→ₗ[R] N'\nv : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nσ : Equiv.Perm ι\n⊢ g v = (Equiv.Perm.sign σ * Equiv.Perm.sign σ) • g v","tactic":"rw [g.map_perm, smul_smul]","premises":[{"full_name":"AlternatingMap.map_perm","def_path":"Mathlib/LinearAlgebra/Alternating/Basic.lean","def_pos":[603,8],"def_end_pos":[603,16]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"R : Type u_1\ninst✝¹⁶ : Semiring R\nM : Type u_2\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type u_3\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type u_4\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type u_5\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_6\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_7\nι' : Type u_8\nι'' : Type u_9\nM₂ : Type u_10\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type u_11\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : M [⋀^ι]→ₗ[R] N\ng g₂ : M [⋀^ι]→ₗ[R] N'\ng' : M' [⋀^ι]→ₗ[R] N'\nv : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nσ : Equiv.Perm ι\n⊢ g v = (Equiv.Perm.sign σ * Equiv.Perm.sign σ) • g v","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","commit":"","full_name":"MeasureTheory.ofReal_setIntegral_one_of_measure_ne_top","start":[189,0],"end":[197,33],"file_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","tactics":[{"state_before":"X✝ : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X✝\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X✝ → E\ns✝ t : Set X✝\nμ✝ ν : Measure X✝\nl l' : Filter X✝\nX : Type u_5\nm : MeasurableSpace X\nμ : Measure X\ns : Set X\nhs : μ s ≠ ⊤\n⊢ ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = ENNReal.ofReal (∫ (x : X) in s, ‖1‖ ∂μ)","state_after":"no goals","tactic":"simp only [norm_one]","premises":[{"full_name":"NormOneClass.norm_one","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[155,2],"def_end_pos":[155,10]}]},{"state_before":"X✝ : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X✝\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X✝ → E\ns✝ t : Set X✝\nμ✝ ν : Measure X✝\nl l' : Filter X✝\nX : Type u_5\nm : MeasurableSpace X\nμ : Measure X\ns : Set X\nhs : μ s ≠ ⊤\n⊢ ENNReal.ofReal (∫ (x : X) in s, ‖1‖ ∂μ) = ∫⁻ (x : X) in s, 1 ∂μ","state_after":"X✝ : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X✝\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X✝ → E\ns✝ t : Set X✝\nμ✝ ν : Measure X✝\nl l' : Filter X✝\nX : Type u_5\nm : MeasurableSpace X\nμ : Measure X\ns : Set X\nhs : μ s ≠ ⊤\n⊢ ∫⁻ (x : X) in s, ↑‖1‖₊ ∂μ = ∫⁻ (x : X) in s, 1 ∂μ","tactic":"rw [ofReal_integral_norm_eq_lintegral_nnnorm (integrableOn_const.2 (Or.inr hs.lt_top))]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.integrableOn_const","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[97,8],"def_end_pos":[97,26]},{"full_name":"MeasureTheory.ofReal_integral_norm_eq_lintegral_nnnorm","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1060,8],"def_end_pos":[1060,48]},{"full_name":"Ne.lt_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[132,8],"def_end_pos":[132,17]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]}]},{"state_before":"X✝ : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X✝\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X✝ → E\ns✝ t : Set X✝\nμ✝ ν : Measure X✝\nl l' : Filter X✝\nX : Type u_5\nm : MeasurableSpace X\nμ : Measure X\ns : Set X\nhs : μ s ≠ ⊤\n⊢ ∫⁻ (x : X) in s, ↑‖1‖₊ ∂μ = ∫⁻ (x : X) in s, 1 ∂μ","state_after":"no goals","tactic":"simp only [nnnorm_one, ENNReal.coe_one]","premises":[{"full_name":"ENNReal.coe_one","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[222,27],"def_end_pos":[222,34]},{"full_name":"nnnorm_one","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[162,8],"def_end_pos":[162,18]}]}]}
{"url":"Mathlib/Data/Matrix/Kronecker.lean","commit":"","full_name":"Matrix.natCast_kronecker","start":[311,0],"end":[317,36],"file_path":"Mathlib/Data/Matrix/Kronecker.lean","tactics":[{"state_before":"R : Type u_1\nα : Type u_2\nα' : Type u_3\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_6\nγ' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst✝¹ : NonAssocSemiring α\ninst✝ : DecidableEq l\na : ℕ\nB : Matrix m n α\n⊢ (reindex (Equiv.prodComm m l) (Equiv.prodComm n l)) (blockDiagonal fun i => ↑a •> B) =\n    (reindex (Equiv.prodComm m l) (Equiv.prodComm n l)) (blockDiagonal fun x => a •> B)","state_after":"case h.e'_6.h.e'_7\nR : Type u_1\nα : Type u_2\nα' : Type u_3\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_6\nγ' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst✝¹ : NonAssocSemiring α\ninst✝ : DecidableEq l\na : ℕ\nB : Matrix m n α\n⊢ (fun i => ↑a •> B) = fun x => a •> B","tactic":"congr! 2","premises":[]},{"state_before":"case h.e'_6.h.e'_7\nR : Type u_1\nα : Type u_2\nα' : Type u_3\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_6\nγ' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst✝¹ : NonAssocSemiring α\ninst✝ : DecidableEq l\na : ℕ\nB : Matrix m n α\n⊢ (fun i => ↑a •> B) = fun x => a •> B","state_after":"case h.e'_6.h.e'_7.h.a\nR : Type u_1\nα : Type u_2\nα' : Type u_3\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_6\nγ' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst✝¹ : NonAssocSemiring α\ninst✝ : DecidableEq l\na : ℕ\nB : Matrix m n α\nx✝ : l\ni✝ : m\nj✝ : n\n⊢ (↑a • B) i✝ j✝ = (a • B) i✝ j✝","tactic":"ext","premises":[]},{"state_before":"case h.e'_6.h.e'_7.h.a\nR : Type u_1\nα : Type u_2\nα' : Type u_3\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_6\nγ' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst✝¹ : NonAssocSemiring α\ninst✝ : DecidableEq l\na : ℕ\nB : Matrix m n α\nx✝ : l\ni✝ : m\nj✝ : n\n⊢ (↑a • B) i✝ j✝ = (a • B) i✝ j✝","state_after":"no goals","tactic":"simp [(Nat.cast_commute a _).eq]","premises":[{"full_name":"Commute.eq","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[52,18],"def_end_pos":[52,20]},{"full_name":"Nat.cast_commute","def_path":"Mathlib/Data/Nat/Cast/Commute.lean","def_pos":[22,8],"def_end_pos":[22,20]}]}]}
{"url":"Mathlib/RingTheory/WittVector/Basic.lean","commit":"","full_name":"_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_one","start":[173,0],"end":[174,22],"file_path":"Mathlib/RingTheory/WittVector/Basic.lean","tactics":[{"state_before":"p : ℕ\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type u_4\nβ : Type u_5\nx y : 𝕎 R\n⊢ WittVector.ghostFun 1 = 1","state_after":"no goals","tactic":"ghost_fun_tac 1, ![]","premises":[{"full_name":"Add.add","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1310,2],"def_end_pos":[1310,5]},{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.uncurry","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[189,4],"def_end_pos":[189,11]},{"full_name":"HAdd.hAdd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1180,2],"def_end_pos":[1180,6]},{"full_name":"HMul.hMul","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1199,2],"def_end_pos":[1199,6]},{"full_name":"HPow.hPow","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1237,2],"def_end_pos":[1237,6]},{"full_name":"HSMul.hSMul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[78,2],"def_end_pos":[78,7]},{"full_name":"HSub.hSub","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1190,2],"def_end_pos":[1190,6]},{"full_name":"Matrix.vecEmpty","def_path":"Mathlib/Data/Fin/VecNotation.lean","def_pos":[48,4],"def_end_pos":[48,12]},{"full_name":"Mul.mul","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1320,2],"def_end_pos":[1320,5]},{"full_name":"MvPolynomial.aeval_bind₁","def_path":"Mathlib/Algebra/MvPolynomial/Monad.lean","def_pos":[251,8],"def_end_pos":[251,19]},{"full_name":"MvPolynomial.aeval_rename","def_path":"Mathlib/Algebra/MvPolynomial/Rename.lean","def_pos":[173,8],"def_end_pos":[173,20]},{"full_name":"Neg.neg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1329,2],"def_end_pos":[1329,5]},{"full_name":"OfNat.ofNat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1105,2],"def_end_pos":[1105,7]},{"full_name":"One.one","def_path":"Mathlib/Algebra/Group/ZeroOne.lean","def_pos":[25,2],"def_end_pos":[25,5]},{"full_name":"Pow.pow","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1358,2],"def_end_pos":[1358,5]},{"full_name":"SMul.smul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[101,2],"def_end_pos":[101,6]},{"full_name":"Sub.sub","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1315,2],"def_end_pos":[1315,5]},{"full_name":"WittVector.eval","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[159,4],"def_end_pos":[159,8]},{"full_name":"WittVector.peval","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[146,4],"def_end_pos":[146,9]},{"full_name":"WittVector.wittNSMul","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[116,4],"def_end_pos":[116,13]},{"full_name":"WittVector.wittOne","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[108,4],"def_end_pos":[108,11]},{"full_name":"WittVector.wittZSMul","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[120,4],"def_end_pos":[120,13]},{"full_name":"WittVector.wittZero","def_path":"Mathlib/RingTheory/WittVector/Defs.lean","def_pos":[104,4],"def_end_pos":[104,12]},{"full_name":"Zero.zero","def_path":"Mathlib/Algebra/Group/ZeroOne.lean","def_pos":[13,2],"def_end_pos":[13,6]},{"full_name":"_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun","def_path":"Mathlib/RingTheory/WittVector/Basic.lean","def_pos":[140,12],"def_end_pos":[140,20]},{"full_name":"wittStructureInt_prop","def_path":"Mathlib/RingTheory/WittVector/StructurePolynomial.lean","def_pos":[290,8],"def_end_pos":[290,29]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","commit":"","full_name":"Real.logb_eq_zero","start":[370,0],"end":[373,7],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","tactics":[{"state_before":"b x y : ℝ\n⊢ logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1","state_after":"b x y : ℝ\n⊢ (x = 0 ∨ x = 1 ∨ x = -1) ∨ b = 0 ∨ b = 1 ∨ b = -1 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1","tactic":"simp_rw [logb, div_eq_zero_iff, log_eq_zero]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real.log_eq_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[238,8],"def_end_pos":[238,19]},{"full_name":"Real.logb","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","def_pos":[38,18],"def_end_pos":[38,22]},{"full_name":"div_eq_zero_iff","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[246,8],"def_end_pos":[246,23]}]},{"state_before":"b x y : ℝ\n⊢ (x = 0 ∨ x = 1 ∨ x = -1) ∨ b = 0 ∨ b = 1 ∨ b = -1 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1","state_after":"no goals","tactic":"tauto","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Ascoli.lean","commit":"","full_name":"EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi","start":[384,0],"end":[399,51],"file_path":"Mathlib/Topology/UniformSpace/Ascoli.lean","tactics":[{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝² : TopologicalSpace X\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\n𝔖_covers : ⋃₀ 𝔖 = univ\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\n⊢ IsClosed (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F)) ↔ IsClosed (range F)","state_after":"no goals","tactic":"simp_rw [isClosed_iff_clusterPt, ← Filter.map_top, ← mapClusterPt_def,\n    mapClusterPt_iff_ultrafilter, range_comp, (UniformOnFun.ofFun 𝔖).surjective.forall,\n    ← EquicontinuousOn.tendsto_uniformOnFun_iff_pi 𝔖_compact 𝔖_covers F_eqcont,\n    (UniformOnFun.ofFun 𝔖).injective.mem_set_image]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"EquicontinuousOn.tendsto_uniformOnFun_iff_pi","def_path":"Mathlib/Topology/UniformSpace/Ascoli.lean","def_pos":[354,8],"def_end_pos":[354,52]},{"full_name":"Equiv.injective","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[175,18],"def_end_pos":[175,27]},{"full_name":"Equiv.surjective","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[177,18],"def_end_pos":[177,28]},{"full_name":"Filter.map_top","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2065,8],"def_end_pos":[2065,15]},{"full_name":"Function.Injective.mem_set_image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[187,8],"def_end_pos":[187,47]},{"full_name":"Function.Surjective.forall","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[164,18],"def_end_pos":[164,35]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.range_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[621,8],"def_end_pos":[621,18]},{"full_name":"UniformOnFun.ofFun","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[176,4],"def_end_pos":[176,22]},{"full_name":"isClosed_iff_clusterPt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1169,8],"def_end_pos":[1169,30]},{"full_name":"mapClusterPt_def","def_path":"Mathlib/Topology/Basic.lean","def_pos":[937,8],"def_end_pos":[937,24]},{"full_name":"mapClusterPt_iff_ultrafilter","def_path":"Mathlib/Topology/Basic.lean","def_pos":[945,8],"def_end_pos":[945,36]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Basic.lean","commit":"","full_name":"EuclideanGeometry.dist_reflection","start":[559,0],"end":[564,35],"file_path":"Mathlib/Geometry/Euclidean/Basic.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np₁ p₂ : P\n⊢ dist p₁ ((reflection s) p₂) = dist ((reflection s) p₁) p₂","state_after":"V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np₁ p₂ : P\n⊢ dist ((reflection s) ((reflection s) p₁)) ((reflection s) p₂) = dist ((reflection s) p₁) p₂","tactic":"conv_lhs => rw [← reflection_reflection s p₁]","premises":[{"full_name":"EuclideanGeometry.reflection_reflection","def_path":"Mathlib/Geometry/Euclidean/Basic.lean","def_pos":[505,8],"def_end_pos":[505,29]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np₁ p₂ : P\n⊢ dist ((reflection s) ((reflection s) p₁)) ((reflection s) p₂) = dist ((reflection s) p₁) p₂","state_after":"no goals","tactic":"exact (reflection s).dist_map _ _","premises":[{"full_name":"AffineIsometryEquiv.dist_map","def_path":"Mathlib/Analysis/NormedSpace/AffineIsometry.lean","def_pos":[529,8],"def_end_pos":[529,16]},{"full_name":"EuclideanGeometry.reflection","def_path":"Mathlib/Geometry/Euclidean/Basic.lean","def_pos":[474,4],"def_end_pos":[474,14]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/Generator.lean","commit":"","full_name":"CategoryTheory.Abelian.has_injective_coseparator","start":[33,0],"end":[50,37],"file_path":"Mathlib/CategoryTheory/Abelian/Generator.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\n⊢ ∃ G, Injective G ∧ IsCoseparator G","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis : WellPowered C\n⊢ ∃ G, Injective G ∧ IsCoseparator G","tactic":"haveI : WellPowered C := wellPowered_of_isDetector G hG.isDetector","premises":[{"full_name":"CategoryTheory.IsSeparator.isDetector","def_path":"Mathlib/CategoryTheory/Generator.lean","def_pos":[389,8],"def_end_pos":[389,30]},{"full_name":"CategoryTheory.WellPowered","def_path":"Mathlib/CategoryTheory/Subobject/WellPowered.lean","def_pos":[40,6],"def_end_pos":[40,17]},{"full_name":"CategoryTheory.wellPowered_of_isDetector","def_path":"Mathlib/CategoryTheory/Generator.lean","def_pos":[568,8],"def_end_pos":[568,33]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis : WellPowered C\n⊢ ∃ G, Injective G ∧ IsCoseparator G","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\n⊢ ∃ G, Injective G ∧ IsCoseparator G","tactic":"haveI : HasProductsOfShape (Subobject (op G)) C := hasProductsOfShape_of_small _ _","premises":[{"full_name":"CategoryTheory.Limits.HasProductsOfShape","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[166,7],"def_end_pos":[166,25]},{"full_name":"CategoryTheory.Limits.hasProductsOfShape_of_small","def_path":"Mathlib/CategoryTheory/Limits/EssentiallySmall.lean","def_pos":[37,8],"def_end_pos":[37,35]},{"full_name":"CategoryTheory.Subobject","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[95,4],"def_end_pos":[95,13]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\n⊢ ∃ G, Injective G ∧ IsCoseparator G","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\n⊢ ∃ G, Injective G ∧ IsCoseparator G","tactic":"let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P)","premises":[{"full_name":"CategoryTheory.Injective.under","def_path":"Mathlib/CategoryTheory/Preadditive/Injective.lean","def_pos":[201,4],"def_end_pos":[201,9]},{"full_name":"CategoryTheory.Limits.piObj","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[178,7],"def_end_pos":[178,12]},{"full_name":"CategoryTheory.Subobject","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[95,4],"def_end_pos":[95,13]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\n⊢ ∃ G, Injective G ∧ IsCoseparator G","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\n⊢ f = 0","tactic":"refine ⟨T, inferInstance, (Preadditive.isCoseparator_iff _).2 fun X Y f hf => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"CategoryTheory.Preadditive.isCoseparator_iff","def_path":"Mathlib/CategoryTheory/Preadditive/Generator.lean","def_pos":[42,8],"def_end_pos":[42,37]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\n⊢ f = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\n⊢ h ≫ f = 0","tactic":"refine (Preadditive.isSeparator_iff _).1 hG _ fun h => ?_","premises":[{"full_name":"CategoryTheory.Preadditive.isSeparator_iff","def_path":"Mathlib/CategoryTheory/Preadditive/Generator.lean","def_pos":[36,8],"def_end_pos":[36,35]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\n⊢ h ≫ f = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\n⊢ factorThruImage (h ≫ f) = 0","tactic":"suffices hh : factorThruImage (h ≫ f) = 0 by\n    rw [← Limits.image.fac (h ≫ f), hh, zero_comp]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.factorThruImage","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[290,4],"def_end_pos":[290,19]},{"full_name":"CategoryTheory.Limits.image.fac","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[299,8],"def_end_pos":[299,17]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\n⊢ factorThruImage (h ≫ f) = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\n⊢ factorThruImage (h ≫ f) = 0","tactic":"let R := Subobject.mk (factorThruImage (h ≫ f)).op","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Limits.factorThruImage","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[290,4],"def_end_pos":[290,19]},{"full_name":"CategoryTheory.Subobject.mk","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[107,4],"def_end_pos":[107,6]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\n⊢ factorThruImage (h ≫ f) = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\nq₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) :=\n  (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\n⊢ factorThruImage (h ≫ f) = 0","tactic":"let q₁ : image (h ≫ f) ⟶ unop R :=\n    (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.unop","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[410,4],"def_end_pos":[410,8]},{"full_name":"CategoryTheory.Limits.factorThruImage","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[290,4],"def_end_pos":[290,19]},{"full_name":"CategoryTheory.Limits.image","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[276,4],"def_end_pos":[276,9]},{"full_name":"CategoryTheory.Subobject.underlyingIso","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[182,18],"def_end_pos":[182,31]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\nq₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) :=\n  (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\n⊢ factorThruImage (h ≫ f) = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\nq₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) :=\n  (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\nq₂ : unop (Subobject.underlying.obj R) ⟶ ∏ᶜ fun P => unop (Subobject.underlying.obj P) :=\n  section_ (Pi.π (fun P => unop (Subobject.underlying.obj P)) R)\n⊢ factorThruImage (h ≫ f) = 0","tactic":"let q₂ : unop (R : Cᵒᵖ) ⟶ piObj fun P : Subobject (op G) => unop P :=\n    section_ (Pi.π (fun P : Subobject (op G) => (unop P : C)) R)","premises":[{"full_name":"CategoryTheory.Limits.Pi.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[194,7],"def_end_pos":[194,11]},{"full_name":"CategoryTheory.Limits.piObj","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[178,7],"def_end_pos":[178,12]},{"full_name":"CategoryTheory.Subobject","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[95,4],"def_end_pos":[95,13]},{"full_name":"CategoryTheory.section_","def_path":"Mathlib/CategoryTheory/EpiMono.lean","def_pos":[110,18],"def_end_pos":[110,26]},{"full_name":"Opposite","def_path":"Mathlib/Data/Opposite.lean","def_pos":[33,10],"def_end_pos":[33,18]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\nq₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) :=\n  (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\nq₂ : unop (Subobject.underlying.obj R) ⟶ ∏ᶜ fun P => unop (Subobject.underlying.obj P) :=\n  section_ (Pi.π (fun P => unop (Subobject.underlying.obj P)) R)\n⊢ factorThruImage (h ≫ f) = 0","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\nq₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) :=\n  (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\nq₂ : unop (Subobject.underlying.obj R) ⟶ ∏ᶜ fun P => unop (Subobject.underlying.obj P) :=\n  section_ (Pi.π (fun P => unop (Subobject.underlying.obj P)) R)\nq : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ Injective.ι (∏ᶜ fun P => unop (Subobject.underlying.obj P))\n⊢ factorThruImage (h ≫ f) = 0","tactic":"let q : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ Injective.ι _","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Injective.ι","def_path":"Mathlib/CategoryTheory/Preadditive/Injective.lean","def_pos":[210,4],"def_end_pos":[210,5]},{"full_name":"CategoryTheory.Limits.image","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[276,4],"def_end_pos":[276,9]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ T), f ≫ h = 0\nh : G ⟶ X\nR : Subobject (op G) := Subobject.mk (factorThruImage (h ≫ f)).op\nq₁ : image (h ≫ f) ⟶ unop (Subobject.underlying.obj R) :=\n  (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\nq₂ : unop (Subobject.underlying.obj R) ⟶ ∏ᶜ fun P => unop (Subobject.underlying.obj P) :=\n  section_ (Pi.π (fun P => unop (Subobject.underlying.obj P)) R)\nq : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ Injective.ι (∏ᶜ fun P => unop (Subobject.underlying.obj P))\n⊢ factorThruImage (h ≫ f) = 0","state_after":"no goals","tactic":"exact zero_of_comp_mono q\n    (by rw [← Injective.comp_factorThru q (Limits.image.ι (h ≫ f)), Limits.image.fac_assoc,\n      Category.assoc, hf, comp_zero])","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Injective.comp_factorThru","def_path":"Mathlib/CategoryTheory/Preadditive/Injective.lean","def_pos":[76,8],"def_end_pos":[76,23]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.Limits.image.ι","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Images.lean","def_pos":[280,4],"def_end_pos":[280,11]},{"full_name":"CategoryTheory.Limits.zero_of_comp_mono","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[126,8],"def_end_pos":[126,25]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean","commit":"","full_name":"CategoryTheory.FreeMonoidalCategory.tensorFunc_obj_map","start":[169,0],"end":[176,6],"file_path":"Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean","tactics":[{"state_before":"C : Type u\nZ : F C\nn n' : (Discrete ∘ NormalMonoidalObject) C\nf : n ⟶ n'\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z","state_after":"case mk\nC : Type u\nZ : F C\nn' : (Discrete ∘ NormalMonoidalObject) C\nas✝ : NormalMonoidalObject C\nf : { as := as✝ } ⟶ n'\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z","tactic":"cases n","premises":[]},{"state_before":"case mk\nC : Type u\nZ : F C\nn' : (Discrete ∘ NormalMonoidalObject) C\nas✝ : NormalMonoidalObject C\nf : { as := as✝ } ⟶ n'\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z","state_after":"case mk.mk\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nf : { as := as✝¹ } ⟶ { as := as✝ }\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z","tactic":"cases n'","premises":[]},{"state_before":"case mk.mk\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nf : { as := as✝¹ } ⟶ { as := as✝ }\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z","state_after":"case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : { as := as✝¹ }.as = { as := as✝ }.as\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ▷ Z","tactic":"rcases f with ⟨⟨h⟩⟩","premises":[]},{"state_before":"case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : { as := as✝¹ }.as = { as := as✝ }.as\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ▷ Z","state_after":"case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : as✝¹ = as✝\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ▷ Z","tactic":"dsimp at h","premises":[]},{"state_before":"case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : as✝¹ = as✝\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ▷ Z","state_after":"case mk.mk.up.up\nC : Type u\nZ : F C\nas✝ : NormalMonoidalObject C\n⊢ ((tensorFunc C).obj Z).map { down := { down := ⋯ } } = inclusion.map { down := { down := ⋯ } } ▷ Z","tactic":"subst h","premises":[]},{"state_before":"case mk.mk.up.up\nC : Type u\nZ : F C\nas✝ : NormalMonoidalObject C\n⊢ ((tensorFunc C).obj Z).map { down := { down := ⋯ } } = inclusion.map { down := { down := ⋯ } } ▷ Z","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Vector/Basic.lean","commit":"","full_name":"Mathlib.Vector.id_traverse","start":[621,0],"end":[624,17],"file_path":"Mathlib/Data/Vector/Basic.lean","tactics":[{"state_before":"n : ℕ\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\n⊢ ∀ (x : Vector α n), Vector.traverse pure x = x","state_after":"case mk\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : List α\n⊢ Vector.traverse pure ⟨x, ⋯⟩ = ⟨x, ⋯⟩","tactic":"rintro ⟨x, rfl⟩","premises":[]},{"state_before":"case mk\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : List α\n⊢ Vector.traverse pure ⟨x, ⋯⟩ = ⟨x, ⋯⟩","state_after":"case mk\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : List α\n⊢ Mathlib.Vector.traverseAux pure x = ⟨x, ⋯⟩","tactic":"dsimp [Vector.traverse, cast]","premises":[{"full_name":"Mathlib.Vector.traverse","def_path":"Mathlib/Data/Vector/Basic.lean","def_pos":[609,14],"def_end_pos":[609,22]},{"full_name":"cast","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[349,20],"def_end_pos":[349,24]}]},{"state_before":"case mk\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : List α\n⊢ Mathlib.Vector.traverseAux pure x = ⟨x, ⋯⟩","state_after":"case mk.nil\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\n⊢ Mathlib.Vector.traverseAux pure [] = ⟨[], ⋯⟩\n\ncase mk.cons\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : α\nxs : List α\nIH : Mathlib.Vector.traverseAux pure xs = ⟨xs, ⋯⟩\n⊢ Mathlib.Vector.traverseAux pure (x :: xs) = ⟨x :: xs, ⋯⟩","tactic":"induction' x with x xs IH","premises":[]},{"state_before":"case mk.cons\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : α\nxs : List α\nIH : Mathlib.Vector.traverseAux pure xs = ⟨xs, ⋯⟩\n⊢ Mathlib.Vector.traverseAux pure (x :: xs) = ⟨x :: xs, ⋯⟩","state_after":"case mk.cons\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : α\nxs : List α\nIH : Mathlib.Vector.traverseAux pure xs = ⟨xs, ⋯⟩\n⊢ (Seq.seq (cons x) fun x => ⟨xs, ⋯⟩) = ⟨x :: xs, ⋯⟩","tactic":"simp! [IH]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Lean.Meta.Simp.Config.arith","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[182,2],"def_end_pos":[182,7]},{"full_name":"Lean.Meta.Simp.Config.beta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[157,2],"def_end_pos":[157,6]},{"full_name":"Lean.Meta.Simp.Config.contextual","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[136,2],"def_end_pos":[136,12]},{"full_name":"Lean.Meta.Simp.Config.decide","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[180,2],"def_end_pos":[180,8]},{"full_name":"Lean.Meta.Simp.Config.dsimp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[193,2],"def_end_pos":[193,7]},{"full_name":"Lean.Meta.Simp.Config.eta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[162,2],"def_end_pos":[162,5]},{"full_name":"Lean.Meta.Simp.Config.etaStruct","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[167,2],"def_end_pos":[167,11]},{"full_name":"Lean.Meta.Simp.Config.failIfUnchanged","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[198,2],"def_end_pos":[198,17]},{"full_name":"Lean.Meta.Simp.Config.ground","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[205,2],"def_end_pos":[205,8]},{"full_name":"Lean.Meta.Simp.Config.implicitDefEqProofs","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[228,2],"def_end_pos":[228,21]},{"full_name":"Lean.Meta.Simp.Config.index","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"Lean.Meta.Simp.Config.iota","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[171,2],"def_end_pos":[171,6]},{"full_name":"Lean.Meta.Simp.Config.maxDischargeDepth","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[131,2],"def_end_pos":[131,19]},{"full_name":"Lean.Meta.Simp.Config.maxSteps","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[126,2],"def_end_pos":[126,10]},{"full_name":"Lean.Meta.Simp.Config.memoize","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[140,2],"def_end_pos":[140,9]},{"full_name":"Lean.Meta.Simp.Config.proj","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[175,2],"def_end_pos":[175,6]},{"full_name":"Lean.Meta.Simp.Config.singlePass","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[146,2],"def_end_pos":[146,12]},{"full_name":"Lean.Meta.Simp.Config.unfoldPartialApp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[210,2],"def_end_pos":[210,18]},{"full_name":"Lean.Meta.Simp.Config.zeta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[152,2],"def_end_pos":[152,6]},{"full_name":"Lean.Meta.Simp.Config.zetaDelta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[215,2],"def_end_pos":[215,11]}]},{"state_before":"case mk.cons\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nx : α\nxs : List α\nIH : Mathlib.Vector.traverseAux pure xs = ⟨xs, ⋯⟩\n⊢ (Seq.seq (cons x) fun x => ⟨xs, ⋯⟩) = ⟨x :: xs, ⋯⟩","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/Algebra/UniformGroup.lean","commit":"","full_name":"uniformity_eq_comap_neg_add_nhds_zero_swapped","start":[277,0],"end":[281,5],"file_path":"Mathlib/Topology/Algebra/UniformGroup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (fun x => x.2⁻¹ * x.1) (𝓝 1)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ comap ((fun x => x.1⁻¹ * x.2) ∘ Prod.swap) (𝓝 1) = comap (fun x => x.2⁻¹ * x.1) (𝓝 1)","tactic":"rw [← comap_swap_uniformity, uniformity_eq_comap_inv_mul_nhds_one, comap_comap]","premises":[{"full_name":"Filter.comap_comap","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1934,8],"def_end_pos":[1934,19]},{"full_name":"comap_swap_uniformity","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[494,8],"def_end_pos":[494,29]},{"full_name":"uniformity_eq_comap_inv_mul_nhds_one","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[271,8],"def_end_pos":[271,44]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ comap ((fun x => x.1⁻¹ * x.2) ∘ Prod.swap) (𝓝 1) = comap (fun x => x.2⁻¹ * x.1) (𝓝 1)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Nat/Bits.lean","commit":"","full_name":"Nat.shiftLeft'_false","start":[139,0],"end":[145,68],"file_path":"Mathlib/Data/Nat/Bits.lean","tactics":[{"state_before":"m n✝ n : ℕ\n⊢ shiftLeft' false m (n + 1) = m <<< (n + 1)","state_after":"m n✝ n : ℕ\nthis : 2 * (m * 2 ^ n) = 2 ^ (n + 1) * m\n⊢ shiftLeft' false m (n + 1) = m <<< (n + 1)","tactic":"have : 2 * (m * 2^n) = 2^(n+1)*m := by\n      rw [Nat.mul_comm, Nat.mul_assoc, ← Nat.pow_succ]; simp","premises":[{"full_name":"Nat.mul_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[245,18],"def_end_pos":[245,27]},{"full_name":"Nat.mul_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[217,18],"def_end_pos":[217,26]},{"full_name":"Nat.pow_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[749,18],"def_end_pos":[749,26]}]},{"state_before":"m n✝ n : ℕ\nthis : 2 * (m * 2 ^ n) = 2 ^ (n + 1) * m\n⊢ shiftLeft' false m (n + 1) = m <<< (n + 1)","state_after":"no goals","tactic":"simp [shiftLeft_eq, shiftLeft', bit_val, shiftLeft'_false, this]","premises":[{"full_name":"Nat.bit_val","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[118,6],"def_end_pos":[118,13]},{"full_name":"Nat.shiftLeft'","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[135,4],"def_end_pos":[135,14]},{"full_name":"Nat.shiftLeft_eq","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Basic.lean","def_pos":[54,8],"def_end_pos":[54,20]}]}]}
{"url":"Mathlib/Algebra/Group/Submonoid/Membership.lean","commit":"","full_name":"multiset_sum_mem","start":[75,0],"end":[83,23],"file_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","tactics":[{"state_before":"M✝ : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : Monoid M✝\ninst✝⁴ : SetLike B M✝\ninst✝³ : SubmonoidClass B M✝\nS : B\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nm : Multiset M\nhm : ∀ a ∈ m, a ∈ S\n⊢ m.prod ∈ S","state_after":"case intro\nM✝ : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : Monoid M✝\ninst✝⁴ : SetLike B M✝\ninst✝³ : SubmonoidClass B M✝\nS : B\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nm : Multiset ↥S\n⊢ (Multiset.map Subtype.val m).prod ∈ S","tactic":"lift m to Multiset S using hm","premises":[{"full_name":"Multiset","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[28,4],"def_end_pos":[28,12]}]},{"state_before":"case intro\nM✝ : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : Monoid M✝\ninst✝⁴ : SetLike B M✝\ninst✝³ : SubmonoidClass B M✝\nS : B\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nm : Multiset ↥S\n⊢ (Multiset.map Subtype.val m).prod ∈ S","state_after":"case intro\nM✝ : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : Monoid M✝\ninst✝⁴ : SetLike B M✝\ninst✝³ : SubmonoidClass B M✝\nS : B\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nm : Multiset ↥S\n⊢ ↑m.prod ∈ S","tactic":"rw [← coe_multiset_prod]","premises":[{"full_name":"SubmonoidClass.coe_multiset_prod","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[55,8],"def_end_pos":[55,25]}]},{"state_before":"case intro\nM✝ : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : Monoid M✝\ninst✝⁴ : SetLike B M✝\ninst✝³ : SubmonoidClass B M✝\nS : B\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nm : Multiset ↥S\n⊢ ↑m.prod ∈ S","state_after":"no goals","tactic":"exact m.prod.coe_prop","premises":[{"full_name":"Multiset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[38,4],"def_end_pos":[38,8]},{"full_name":"Subtype.coe_prop","def_path":"Mathlib/Data/Subtype.lean","def_pos":[227,8],"def_end_pos":[227,16]}]}]}
{"url":"Mathlib/Analysis/RCLike/Basic.lean","commit":"","full_name":"RCLike.inv_I","start":[489,0],"end":[493,37],"file_path":"Mathlib/Analysis/RCLike/Basic.lean","tactics":[{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nz : K\n⊢ I⁻¹ = -I","state_after":"case pos\nK : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nz : K\nh : I = 0\n⊢ I⁻¹ = -I\n\ncase neg\nK : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nz : K\nh : ¬I = 0\n⊢ I⁻¹ = -I","tactic":"by_cases h : (I : K) = 0","premises":[{"full_name":"RCLike.I","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[57,2],"def_end_pos":[57,3]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Set/Pointwise/Basic.lean","commit":"","full_name":"Set.neg_subset","start":[224,0],"end":[225,75],"file_path":"Mathlib/Data/Set/Pointwise/Basic.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝ : InvolutiveInv α\ns t : Set α\na : α\n⊢ s⁻¹ ⊆ t ↔ s ⊆ t⁻¹","state_after":"no goals","tactic":"rw [← inv_subset_inv, inv_inv]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.inv_subset_inv","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[221,8],"def_end_pos":[221,22]},{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]}]}]}
{"url":"Mathlib/Topology/UrysohnsLemma.lean","commit":"","full_name":"Urysohns.CU.continuous_lim","start":[249,0],"end":[284,36],"file_path":"Mathlib/Topology/UrysohnsLemma.lean","tactics":[{"state_before":"X : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\n⊢ Continuous c.lim","state_after":"case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\n⊢ Continuous c.lim","tactic":"obtain ⟨h0, h1234, h1⟩ : 0 < (2⁻¹ : ℝ) ∧ (2⁻¹ : ℝ) < 3 / 4 ∧ (3 / 4 : ℝ) < 1 := by norm_num","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\n⊢ Continuous c.lim","state_after":"case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, c.lim x_1 ∈ Metric.closedBall (c.lim x) ((3 / 4) ^ n)","tactic":"refine\n    continuous_iff_continuousAt.2 fun x =>\n      (Metric.nhds_basis_closedBall_pow (h0.trans h1234) h1).tendsto_right_iff.2 fun n _ => ?_","premises":[{"full_name":"Filter.HasBasis.tendsto_right_iff","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[766,8],"def_end_pos":[766,34]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Metric.nhds_basis_closedBall_pow","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[853,8],"def_end_pos":[853,33]},{"full_name":"continuous_iff_continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1427,8],"def_end_pos":[1427,35]}]},{"state_before":"case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, c.lim x_1 ∈ Metric.closedBall (c.lim x) ((3 / 4) ^ n)","state_after":"case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n","tactic":"simp only [Metric.mem_closedBall]","premises":[{"full_name":"Metric.mem_closedBall","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[405,16],"def_end_pos":[405,30]}]},{"state_before":"case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n","state_after":"case intro.intro.zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nc : CU P\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ 0\n\ncase intro.intro.succ\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n\nc : CU P\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ (n + 1)","tactic":"induction' n with n ihn generalizing c","premises":[]}]}
{"url":"Mathlib/FieldTheory/SplittingField/Construction.lean","commit":"","full_name":"Polynomial.natDegree_removeFactor'","start":[102,0],"end":[103,89],"file_path":"Mathlib/FieldTheory/SplittingField/Construction.lean","tactics":[{"state_before":"F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\nf : K[X]\nn : ℕ\nhfn : f.natDegree = n + 1\n⊢ f.removeFactor.natDegree = n","state_after":"no goals","tactic":"rw [natDegree_removeFactor, hfn, n.add_sub_cancel]","premises":[{"full_name":"Nat.add_sub_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[904,26],"def_end_pos":[904,40]},{"full_name":"Polynomial.natDegree_removeFactor","def_path":"Mathlib/FieldTheory/SplittingField/Construction.lean","def_pos":[97,8],"def_end_pos":[97,30]}]}]}
{"url":"Mathlib/Order/Filter/Pointwise.lean","commit":"","full_name":"Filter.mul_top_of_one_le","start":[546,0],"end":[551,71],"file_path":"Mathlib/Order/Filter/Pointwise.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\n⊢ f * ⊤ = ⊤","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns : Set α\n⊢ s ∈ f * ⊤ → s ∈ ⊤","tactic":"refine top_le_iff.1 fun s => ?_","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"top_le_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[110,8],"def_end_pos":[110,18]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns : Set α\n⊢ s ∈ f * ⊤ → s ∈ ⊤","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns : Set α\n⊢ (∃ t₁ ∈ f, t₁ * univ ⊆ s) → s = univ","tactic":"simp only [mem_mul, mem_top, exists_and_left, exists_eq_left]","premises":[{"full_name":"Filter.mem_mul","def_path":"Mathlib/Order/Filter/Pointwise.lean","def_pos":[247,8],"def_end_pos":[247,15]},{"full_name":"Filter.mem_top","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[430,8],"def_end_pos":[430,15]},{"full_name":"exists_and_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[288,16],"def_end_pos":[288,31]},{"full_name":"exists_eq_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[282,16],"def_end_pos":[282,30]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns : Set α\n⊢ (∃ t₁ ∈ f, t₁ * univ ⊆ s) → s = univ","state_after":"case intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : t * univ ⊆ s\n⊢ s = univ","tactic":"rintro ⟨t, ht, hs⟩","premises":[]},{"state_before":"case intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : t * univ ⊆ s\n⊢ s = univ","state_after":"no goals","tactic":"rwa [mul_univ_of_one_mem (mem_one.1 <| hf ht), univ_subset_iff] at hs","premises":[{"full_name":"Filter.mem_one","def_path":"Mathlib/Order/Filter/Pointwise.lean","def_pos":[78,8],"def_end_pos":[78,15]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mul_univ_of_one_mem","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[741,8],"def_end_pos":[741,27]},{"full_name":"Set.univ_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[553,8],"def_end_pos":[553,23]}]}]}
{"url":"Mathlib/Order/Filter/ListTraverse.lean","commit":"","full_name":"Filter.mem_traverse_iff","start":[34,0],"end":[49,51],"file_path":"Mathlib/Order/Filter/ListTraverse.lean","tactics":[{"state_before":"α β γ : Type u\nf : β → Filter α\ns : γ → Set α\nfs : List β\nt : Set (List α)\n⊢ t ∈ traverse f fs ↔ ∃ us, Forall₂ (fun b s => s ∈ f b) fs us ∧ sequence us ⊆ t","state_after":"case mp\nα β γ : Type u\nf : β → Filter α\ns : γ → Set α\nfs : List β\nt : Set (List α)\n⊢ t ∈ traverse f fs → ∃ us, Forall₂ (fun b s => s ∈ f b) fs us ∧ sequence us ⊆ t\n\ncase mpr\nα β γ : Type u\nf : β → Filter α\ns : γ → Set α\nfs : List β\nt : Set (List α)\n⊢ (∃ us, Forall₂ (fun b s => s ∈ f b) fs us ∧ sequence us ⊆ t) → t ∈ traverse f fs","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean","commit":"","full_name":"MvQPF.Cofix.ext","start":[333,0],"end":[334,42],"file_path":"Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean","tactics":[{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx y : Cofix F α\nh : x.dest = y.dest\n⊢ x = y","state_after":"no goals","tactic":"rw [← Cofix.mk_dest x, h, Cofix.mk_dest]","premises":[{"full_name":"MvQPF.Cofix.mk_dest","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean","def_pos":[308,8],"def_end_pos":[308,21]}]}]}
{"url":"Mathlib/Data/Sign.lean","commit":"","full_name":"SignType.map_cast'","start":[226,0],"end":[230,51],"file_path":"Mathlib/Data/Sign.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁵ : Zero α\ninst✝⁴ : One α\ninst✝³ : Neg α\nβ : Type u_2\ninst✝² : One β\ninst✝¹ : Neg β\ninst✝ : Zero β\nf : α → β\nh₁ : f 1 = 1\nh₂ : f 0 = 0\nh₃ : f (-1) = -1\ns : SignType\n⊢ f ↑s = ↑s","state_after":"no goals","tactic":"cases s <;> simp only [SignType.cast, h₁, h₂, h₃]","premises":[{"full_name":"SignType.cast","def_path":"Mathlib/Data/Sign.lean","def_pos":[215,4],"def_end_pos":[215,8]}]}]}
{"url":"Mathlib/Algebra/Ring/Ext.lean","commit":"","full_name":"Ring.toNonAssocRing_injective","start":[382,0],"end":[385,15],"file_path":"Mathlib/Algebra/Ring/Ext.lean","tactics":[{"state_before":"R : Type u\n⊢ Function.Injective (@toNonAssocRing R)","state_after":"R : Type u\na₁✝ a₂✝ : Ring R\na✝ : toNonAssocRing = toNonAssocRing\n⊢ a₁✝ = a₂✝","tactic":"intro _ _ _","premises":[]},{"state_before":"R : Type u\na₁✝ a₂✝ : Ring R\na✝ : toNonAssocRing = toNonAssocRing\n⊢ a₁✝ = a₂✝","state_after":"no goals","tactic":"ext <;> congr","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","commit":"","full_name":"AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial","start":[721,0],"end":[726,69],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\ninst✝ : Nontrivial P\n⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤","state_after":"case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\ninst✝ : Nontrivial P\nhs : s = ∅\n⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤\n\ncase inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\ninst✝ : Nontrivial P\nhs : s.Nonempty\n⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤","tactic":"rcases s.eq_empty_or_nonempty with hs | hs","premises":[{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]}]}]}
{"url":"Mathlib/Topology/ClopenBox.lean","commit":"","full_name":"TopologicalSpace.Clopens.countable_iff_second_countable","start":[77,0],"end":[92,22],"file_path":"Mathlib/Topology/ClopenBox.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : CompactSpace Y\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : TotallyDisconnectedSpace X\n⊢ Countable (Clopens X) ↔ SecondCountableTopology X","state_after":"case refine_1\nX : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : CompactSpace Y\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : TotallyDisconnectedSpace X\nh : Countable (Clopens X)\n⊢ {s | IsClopen s}.Countable\n\ncase refine_2\nX : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : CompactSpace Y\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : TotallyDisconnectedSpace X\nh : Countable (Clopens X)\n⊢ inst✝⁵ = generateFrom {s | IsClopen s}\n\ncase refine_3\nX : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : CompactSpace Y\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : TotallyDisconnectedSpace X\nh : SecondCountableTopology X\n⊢ Countable (Clopens X)","tactic":"refine ⟨fun h ↦ ⟨{s : Set X | IsClopen s}, ?_, ?_⟩, fun h ↦ ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"IsClopen","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[98,4],"def_end_pos":[98,12]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.shiftFunctorZero_op_hom_app","start":[85,0],"end":[89,5],"file_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (shiftFunctorZero Cᵒᵖ ℤ).hom.app X =\n    (shiftFunctorOpIso C 0 0 ⋯).hom.app X ≫ ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op","state_after":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (pullbackShiftIso (OppositeShift C ℤ)\n              (AddMonoidHom.mk' (fun n => -n) CategoryTheory.Pretriangulated.Opposite.OppositeShiftAux.proof_1) 0 0\n              ⋯).hom.app\n        X ≫\n      ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op =\n    (shiftFunctorOpIso C 0 0 ⋯).hom.app X ≫ ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op","tactic":"erw [@pullbackShiftFunctorZero_hom_app (OppositeShift C ℤ), oppositeShiftFunctorZero_hom_app]","premises":[{"full_name":"CategoryTheory.OppositeShift","def_path":"Mathlib/CategoryTheory/Shift/Opposite.lean","def_pos":[52,4],"def_end_pos":[52,17]},{"full_name":"CategoryTheory.oppositeShiftFunctorZero_hom_app","def_path":"Mathlib/CategoryTheory/Shift/Opposite.lean","def_pos":[78,6],"def_end_pos":[78,38]},{"full_name":"CategoryTheory.pullbackShiftFunctorZero_hom_app","def_path":"Mathlib/CategoryTheory/Shift/Pullback.lean","def_pos":[70,6],"def_end_pos":[70,38]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (pullbackShiftIso (OppositeShift C ℤ)\n              (AddMonoidHom.mk' (fun n => -n) CategoryTheory.Pretriangulated.Opposite.OppositeShiftAux.proof_1) 0 0\n              ⋯).hom.app\n        X ≫\n      ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op =\n    (shiftFunctorOpIso C 0 0 ⋯).hom.app X ≫ ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","commit":"","full_name":"MeasureTheory.Measure.nullMeasurableSet_prod","start":[671,0],"end":[678,45],"file_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","tactics":[{"state_before":"α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0","state_after":"case inl\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s = 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0\n\ncase inr\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0","tactic":"rcases eq_or_ne (μ s) 0 with hs | hs","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0","state_after":"case inr.inl\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\nht : ν t = 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0\n\ncase inr.inr\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\nht : ν t ≠ 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0","tactic":"rcases eq_or_ne (ν t) 0 with ht | ht","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\nht : ν t ≠ 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0","state_after":"no goals","tactic":"simp [*, nullMeasurableSet_prod_of_ne_zero]","premises":[{"full_name":"MeasureTheory.Measure.nullMeasurableSet_prod_of_ne_zero","def_path":"Mathlib/MeasureTheory/Constructions/Prod/Basic.lean","def_pos":[667,6],"def_end_pos":[667,39]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finsum_mem_union_inter'","start":[657,0],"end":[668,77],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : (s ∩ mulSupport f).Finite\nht : (t ∩ mulSupport f).Finite\n⊢ (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t), f i =\n    (∏ᶠ (i : α) (_ : i ∈ s), f i) * ∏ᶠ (i : α) (_ : i ∈ t), f i","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : (s ∩ mulSupport f).Finite\nht : (t ∩ mulSupport f).Finite\n⊢ (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t ∩ mulSupport f), f i =\n    (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f i","tactic":"rw [← finprod_mem_inter_mulSupport f s, ← finprod_mem_inter_mulSupport f t, ←\n    finprod_mem_union_inter hs ht, ← union_inter_distrib_right, finprod_mem_inter_mulSupport, ←\n    finprod_mem_inter_mulSupport f (s ∩ t)]","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.union_inter_distrib_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[834,8],"def_end_pos":[834,33]},{"full_name":"finprod_mem_inter_mulSupport","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[456,8],"def_end_pos":[456,36]},{"full_name":"finprod_mem_union_inter","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[650,8],"def_end_pos":[650,31]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : (s ∩ mulSupport f).Finite\nht : (t ∩ mulSupport f).Finite\n⊢ (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t ∩ mulSupport f), f i =\n    (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f i","state_after":"case e_a.e_f\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : (s ∩ mulSupport f).Finite\nht : (t ∩ mulSupport f).Finite\n⊢ (fun i => ∏ᶠ (_ : i ∈ s ∩ t ∩ mulSupport f), f i) = fun i => ∏ᶠ (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f i","tactic":"congr 2","premises":[]},{"state_before":"case e_a.e_f\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : (s ∩ mulSupport f).Finite\nht : (t ∩ mulSupport f).Finite\n⊢ (fun i => ∏ᶠ (_ : i ∈ s ∩ t ∩ mulSupport f), f i) = fun i => ∏ᶠ (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f i","state_after":"no goals","tactic":"rw [inter_left_comm, inter_assoc, inter_assoc, inter_self, inter_left_comm]","premises":[{"full_name":"Set.inter_assoc","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[745,8],"def_end_pos":[745,19]},{"full_name":"Set.inter_left_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[754,8],"def_end_pos":[754,23]},{"full_name":"Set.inter_self","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[731,8],"def_end_pos":[731,18]}]}]}
{"url":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","commit":"","full_name":"lp.norm_le_of_forall_le","start":[526,0],"end":[530,37],"file_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E ⊤)\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\n⊢ ‖f‖ ≤ C","state_after":"case inl\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E ⊤)\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\nh✝ : IsEmpty α\n⊢ ‖f‖ ≤ C\n\ncase inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E ⊤)\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\nh✝ : Nonempty α\n⊢ ‖f‖ ≤ C","tactic":"cases isEmpty_or_nonempty α","premises":[{"full_name":"isEmpty_or_nonempty","def_path":"Mathlib/Logic/IsEmpty.lean","def_pos":[195,8],"def_end_pos":[195,27]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean","commit":"","full_name":"EuclideanGeometry.cos_oangle_right_mul_dist_of_oangle_eq_pi_div_two","start":[621,0],"end":[627,94],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\n⊢ (∡ p₂ p₃ p₁).cos * dist p₁ p₃ = dist p₃ p₂","state_after":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₂ p₃ p₁).sign = 1\n⊢ (∡ p₂ p₃ p₁).cos * dist p₁ p₃ = dist p₃ p₂","tactic":"have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]","premises":[{"full_name":"EuclideanGeometry.oangle","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[40,4],"def_end_pos":[40,10]},{"full_name":"EuclideanGeometry.oangle_rotate_sign","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[406,8],"def_end_pos":[406,26]},{"full_name":"Real.Angle.sign","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[712,4],"def_end_pos":[712,8]},{"full_name":"Real.Angle.sign_coe_pi_div_two","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[812,8],"def_end_pos":[812,27]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₂ p₃ p₁).sign = 1\n⊢ (∡ p₂ p₃ p₁).cos * dist p₁ p₃ = dist p₃ p₂","state_after":"no goals","tactic":"rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,\n    cos_angle_mul_dist_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]","premises":[{"full_name":"EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_two","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[363,8],"def_end_pos":[363,51]},{"full_name":"EuclideanGeometry.cos_angle_mul_dist_of_angle_eq_pi_div_two","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[410,8],"def_end_pos":[410,49]},{"full_name":"EuclideanGeometry.oangle_eq_angle_of_sign_eq_one","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[342,8],"def_end_pos":[342,38]},{"full_name":"Real.Angle.cos_coe","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[278,8],"def_end_pos":[278,15]}]}]}
{"url":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","commit":"","full_name":"MeasureTheory.condexpL1_mono","start":[545,0],"end":[551,85],"file_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁵ : RCLike 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : NormedAddCommGroup F'\ninst✝¹¹ : NormedSpace 𝕜 F'\ninst✝¹⁰ : NormedSpace ℝ F'\ninst✝⁹ : CompleteSpace F'\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup G'\ninst✝⁶ : NormedSpace ℝ G'\ninst✝⁵ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝⁴ : SigmaFinite (μ.trim hm)\nf✝ g✝ : α → F'\ns : Set α\nE : Type u_8\ninst✝³ : NormedLatticeAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OrderedSMul ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\nhfg : f ≤ᶠ[ae μ] g\n⊢ ↑↑(condexpL1 hm μ f) ≤ᶠ[ae μ] ↑↑(condexpL1 hm μ g)","state_after":"α : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁵ : RCLike 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : NormedAddCommGroup F'\ninst✝¹¹ : NormedSpace 𝕜 F'\ninst✝¹⁰ : NormedSpace ℝ F'\ninst✝⁹ : CompleteSpace F'\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup G'\ninst✝⁶ : NormedSpace ℝ G'\ninst✝⁵ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝⁴ : SigmaFinite (μ.trim hm)\nf✝ g✝ : α → F'\ns : Set α\nE : Type u_8\ninst✝³ : NormedLatticeAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OrderedSMul ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\nhfg : f ≤ᶠ[ae μ] g\n⊢ condexpL1 hm μ f ≤ condexpL1 hm μ g","tactic":"rw [coeFn_le]","premises":[{"full_name":"MeasureTheory.Lp.coeFn_le","def_path":"Mathlib/MeasureTheory/Function/LpOrder.lean","def_pos":[38,8],"def_end_pos":[38,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁵ : RCLike 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : NormedAddCommGroup F'\ninst✝¹¹ : NormedSpace 𝕜 F'\ninst✝¹⁰ : NormedSpace ℝ F'\ninst✝⁹ : CompleteSpace F'\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup G'\ninst✝⁶ : NormedSpace ℝ G'\ninst✝⁵ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝⁴ : SigmaFinite (μ.trim hm)\nf✝ g✝ : α → F'\ns : Set α\nE : Type u_8\ninst✝³ : NormedLatticeAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OrderedSMul ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\nhfg : f ≤ᶠ[ae μ] g\n⊢ condexpL1 hm μ f ≤ condexpL1 hm μ g","state_after":"α : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁵ : RCLike 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : NormedAddCommGroup F'\ninst✝¹¹ : NormedSpace 𝕜 F'\ninst✝¹⁰ : NormedSpace ℝ F'\ninst✝⁹ : CompleteSpace F'\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup G'\ninst✝⁶ : NormedSpace ℝ G'\ninst✝⁵ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝⁴ : SigmaFinite (μ.trim hm)\nf✝ g✝ : α → F'\ns : Set α\nE : Type u_8\ninst✝³ : NormedLatticeAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OrderedSMul ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\nhfg : f ≤ᶠ[ae μ] g\nh_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), 0 ≤ x → 0 ≤ (condexpInd E hm μ s) x\n⊢ condexpL1 hm μ f ≤ condexpL1 hm μ g","tactic":"have h_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x : E, 0 ≤ x → 0 ≤ condexpInd E hm μ s x :=\n    fun s hs hμs x hx => condexpInd_nonneg hs hμs.ne x hx","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"MeasureTheory.condexpInd","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","def_pos":[243,4],"def_end_pos":[243,14]},{"full_name":"MeasureTheory.condexpInd_nonneg","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","def_pos":[327,8],"def_end_pos":[327,25]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁵ : RCLike 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : NormedAddCommGroup F'\ninst✝¹¹ : NormedSpace 𝕜 F'\ninst✝¹⁰ : NormedSpace ℝ F'\ninst✝⁹ : CompleteSpace F'\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup G'\ninst✝⁶ : NormedSpace ℝ G'\ninst✝⁵ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝⁴ : SigmaFinite (μ.trim hm)\nf✝ g✝ : α → F'\ns : Set α\nE : Type u_8\ninst✝³ : NormedLatticeAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OrderedSMul ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\nhfg : f ≤ᶠ[ae μ] g\nh_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), 0 ≤ x → 0 ≤ (condexpInd E hm μ s) x\n⊢ condexpL1 hm μ f ≤ condexpL1 hm μ g","state_after":"no goals","tactic":"exact setToFun_mono (dominatedFinMeasAdditive_condexpInd E hm μ) h_nonneg hf hg hfg","premises":[{"full_name":"MeasureTheory.dominatedFinMeasAdditive_condexpInd","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","def_pos":[297,8],"def_end_pos":[297,43]},{"full_name":"MeasureTheory.setToFun_mono","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[1331,8],"def_end_pos":[1331,21]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Splits.lean","commit":"","full_name":"Polynomial.adjoin_rootSet_eq_range","start":[287,0],"end":[291,63],"file_path":"Mathlib/Algebra/Polynomial/Splits.lean","tactics":[{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\ninst✝¹ : Algebra R K\ninst✝ : Algebra R L\np : R[X]\nh : Splits (algebraMap R K) p\nf : K →ₐ[R] L\n⊢ Algebra.adjoin R (p.rootSet L) = f.range ↔ Algebra.adjoin R (p.rootSet K) = ⊤","state_after":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\ninst✝¹ : Algebra R K\ninst✝ : Algebra R L\np : R[X]\nh : Splits (algebraMap R K) p\nf : K →ₐ[R] L\n⊢ Subalgebra.map f (Algebra.adjoin R (p.rootSet K)) = Subalgebra.map f ⊤ ↔ Algebra.adjoin R (p.rootSet K) = ⊤","tactic":"rw [← image_rootSet h f, Algebra.adjoin_image, ← Algebra.map_top]","premises":[{"full_name":"Algebra.adjoin_image","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[201,8],"def_end_pos":[201,20]},{"full_name":"Algebra.map_top","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[750,8],"def_end_pos":[750,15]},{"full_name":"Polynomial.image_rootSet","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[280,8],"def_end_pos":[280,21]}]},{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\ninst✝¹ : Algebra R K\ninst✝ : Algebra R L\np : R[X]\nh : Splits (algebraMap R K) p\nf : K →ₐ[R] L\n⊢ Subalgebra.map f (Algebra.adjoin R (p.rootSet K)) = Subalgebra.map f ⊤ ↔ Algebra.adjoin R (p.rootSet K) = ⊤","state_after":"no goals","tactic":"exact (Subalgebra.map_injective f.toRingHom.injective).eq_iff","premises":[{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"RingHom.injective","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[214,18],"def_end_pos":[214,27]},{"full_name":"Subalgebra.map_injective","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[363,8],"def_end_pos":[363,21]}]}]}
{"url":"Mathlib/Combinatorics/Configuration.lean","commit":"","full_name":"Configuration.HasLines.exists_bijective_of_card_eq","start":[240,0],"end":[247,94],"file_path":"Mathlib/Combinatorics/Configuration.lean","tactics":[{"state_before":"P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nh : Fintype.card P = Fintype.card L\n⊢ ∃ f, Function.Bijective f ∧ ∀ (l : L), pointCount P l = lineCount L (f l)","state_after":"no goals","tactic":"classical\n    obtain ⟨f, hf1, hf2⟩ := Nondegenerate.exists_injective_of_card_le (ge_of_eq h)\n    have hf3 := (Fintype.bijective_iff_injective_and_card f).mpr ⟨hf1, h.symm⟩\n    exact ⟨f, hf3, fun l ↦ (sum_eq_sum_iff_of_le fun l _ ↦ pointCount_le_lineCount (hf2 l)).1\n          ((hf3.sum_comp _).trans (sum_lineCount_eq_sum_pointCount P L)).symm _ <| mem_univ _⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Configuration.HasLines.pointCount_le_lineCount","def_path":"Mathlib/Combinatorics/Configuration.lean","def_pos":[188,8],"def_end_pos":[188,40]},{"full_name":"Configuration.Nondegenerate.exists_injective_of_card_le","def_path":"Mathlib/Combinatorics/Configuration.lean","def_pos":[117,8],"def_end_pos":[117,49]},{"full_name":"Configuration.sum_lineCount_eq_sum_pointCount","def_path":"Mathlib/Combinatorics/Configuration.lean","def_pos":[175,8],"def_end_pos":[175,39]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Finset.sum_eq_sum_iff_of_le","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[452,2],"def_end_pos":[452,13]},{"full_name":"Fintype.bijective_iff_injective_and_card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[600,8],"def_end_pos":[600,40]},{"full_name":"Function.Bijective.sum_comp","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1936,2],"def_end_pos":[1936,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"ge_of_eq","def_path":"Mathlib/Order/Basic.lean","def_pos":[287,8],"def_end_pos":[287,16]}]}]}
{"url":"Mathlib/Data/Real/ConjExponents.lean","commit":"","full_name":"Real.isConjExponent_iff_eq_conjExponent","start":[113,0],"end":[114,59],"file_path":"Mathlib/Data/Real/ConjExponents.lean","tactics":[{"state_before":"a b p q : ℝ\nh✝ : p.IsConjExponent q\nhp : 1 < p\nh : q = p / (p - 1)\n⊢ p⁻¹ + q⁻¹ = 1","state_after":"no goals","tactic":"field_simp [h]","premises":[]}]}
{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.le_one_iff_eq_zero_or_eq_one","start":[139,0],"end":[139,94],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\n⊢ ∀ {n : ℕ}, n ≤ 1 ↔ n = 0 ∨ n = 1","state_after":"no goals","tactic":"simp [le_succ_iff]","premises":[{"full_name":"Nat.le_succ_iff","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[111,6],"def_end_pos":[111,17]}]}]}
{"url":"Mathlib/Data/Multiset/Nodup.lean","commit":"","full_name":"Multiset.nodup_iff_count_le_one","start":[67,0],"end":[70,37],"file_path":"Mathlib/Data/Multiset/Nodup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t : Multiset α\na : α\ninst✝ : DecidableEq α\ns : Multiset α\n_l : List α\n⊢ Nodup (Quot.mk Setoid.r _l) ↔ ∀ (a : α), count a (Quot.mk Setoid.r _l) ≤ 1","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t : Multiset α\na : α\ninst✝ : DecidableEq α\ns : Multiset α\n_l : List α\n⊢ _l.Nodup ↔ ∀ (a : α), List.count a _l ≤ 1","tactic":"simp only [quot_mk_to_coe'', coe_nodup, mem_coe, coe_count]","premises":[{"full_name":"Multiset.coe_count","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[2066,8],"def_end_pos":[2066,17]},{"full_name":"Multiset.coe_nodup","def_path":"Mathlib/Data/Multiset/Nodup.lean","def_pos":[27,8],"def_end_pos":[27,17]},{"full_name":"Multiset.mem_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[211,8],"def_end_pos":[211,15]},{"full_name":"Multiset.quot_mk_to_coe''","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[51,8],"def_end_pos":[51,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t : Multiset α\na : α\ninst✝ : DecidableEq α\ns : Multiset α\n_l : List α\n⊢ _l.Nodup ↔ ∀ (a : α), List.count a _l ≤ 1","state_after":"no goals","tactic":"exact List.nodup_iff_count_le_one","premises":[{"full_name":"List.nodup_iff_count_le_one","def_path":"Mathlib/Data/List/Nodup.lean","def_pos":[163,8],"def_end_pos":[163,30]}]}]}
{"url":"Mathlib/Algebra/Polynomial/BigOperators.lean","commit":"","full_name":"Polynomial.degree_list_sum_le","start":[61,0],"end":[72,12],"file_path":"Mathlib/Algebra/Polynomial/BigOperators.lean","tactics":[{"state_before":"R : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\nl : List S[X]\n⊢ l.sum.degree ≤ (List.map natDegree l).maximum","state_after":"case pos\nR : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\nl : List S[X]\nh : l.sum = 0\n⊢ l.sum.degree ≤ (List.map natDegree l).maximum\n\ncase neg\nR : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\nl : List S[X]\nh : ¬l.sum = 0\n⊢ l.sum.degree ≤ (List.map natDegree l).maximum","tactic":"by_cases h : l.sum = 0","premises":[{"full_name":"List.sum","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[36,2],"def_end_pos":[36,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","commit":"","full_name":"NumberField.mixedEmbedding.convexBodySumFun_eq_zero_iff","start":[314,0],"end":[322,41],"file_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\n⊢ convexBodySumFun x = 0 ↔ x = 0","state_after":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\n⊢ (∀ i ∈ Finset.univ, ↑i.mult * (normAtPlace i) x = 0) ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0","tactic":"rw [← normAtPlace_eq_zero, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg fun _ _ =>\n    mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]","premises":[{"full_name":"Finset.sum_eq_zero_iff_of_nonneg","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[155,14],"def_end_pos":[155,39]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.cast_pos","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[53,8],"def_end_pos":[53,16]},{"full_name":"NumberField.InfinitePlace.mult_pos","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[424,8],"def_end_pos":[424,16]},{"full_name":"NumberField.mixedEmbedding.convexBodySumFun","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean","def_pos":[279,21],"def_end_pos":[279,37]},{"full_name":"NumberField.mixedEmbedding.normAtPlace_eq_zero","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","def_pos":[324,8],"def_end_pos":[324,27]},{"full_name":"NumberField.mixedEmbedding.normAtPlace_nonneg","def_path":"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean","def_pos":[281,8],"def_end_pos":[281,26]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\n⊢ (∀ i ∈ Finset.univ, ↑i.mult * (normAtPlace i) x = 0) ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0","state_after":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\n⊢ (∀ w ∈ Finset.univ, (normAtPlace w) x = 0) ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0","tactic":"conv =>\n    enter [1, w, hw]\n    rw [mul_left_mem_nonZeroDivisors_eq_zero_iff\n      (mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_zero)]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.cast_ne_zero","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[76,8],"def_end_pos":[76,20]},{"full_name":"NumberField.InfinitePlace.mult_ne_zero","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[429,8],"def_end_pos":[429,20]},{"full_name":"mem_nonZeroDivisors_iff_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[199,8],"def_end_pos":[199,39]},{"full_name":"mul_left_mem_nonZeroDivisors_eq_zero_iff","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[123,8],"def_end_pos":[123,48]}]},{"state_before":"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\n⊢ (∀ w ∈ Finset.univ, (normAtPlace w) x = 0) ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0","state_after":"no goals","tactic":"simp_rw [Finset.mem_univ, true_implies]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"true_implies","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[133,16],"def_end_pos":[133,28]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Basic.lean","commit":"","full_name":"Metric.uniformEmbedding_bot_of_pairwise_le_dist","start":[146,0],"end":[151,78],"file_path":"Mathlib/Topology/MetricSpace/Basic.lean","tactics":[{"state_before":"α : Type u\nβ✝ : Type v\nX : Type u_1\nι : Type u_2\ninst✝¹ : PseudoMetricSpace α\nγ : Type w\ninst✝ : MetricSpace γ\nx : γ\ns : Set γ\nβ : Type u_3\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y => ε ≤ dist (f x) (f y)\n⊢ UniformSpace α","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"α : Type u\nβ✝ : Type v\nX : Type u_1\nι : Type u_2\ninst✝¹ : PseudoMetricSpace α\nγ : Type w\ninst✝ : MetricSpace γ\nx : γ\ns : Set γ\nβ : Type u_3\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y => ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y => (f x, f y) ∉ {p | dist p.1 p.2 < ε}","state_after":"no goals","tactic":"simpa using hf","premises":[]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"OpenEmbedding.singleton_smoothManifoldWithCorners","start":[705,0],"end":[710,75],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : Nonempty M\nf : M → H\nh : OpenEmbedding f\n⊢ (toPartialHomeomorph f h).source = univ","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/SingleObj.lean","commit":"","full_name":"MulEquiv.toSingleObjEquiv_unitIso_inv","start":[200,0],"end":[212,14],"file_path":"Mathlib/CategoryTheory/SingleObj.lean","tactics":[{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ 𝟭 (SingleObj M) = e.toMonoidHom.toFunctor ⋙ e.symm.toMonoidHom.toFunctor","state_after":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (MonoidHom.id M).toFunctor = (e.symm.toMonoidHom.comp e.toMonoidHom).toFunctor","tactic":"rw [← MonoidHom.comp_toFunctor, ← MonoidHom.id_toFunctor]","premises":[{"full_name":"MonoidHom.comp_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[184,8],"def_end_pos":[184,22]},{"full_name":"MonoidHom.id_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[191,8],"def_end_pos":[191,20]}]},{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (MonoidHom.id M).toFunctor = (e.symm.toMonoidHom.comp e.toMonoidHom).toFunctor","state_after":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ MonoidHom.id M = e.symm.toMonoidHom.comp e.toMonoidHom","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ MonoidHom.id M = e.symm.toMonoidHom.comp e.toMonoidHom","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.symm.toMonoidHom.toFunctor ⋙ e.toMonoidHom.toFunctor = 𝟭 (SingleObj N)","state_after":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (e.toMonoidHom.comp e.symm.toMonoidHom).toFunctor = (MonoidHom.id N).toFunctor","tactic":"rw [← MonoidHom.comp_toFunctor, ← MonoidHom.id_toFunctor]","premises":[{"full_name":"MonoidHom.comp_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[184,8],"def_end_pos":[184,22]},{"full_name":"MonoidHom.id_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[191,8],"def_end_pos":[191,20]}]},{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (e.toMonoidHom.comp e.symm.toMonoidHom).toFunctor = (MonoidHom.id N).toFunctor","state_after":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.toMonoidHom.comp e.symm.toMonoidHom = MonoidHom.id N","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.toMonoidHom.comp e.symm.toMonoidHom = MonoidHom.id N","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/End.lean","commit":"","full_name":"CategoryTheory.μ_inv_naturalityᵣ","start":[166,0],"end":[171,6],"file_path":"Mathlib/CategoryTheory/Monoidal/End.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nM : Type u_1\ninst✝¹ : Category.{u_2, u_1} M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm n n' : M\ng : n ⟶ n'\nX : C\n⊢ (F.μIso m n).inv.app X ≫ (F.map g).app ((F.obj m).obj X) = (F.map (m ◁ g)).app X ≫ (F.μIso m n').inv.app X","state_after":"C : Type u\ninst✝² : Category.{v, u} C\nM : Type u_1\ninst✝¹ : Category.{u_2, u_1} M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm n n' : M\ng : n ⟶ n'\nX : C\n⊢ (F.map g).app ((F.obj m).obj X) ≫ inv ((F.μIso m n').inv.app X) = inv ((F.μIso m n).inv.app X) ≫ (F.map (m ◁ g)).app X","tactic":"rw [← IsIso.comp_inv_eq, Category.assoc, ← IsIso.eq_inv_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.IsIso.comp_inv_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[381,8],"def_end_pos":[381,19]},{"full_name":"CategoryTheory.IsIso.eq_inv_comp","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[377,8],"def_end_pos":[377,19]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nM : Type u_1\ninst✝¹ : Category.{u_2, u_1} M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm n n' : M\ng : n ⟶ n'\nX : C\n⊢ (F.map g).app ((F.obj m).obj X) ≫ inv ((F.μIso m n').inv.app X) = inv ((F.μIso m n).inv.app X) ≫ (F.map (m ◁ g)).app X","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.erase_lt","start":[955,0],"end":[958,47],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝ : DecidableEq α\ns✝ t : Multiset α\na✝ b a : α\ns : Multiset α\nh : a ∈ s\n⊢ s.erase a < s","state_after":"no goals","tactic":"simpa [h] using lt_cons_self (s.erase a) a","premises":[{"full_name":"Multiset.erase","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[890,4],"def_end_pos":[890,9]},{"full_name":"Multiset.lt_cons_self","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[498,8],"def_end_pos":[498,20]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Laurent.lean","commit":"","full_name":"LaurentPolynomial.toLaurent_reverse","start":[543,0],"end":[549,30],"file_path":"Mathlib/Algebra/Polynomial/Laurent.lean","tactics":[{"state_before":"R✝ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\np : R[X]\n⊢ toLaurent p.reverse = invert (toLaurent p) * T ↑p.natDegree","state_after":"R✝ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\np : R[X]\na✝ : Nontrivial R\n⊢ toLaurent p.reverse = invert (toLaurent p) * T ↑p.natDegree","tactic":"nontriviality R","premises":[]},{"state_before":"R✝ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\np : R[X]\na✝ : Nontrivial R\n⊢ toLaurent p.reverse = invert (toLaurent p) * T ↑p.natDegree","state_after":"case M0\nR✝ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\na✝ : Nontrivial R\n⊢ toLaurent (reverse 0) = invert (toLaurent 0) * T ↑(natDegree 0)\n\ncase MC\nR✝ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\na✝² : Nontrivial R\np : R[X]\nt : R\na✝¹ : p.coeff 0 = 0\na✝ : t ≠ 0\nih : toLaurent p.reverse = invert (toLaurent p) * T ↑p.natDegree\n⊢ toLaurent (p + Polynomial.C t).reverse = invert (toLaurent (p + Polynomial.C t)) * T ↑(p + Polynomial.C t).natDegree\n\ncase MX\nR✝ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\na✝ : Nontrivial R\np : R[X]\nhp : p ≠ 0\nih : toLaurent p.reverse = invert (toLaurent p) * T ↑p.natDegree\n⊢ toLaurent (p * X).reverse = invert (toLaurent (p * X)) * T ↑(p * X).natDegree","tactic":"induction' p using Polynomial.recOnHorner with p t _ _ ih p hp ih","premises":[{"full_name":"Polynomial.recOnHorner","def_path":"Mathlib/Algebra/Polynomial/Inductions.lean","def_pos":[133,18],"def_end_pos":[133,29]}]}]}
{"url":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","commit":"","full_name":"UV.sup_sdiff_mem_of_mem_compression","start":[230,0],"end":[247,52],"file_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\ninst✝ : DecidableEq α\nha : a ∈ 𝓒 u v s\nhva : v ≤ a\nhua : Disjoint u a\n⊢ (a ⊔ u) \\ v ∈ s","state_after":"α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\ninst✝ : DecidableEq α\nha : a ∈ s ∧ (a ⊔ u) \\ v ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a\nhva : v ≤ a\nhua : Disjoint u a\n⊢ (a ⊔ u) \\ v ∈ s","tactic":"rw [mem_compression, compress_of_disjoint_of_le hua hva] at ha","premises":[{"full_name":"UV.compress_of_disjoint_of_le","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[81,8],"def_end_pos":[81,34]},{"full_name":"UV.mem_compression","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[144,8],"def_end_pos":[144,23]}]},{"state_before":"α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\ninst✝ : DecidableEq α\nha : a ∈ s ∧ (a ⊔ u) \\ v ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a\nhva : v ≤ a\nhua : Disjoint u a\n⊢ (a ⊔ u) \\ v ∈ s","state_after":"case inl.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\ninst✝ : DecidableEq α\nhva : v ≤ a\nhua : Disjoint u a\nleft✝ : a ∈ s\nha : (a ⊔ u) \\ v ∈ s\n⊢ (a ⊔ u) \\ v ∈ s\n\ncase inr.intro.intro.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\n⊢ (compress u v b ⊔ u) \\ v ∈ s","tactic":"obtain ⟨_, ha⟩ | ⟨_, b, hb, rfl⟩ := ha","premises":[]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\n⊢ (compress u v b ⊔ u) \\ v ∈ s","state_after":"case inr.intro.intro.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\nhu : u = ⊥\n⊢ (compress u v b ⊔ u) \\ v ∈ s","tactic":"have hu : u = ⊥ := by\n    suffices Disjoint u (u \\ v) by rwa [(hua.mono_right hva).sdiff_eq_left, disjoint_self] at this\n    refine hua.mono_right ?_\n    rw [← compress_idem, compress_of_disjoint_of_le hua hva]\n    exact sdiff_le_sdiff_right le_sup_right","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Disjoint","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[40,4],"def_end_pos":[40,12]},{"full_name":"Disjoint.mono_right","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[69,8],"def_end_pos":[69,27]},{"full_name":"Disjoint.sdiff_eq_left","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[524,8],"def_end_pos":[524,30]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"UV.compress_idem","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[107,8],"def_end_pos":[107,21]},{"full_name":"UV.compress_of_disjoint_of_le","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[81,8],"def_end_pos":[81,34]},{"full_name":"disjoint_self","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[73,8],"def_end_pos":[73,21]},{"full_name":"le_sup_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[112,8],"def_end_pos":[112,20]},{"full_name":"sdiff_le_sdiff_right","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[501,8],"def_end_pos":[501,28]}]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\nhu : u = ⊥\n⊢ (compress u v b ⊔ u) \\ v ∈ s","state_after":"case inr.intro.intro.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\nhu : u = ⊥\nhv : v = ⊥\n⊢ (compress u v b ⊔ u) \\ v ∈ s","tactic":"have hv : v = ⊥ := by\n    rw [← disjoint_self]\n    apply Disjoint.mono_right hva\n    rw [← compress_idem, compress_of_disjoint_of_le hua hva]\n    exact disjoint_sdiff_self_right","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Disjoint.mono_right","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[69,8],"def_end_pos":[69,27]},{"full_name":"UV.compress_idem","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[107,8],"def_end_pos":[107,21]},{"full_name":"UV.compress_of_disjoint_of_le","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[81,8],"def_end_pos":[81,34]},{"full_name":"disjoint_sdiff_self_right","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[198,8],"def_end_pos":[198,33]},{"full_name":"disjoint_self","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[73,8],"def_end_pos":[73,21]}]},{"state_before":"case inr.intro.intro.intro\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\nhu : u = ⊥\nhv : v = ⊥\n⊢ (compress u v b ⊔ u) \\ v ∈ s","state_after":"no goals","tactic":"rwa [hu, hv, compress_self, sup_bot_eq, sdiff_bot]","premises":[{"full_name":"UV.compress_self","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[92,8],"def_end_pos":[92,21]},{"full_name":"sdiff_bot","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[449,8],"def_end_pos":[449,17]},{"full_name":"sup_bot_eq","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[361,8],"def_end_pos":[361,18]}]}]}
{"url":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.prod_le_pow_card","start":[187,0],"end":[192,8],"file_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf✝ g : ι → N\ns✝ t s : Finset ι\nf : ι → N\nn : N\nh : ∀ x ∈ s, f x ≤ n\n⊢ s.prod f ≤ n ^ s.card","state_after":"case refine_1\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf✝ g : ι → N\ns✝ t s : Finset ι\nf : ι → N\nn : N\nh : ∀ x ∈ s, f x ≤ n\n⊢ ∀ x ∈ Multiset.map f s.val, x ≤ n\n\ncase refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf✝ g : ι → N\ns✝ t s : Finset ι\nf : ι → N\nn : N\nh : ∀ x ∈ s, f x ≤ n\n⊢ n ^ Multiset.card (Multiset.map f s.val) ≤ n ^ s.card","tactic":"refine (Multiset.prod_le_pow_card (s.val.map f) n ?_).trans ?_","premises":[{"full_name":"Finset.val","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[135,2],"def_end_pos":[135,5]},{"full_name":"Multiset.map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1012,4],"def_end_pos":[1012,7]},{"full_name":"Multiset.prod_le_pow_card","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean","def_pos":[37,6],"def_end_pos":[37,22]}]}]}
{"url":"Mathlib/Data/Finset/NoncommProd.lean","commit":"","full_name":"Finset.noncommProd_singleton","start":[307,0],"end":[313,22],"file_path":"Mathlib/Data/Finset/NoncommProd.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\na : α\nf : α → β\n⊢ (↑{a}).Pairwise fun a b => Commute (f a) (f b)","state_after":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\na : α\nf : α → β\n⊢ {a}.Pairwise fun a b => Commute (f a) (f b)","tactic":"norm_cast","premises":[]},{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\na : α\nf : α → β\n⊢ {a}.Pairwise fun a b => Commute (f a) (f b)","state_after":"no goals","tactic":"exact Set.pairwise_singleton _ _","premises":[{"full_name":"Set.pairwise_singleton","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[76,8],"def_end_pos":[76,26]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","commit":"","full_name":"HasDerivAt.mul_const","start":[224,0],"end":[227,22],"file_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nG : Type u_1\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_2\n𝔸 : Type u_3\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d✝ : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nhc : HasDerivAt c c' x\nd : 𝔸\n⊢ HasDerivAt (fun y => c y * d) (c' * d) x","state_after":"𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nG : Type u_1\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_2\n𝔸 : Type u_3\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d✝ : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nhc : HasDerivWithinAt c c' univ x\nd : 𝔸\n⊢ HasDerivWithinAt (fun y => c y * d) (c' * d) univ x","tactic":"rw [← hasDerivWithinAt_univ] at *","premises":[{"full_name":"hasDerivWithinAt_univ","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[340,8],"def_end_pos":[340,29]}]},{"state_before":"𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nG : Type u_1\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_2\n𝔸 : Type u_3\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d✝ : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nhc : HasDerivWithinAt c c' univ x\nd : 𝔸\n⊢ HasDerivWithinAt (fun y => c y * d) (c' * d) univ x","state_after":"no goals","tactic":"exact hc.mul_const d","premises":[{"full_name":"HasDerivWithinAt.mul_const","def_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","def_pos":[219,8],"def_end_pos":[219,34]}]}]}
{"url":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","commit":"","full_name":"UpperHalfPlane.ModularGroup.denom_apply","start":[398,0],"end":[400,20],"file_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","tactics":[{"state_before":"g✝ : SL(2, ℤ)\nz✝ : ℍ\nΓ : Subgroup SL(2, ℤ)\ng : SL(2, ℤ)\nz : ℍ\n⊢ denom (↑g) z = ↑(↑g 1 0) * ↑z + ↑(↑g 1 1)","state_after":"no goals","tactic":"simp [denom, coe']","premises":[{"full_name":"UpperHalfPlane.ModularGroup.coe'","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[276,4],"def_end_pos":[276,8]},{"full_name":"UpperHalfPlane.denom","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[180,4],"def_end_pos":[180,9]}]}]}
{"url":"Mathlib/Algebra/Field/Basic.lean","commit":"","full_name":"same_add_div","start":[33,0],"end":[33,91],"file_path":"Mathlib/Algebra/Field/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nK : Type u_3\ninst✝ : DivisionSemiring α\na b c d : α\nh : b ≠ 0\n⊢ (b + a) / b = 1 + a / b","state_after":"no goals","tactic":"rw [← div_self h, add_div]","premises":[{"full_name":"add_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[27,8],"def_end_pos":[27,15]},{"full_name":"div_self","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[251,14],"def_end_pos":[251,22]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.Valid'.node4L_lemma₄","start":[994,0],"end":[995,63],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Preorder α\na b c d : ℕ\nlr₁ : 3 * a ≤ b + c + 1 + d\nmr₂ : b + c + 1 ≤ 3 * d\nmm₁ : b ≤ 3 * c\n⊢ a + b + 1 ≤ 3 * (c + d + 1)","state_after":"no goals","tactic":"omega","premises":[]}]}
{"url":".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean","commit":"","full_name":"Fin.foldl_succ_last","start":[85,0],"end":[90,83],"file_path":".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean","tactics":[{"state_before":"α : Sort u_1\nn : Nat\nf : α → Fin (n + 1) → α\nx : α\n⊢ foldl (n + 1) f x = f (foldl n (fun x x_1 => f x x_1.castSucc) x) (last n)","state_after":"α : Sort u_1\nn : Nat\nf : α → Fin (n + 1) → α\nx : α\n⊢ foldl n (fun x i => f x i.succ) (f x 0) = f (foldl n (fun x x_1 => f x x_1.castSucc) x) (last n)","tactic":"rw [foldl_succ]","premises":[{"full_name":"Fin.foldl_succ","def_path":".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean","def_pos":[82,8],"def_end_pos":[82,18]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","commit":"","full_name":"EuclideanGeometry.oangle_sign_eq_zero_iff_collinear","start":[201,0],"end":[204,73],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\n⊢ (∡ p₁ p₂ p₃).sign = 0 ↔ Collinear ℝ {p₁, p₂, p₃}","state_after":"no goals","tactic":"rw [Real.Angle.sign_eq_zero_iff, oangle_eq_zero_or_eq_pi_iff_collinear]","premises":[{"full_name":"EuclideanGeometry.oangle_eq_zero_or_eq_pi_iff_collinear","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean","def_pos":[196,8],"def_end_pos":[196,45]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Real.Angle.sign_eq_zero_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[744,8],"def_end_pos":[744,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","commit":"","full_name":"CategoryTheory.Limits.sigmaComparison_map_desc","start":[574,0],"end":[580,32],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","tactics":[{"state_before":"β : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\n⊢ sigmaComparison G f ≫ G.map (Sigma.desc g) = Sigma.desc fun j => G.map (g j)","state_after":"case h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\nj : β\n⊢ Sigma.ι (fun b => G.obj (f b)) j ≫ sigmaComparison G f ≫ G.map (Sigma.desc g) =\n    Sigma.ι (fun b => G.obj (f b)) j ≫ Sigma.desc fun j => G.map (g j)","tactic":"ext j","premises":[]},{"state_before":"case h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\nj : β\n⊢ Sigma.ι (fun b => G.obj (f b)) j ≫ sigmaComparison G f ≫ G.map (Sigma.desc g) =\n    Sigma.ι (fun b => G.obj (f b)) j ≫ Sigma.desc fun j => G.map (g j)","state_after":"no goals","tactic":"simp only [Discrete.functor_obj, ι_comp_sigmaComparison_assoc, ← G.map_comp, colimit.ι_desc,\n    Cofan.mk_pt, Cofan.mk_ι_app]","premises":[{"full_name":"CategoryTheory.Discrete.functor_obj","def_path":"Mathlib/CategoryTheory/DiscreteCategory.lean","def_pos":[168,8],"def_end_pos":[168,19]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Limits.Cofan.mk_pt","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[63,9],"def_end_pos":[63,11]},{"full_name":"CategoryTheory.Limits.Cofan.mk_ι_app","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[63,12],"def_end_pos":[63,17]},{"full_name":"CategoryTheory.Limits.colimit.ι_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[691,8],"def_end_pos":[691,22]}]}]}
{"url":"Mathlib/Topology/Separation.lean","commit":"","full_name":"IsCompact.binary_compact_cover","start":[1121,0],"end":[1135,76],"file_path":"Mathlib/Topology/Separation.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","state_after":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","tactic":"have hK' : IsCompact (closure K) := hK.closure","premises":[{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.closure","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1089,18],"def_end_pos":[1089,35]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]}]},{"state_before":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","state_after":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\nthis : SeparatedNhds (closure K \\ U) (closure K \\ V)\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","tactic":"have : SeparatedNhds (closure K \\ U) (closure K \\ V) := by\n    apply SeparatedNhds.of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)\n      (isClosed_closure.sdiff hV)\n    rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]\n    exact hK.closure_subset_of_isOpen (hU.union hV) h2K","premises":[{"full_name":"IsClosed.sdiff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[186,8],"def_end_pos":[186,22]},{"full_name":"IsCompact.closure_subset_of_isOpen","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1083,8],"def_end_pos":[1083,42]},{"full_name":"IsCompact.diff","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[89,8],"def_end_pos":[89,22]},{"full_name":"IsOpen.union","def_path":"Mathlib/Topology/Basic.lean","def_pos":[106,8],"def_end_pos":[106,20]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"SeparatedNhds","def_path":"Mathlib/Topology/Separation.lean","def_pos":[142,4],"def_end_pos":[142,17]},{"full_name":"SeparatedNhds.of_isCompact_isCompact_isClosed","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1111,8],"def_end_pos":[1111,53]},{"full_name":"Set.diff_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1499,8],"def_end_pos":[1499,21]},{"full_name":"Set.diff_inter_diff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1543,8],"def_end_pos":[1543,23]},{"full_name":"Set.disjoint_iff_inter_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1210,8],"def_end_pos":[1210,35]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]}]},{"state_before":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\nthis : SeparatedNhds (closure K \\ U) (closure K \\ V)\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","state_after":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\nthis✝ : SeparatedNhds (closure K \\ U) (closure K \\ V)\nthis : SeparatedNhds (K \\ U) (K \\ V)\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","tactic":"have : SeparatedNhds (K \\ U) (K \\ V) :=\n    this.mono (diff_subset_diff_left (subset_closure)) (diff_subset_diff_left (subset_closure))","premises":[{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"SeparatedNhds","def_path":"Mathlib/Topology/Separation.lean","def_pos":[142,4],"def_end_pos":[142,17]},{"full_name":"SeparatedNhds.mono","def_path":"Mathlib/Topology/Separation.lean","def_pos":[251,8],"def_end_pos":[251,12]},{"full_name":"Set.diff_subset_diff_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1485,8],"def_end_pos":[1485,29]},{"full_name":"subset_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[347,8],"def_end_pos":[347,22]}]},{"state_before":"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\nthis✝ : SeparatedNhds (closure K \\ U) (closure K \\ V)\nthis : SeparatedNhds (K \\ U) (K \\ V)\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","state_after":"case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\nthis : SeparatedNhds (closure K \\ U) (closure K \\ V)\nO₁ O₂ : Set X\nh1O₁ : IsOpen O₁\nh1O₂ : IsOpen O₂\nh2O₁ : K \\ U ⊆ O₁\nh2O₂ : K \\ V ⊆ O₂\nhO : Disjoint O₁ O₂\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","tactic":"rcases this with ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\ninst✝ : R1Space X\nx y : X\nK U V : Set X\nhK : IsCompact K\nhU : IsOpen U\nhV : IsOpen V\nh2K : K ⊆ U ∪ V\nhK' : IsCompact (closure K)\nthis : SeparatedNhds (closure K \\ U) (closure K \\ V)\nO₁ O₂ : Set X\nh1O₁ : IsOpen O₁\nh1O₂ : IsOpen O₂\nh2O₁ : K \\ U ⊆ O₁\nh2O₂ : K \\ V ⊆ O₂\nhO : Disjoint O₁ O₂\n⊢ ∃ K₁ K₂, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂","state_after":"no goals","tactic":"exact ⟨K \\ O₁, K \\ O₂, hK.diff h1O₁, hK.diff h1O₂, diff_subset_comm.mp h2O₁,\n    diff_subset_comm.mp h2O₂, by rw [← diff_inter, hO.inter_eq, diff_empty]⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Disjoint.inter_eq","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1213,8],"def_end_pos":[1213,32]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsCompact.diff","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[89,8],"def_end_pos":[89,22]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"Set.diff_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1503,8],"def_end_pos":[1503,18]},{"full_name":"Set.diff_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1540,8],"def_end_pos":[1540,18]},{"full_name":"Set.diff_subset_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1537,8],"def_end_pos":[1537,24]}]}]}
{"url":"Mathlib/RingTheory/Coprime/Lemmas.lean","commit":"","full_name":"IsCoprime.prod_right","start":[63,0],"end":[64,64],"file_path":"Mathlib/RingTheory/Coprime/Lemmas.lean","tactics":[{"state_before":"R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\n⊢ (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i)","state_after":"no goals","tactic":"simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)","premises":[{"full_name":"IsCoprime.prod_left","def_path":"Mathlib/RingTheory/Coprime/Lemmas.lean","def_pos":[56,8],"def_end_pos":[56,27]},{"full_name":"isCoprime_comm","def_path":"Mathlib/RingTheory/Coprime/Basic.lean","def_pos":[48,8],"def_end_pos":[48,22]}]}]}
{"url":"Mathlib/CategoryTheory/Square.lean","commit":"","full_name":"CategoryTheory.Square.toArrowArrowFunctor'_map_right_left","start":[205,0],"end":[212,51],"file_path":"Mathlib/CategoryTheory/Square.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : Square C\nφ : X✝ ⟶ Y✝\n⊢ Arrow.homMk ⋯ ≫ ((fun sq => Arrow.mk (Arrow.homMk ⋯)) Y✝).hom =\n    ((fun sq => Arrow.mk (Arrow.homMk ⋯)) X✝).hom ≫ Arrow.homMk ⋯","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","commit":"","full_name":"AlgebraicGeometry.PresheafedSpace.restrictTopIso_inv","start":[370,0],"end":[387,9],"file_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","tactics":[{"state_before":"C : Type u_1\ninst✝ : Category.{?u.84251, u_1} C\nX : PresheafedSpace C\n⊢ X.ofRestrict ⋯ ≫ X.toRestrictTop = 𝟙 (X.restrict ⋯)","state_after":"case w.w\nC : Type u_1\ninst✝ : Category.{?u.84251, u_1} C\nX : PresheafedSpace C\nx✝ : (CategoryTheory.forget TopCat).obj ↑(X.restrict ⋯)\n⊢ (X.ofRestrict ⋯ ≫ X.toRestrictTop).base x✝ = (𝟙 (X.restrict ⋯)).base x✝\n\ncase h.w\nC : Type u_1\ninst✝ : Category.{?u.84251, u_1} C\nX : PresheafedSpace C\nU✝ : Opens ↑↑(X.restrict ⋯)\n⊢ ((X.ofRestrict ⋯ ≫ X.toRestrictTop).c ≫ whiskerRight (eqToHom ⋯) (X.restrict ⋯).presheaf).app (op U✝) =\n    (𝟙 (X.restrict ⋯)).c.app (op U✝)","tactic":"ext","premises":[]},{"state_before":"C : Type u_1\ninst✝ : Category.{?u.84251, u_1} C\nX : PresheafedSpace C\n⊢ X.toRestrictTop ≫ X.ofRestrict ⋯ = 𝟙 X","state_after":"case w.w\nC : Type u_1\ninst✝ : Category.{?u.84251, u_1} C\nX : PresheafedSpace C\nx✝ : (CategoryTheory.forget TopCat).obj ↑X\n⊢ (X.toRestrictTop ≫ X.ofRestrict ⋯).base x✝ = (𝟙 X).base x✝\n\ncase h.w\nC : Type u_1\ninst✝ : Category.{?u.84251, u_1} C\nX : PresheafedSpace C\nU✝ : Opens ↑↑X\n⊢ ((X.toRestrictTop ≫ X.ofRestrict ⋯).c ≫ whiskerRight (eqToHom ⋯) X.presheaf).app (op U✝) = (𝟙 X).c.app (op U✝)","tactic":"ext","premises":[]}]}
{"url":"Mathlib/Probability/Process/Stopping.lean","commit":"","full_name":"MeasureTheory.stoppedProcess_eq_of_mem_finset","start":[828,0],"end":[848,21],"file_path":"Mathlib/Probability/Process/Stopping.lean","tactics":[{"state_before":"Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\n⊢ stoppedProcess u τ n =\n    {a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = i}.indicator (u i)","state_after":"case h\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\nω : Ω\n⊢ stoppedProcess u τ n ω =\n    ({a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = i}.indicator (u i)) ω","tactic":"ext ω","premises":[]},{"state_before":"case h\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\nω : Ω\n⊢ stoppedProcess u τ n ω =\n    ({a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = i}.indicator (u i)) ω","state_after":"case h\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\nω : Ω\n⊢ stoppedProcess u τ n ω =\n    {a | n ≤ τ a}.indicator (u n) ω + ∑ c ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = c}.indicator (u c) ω","tactic":"rw [Pi.add_apply, Finset.sum_apply]","premises":[{"full_name":"Finset.sum_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[32,2],"def_end_pos":[32,13]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]}]},{"state_before":"case h\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\nω : Ω\n⊢ stoppedProcess u τ n ω =\n    {a | n ≤ τ a}.indicator (u n) ω + ∑ c ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = c}.indicator (u c) ω","state_after":"case h.inl\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\nω : Ω\nh : n ≤ τ ω\n⊢ stoppedProcess u τ n ω =\n    {a | n ≤ τ a}.indicator (u n) ω + ∑ c ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = c}.indicator (u c) ω\n\ncase h.inr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\nμ : Measure Ω\nτ σ : Ω → ι\nE : Type u_4\np : ℝ≥0∞\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s\nω : Ω\nh : τ ω < n\n⊢ stoppedProcess u τ n ω =\n    {a | n ≤ τ a}.indicator (u n) ω + ∑ c ∈ Finset.filter (fun x => x < n) s, {ω | τ ω = c}.indicator (u c) ω","tactic":"rcases le_or_lt n (τ ω) with h | h","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]}]}
{"url":"Mathlib/Topology/Order/UpperLowerSetTopology.lean","commit":"","full_name":"Topology.IsUpperSet.closure_eq_lowerClosure","start":[237,0],"end":[242,41],"file_path":"Mathlib/Topology/Order/UpperLowerSetTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns✝ s : Set α\n⊢ closure s = ↑(lowerClosure s)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns✝ s : Set α\n⊢ closure s ⊆ ↑(lowerClosure s) ∧ ↑(lowerClosure s) ⊆ closure s","tactic":"rw [subset_antisymm_iff]","premises":[{"full_name":"subset_antisymm_iff","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[566,8],"def_end_pos":[566,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns✝ s : Set α\n⊢ closure s ⊆ ↑(lowerClosure s) ∧ ↑(lowerClosure s) ⊆ closure s","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns✝ s : Set α\n⊢ closure s ⊆ ↑(lowerClosure s)","tactic":"refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Topology.IsUpperSet.isClosed_iff_isLower","def_path":"Mathlib/Topology/Order/UpperLowerSetTopology.lean","def_pos":[233,6],"def_end_pos":[233,26]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]},{"full_name":"lowerClosure_min","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[1278,8],"def_end_pos":[1278,24]},{"full_name":"subset_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[347,8],"def_end_pos":[347,22]}]}]}
{"url":"Mathlib/GroupTheory/Abelianization.lean","commit":"","full_name":"Abelianization.equivOfComm_apply","start":[226,0],"end":[235,11],"file_path":"Mathlib/GroupTheory/Abelianization.lean","tactics":[{"state_before":"G : Type u\ninst✝¹ : Group G\nH : Type u_1\ninst✝ : CommGroup H\n⊢ Function.RightInverse ⇑(lift (MonoidHom.id H)) ⇑of","state_after":"case mk\nG : Type u\ninst✝¹ : Group G\nH : Type u_1\ninst✝ : CommGroup H\nx✝ : Abelianization H\na : H\n⊢ of ((lift (MonoidHom.id H)) (Quot.mk Setoid.r a)) = Quot.mk Setoid.r a","tactic":"rintro ⟨a⟩","premises":[]},{"state_before":"case mk\nG : Type u\ninst✝¹ : Group G\nH : Type u_1\ninst✝ : CommGroup H\nx✝ : Abelianization H\na : H\n⊢ of ((lift (MonoidHom.id H)) (Quot.mk Setoid.r a)) = Quot.mk Setoid.r a","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/SetTheory/Ordinal/Topology.lean","commit":"","full_name":"Ordinal.isClosed_iff_sup","start":[144,0],"end":[151,17],"file_path":"Mathlib/SetTheory/Ordinal/Topology.lean","tactics":[{"state_before":"s : Set Ordinal.{u}\na : Ordinal.{u}\n⊢ IsClosed s ↔ ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s","state_after":"case mpr\ns : Set Ordinal.{u}\na : Ordinal.{u}\n⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) → IsClosed s","tactic":"use fun hs ι hι f hf => (mem_closed_iff_sup hs).2 ⟨ι, hι, f, hf, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ordinal.mem_closed_iff_sup","def_path":"Mathlib/SetTheory/Ordinal/Topology.lean","def_pos":[127,8],"def_end_pos":[127,26]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case mpr\ns : Set Ordinal.{u}\na : Ordinal.{u}\n⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) → IsClosed s","state_after":"case mpr\ns : Set Ordinal.{u}\na : Ordinal.{u}\n⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) → closure s ⊆ s","tactic":"rw [← closure_subset_iff_isClosed]","premises":[{"full_name":"closure_subset_iff_isClosed","def_path":"Mathlib/Topology/Basic.lean","def_pos":[397,8],"def_end_pos":[397,35]}]},{"state_before":"case mpr\ns : Set Ordinal.{u}\na : Ordinal.{u}\n⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) → closure s ⊆ s","state_after":"case mpr\ns : Set Ordinal.{u}\na : Ordinal.{u}\nh : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s\nx : Ordinal.{u}\nhx : x ∈ closure s\n⊢ x ∈ s","tactic":"intro h x hx","premises":[]},{"state_before":"case mpr\ns : Set Ordinal.{u}\na : Ordinal.{u}\nh : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s\nx : Ordinal.{u}\nhx : x ∈ closure s\n⊢ x ∈ s","state_after":"case mpr.intro.intro.intro.intro\ns : Set Ordinal.{u}\na : Ordinal.{u}\nh : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s\nι : Type u\nhι : Nonempty ι\nf : ι → Ordinal.{u}\nhf : ∀ (i : ι), f i ∈ s\nhx : sup f ∈ closure s\n⊢ sup f ∈ s","tactic":"rcases mem_closure_iff_sup.1 hx with ⟨ι, hι, f, hf, rfl⟩","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ordinal.mem_closure_iff_sup","def_path":"Mathlib/SetTheory/Ordinal/Topology.lean","def_pos":[122,8],"def_end_pos":[122,27]}]},{"state_before":"case mpr.intro.intro.intro.intro\ns : Set Ordinal.{u}\na : Ordinal.{u}\nh : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s\nι : Type u\nhι : Nonempty ι\nf : ι → Ordinal.{u}\nhf : ∀ (i : ι), f i ∈ s\nhx : sup f ∈ closure s\n⊢ sup f ∈ s","state_after":"no goals","tactic":"exact h hι f hf","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Finsupp.lean","commit":"","full_name":"Finsupp.lcongr_symm_single","start":[916,0],"end":[919,6],"file_path":"Mathlib/LinearAlgebra/Finsupp.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nι : Type u_7\nκ : Type u_8\ne₁ : ι ≃ κ\ne₂ : M ≃ₗ[R] N\nk : κ\nn : N\n⊢ (lcongr e₁ e₂).symm (single k n) = single (e₁.symm k) (e₂.symm n)","state_after":"α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nι : Type u_7\nκ : Type u_8\ne₁ : ι ≃ κ\ne₂ : M ≃ₗ[R] N\nk : κ\nn : N\n⊢ (lcongr e₁ e₂) ((lcongr e₁ e₂).symm (single k n)) = (lcongr e₁ e₂) (single (e₁.symm k) (e₂.symm n))","tactic":"apply_fun (lcongr e₁ e₂ : (ι →₀ M) → (κ →₀ N)) using (lcongr e₁ e₂).injective","premises":[{"full_name":"Finsupp","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[91,10],"def_end_pos":[91,17]},{"full_name":"Finsupp.lcongr","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[904,4],"def_end_pos":[904,10]},{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]},{"full_name":"LinearEquiv.injective","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[488,18],"def_end_pos":[488,27]}]},{"state_before":"α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nι : Type u_7\nκ : Type u_8\ne₁ : ι ≃ κ\ne₂ : M ≃ₗ[R] N\nk : κ\nn : N\n⊢ (lcongr e₁ e₂) ((lcongr e₁ e₂).symm (single k n)) = (lcongr e₁ e₂) (single (e₁.symm k) (e₂.symm n))","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","commit":"","full_name":"AlgHom.coe_range","start":[510,0],"end":[514,5],"file_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","tactics":[{"state_before":"R' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\n⊢ ↑φ.range = Set.range ⇑φ","state_after":"case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\nx✝ : B\n⊢ x✝ ∈ ↑φ.range ↔ x✝ ∈ Set.range ⇑φ","tactic":"ext","premises":[]},{"state_before":"case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\nx✝ : B\n⊢ x✝ ∈ ↑φ.range ↔ x✝ ∈ Set.range ⇑φ","state_after":"case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\nx✝ : B\n⊢ (∃ x, φ x = x✝) ↔ x✝ ∈ Set.range ⇑φ","tactic":"rw [SetLike.mem_coe, mem_range]","premises":[{"full_name":"AlgHom.mem_range","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[504,8],"def_end_pos":[504,17]},{"full_name":"SetLike.mem_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[168,8],"def_end_pos":[168,15]}]},{"state_before":"case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\nx✝ : B\n⊢ (∃ x, φ x = x✝) ↔ x✝ ∈ Set.range ⇑φ","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/ModelTheory/DirectLimit.lean","commit":"","full_name":"FirstOrder.Language.DirectLimit.equiv_iff","start":[172,0],"end":[182,9],"file_path":"Mathlib/ModelTheory/DirectLimit.lean","tactics":[{"state_before":"L : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\nx y : Σˣ f\ni : ι\nhx : x.fst ≤ i\nhy : y.fst ≤ i\n⊢ x ≈ y ↔ (f x.fst i hx) x.snd = (f y.fst i hy) y.snd","state_after":"case mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ny : Σˣ f\ni : ι\nhy : y.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhx : ⟨fst✝, snd✝⟩.fst ≤ i\n⊢ ⟨fst✝, snd✝⟩ ≈ y ↔ (f ⟨fst✝, snd✝⟩.fst i hx) ⟨fst✝, snd✝⟩.snd = (f y.fst i hy) y.snd","tactic":"cases x","premises":[]},{"state_before":"case mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ny : Σˣ f\ni : ι\nhy : y.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhx : ⟨fst✝, snd✝⟩.fst ≤ i\n⊢ ⟨fst✝, snd✝⟩ ≈ y ↔ (f ⟨fst✝, snd✝⟩.fst i hx) ⟨fst✝, snd✝⟩.snd = (f y.fst i hy) y.snd","state_after":"case mk.mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\n⊢ ⟨fst✝¹, snd✝¹⟩ ≈ ⟨fst✝, snd✝⟩ ↔\n    (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","tactic":"cases y","premises":[]},{"state_before":"case mk.mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\n⊢ ⟨fst✝¹, snd✝¹⟩ ≈ ⟨fst✝, snd✝⟩ ↔\n    (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","state_after":"case mk.mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nxy : ⟨fst✝¹, snd✝¹⟩ ≈ ⟨fst✝, snd✝⟩\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","tactic":"refine ⟨fun xy => ?_, fun xy => ⟨i, hx, hy, xy⟩⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"case mk.mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nxy : ⟨fst✝¹, snd✝¹⟩ ≈ ⟨fst✝, snd✝⟩\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","state_after":"case mk.mk.intro.intro.intro\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","tactic":"obtain ⟨j, _, _, h⟩ := xy","premises":[]},{"state_before":"case mk.mk.intro.intro.intro.intro.intro\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","state_after":"case mk.mk.intro.intro.intro.intro.intro\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","tactic":"have h := congr_arg (f j k jk) h","premises":[]},{"state_before":"case mk.mk.intro.intro.intro.intro.intro\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd","state_after":"case mk.mk.intro.intro.intro.intro.intro.a\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)\n⊢ (f i k ik) ((f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd) = (f i k ik) ((f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd)","tactic":"apply (f i k ik).injective","premises":[{"full_name":"FirstOrder.Language.Embedding.injective","def_path":"Mathlib/ModelTheory/Basic.lean","def_pos":[580,8],"def_end_pos":[580,17]}]},{"state_before":"case mk.mk.intro.intro.intro.intro.intro.a\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)\n⊢ (f i k ik) ((f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd) = (f i k ik) ((f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd)","state_after":"case mk.mk.intro.intro.intro.intro.intro.a\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f fst✝¹ k ⋯) snd✝¹ = (f fst✝ k ⋯) snd✝\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst k ⋯) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst k ⋯) ⟨fst✝, snd✝⟩.snd","tactic":"rw [DirectedSystem.map_map, DirectedSystem.map_map] at *","premises":[{"full_name":"FirstOrder.Language.DirectedSystem.map_map","def_path":"Mathlib/ModelTheory/DirectLimit.lean","def_pos":[49,15],"def_end_pos":[49,22]}]},{"state_before":"case mk.mk.intro.intro.intro.intro.intro.a\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f fst✝¹ k ⋯) snd✝¹ = (f fst✝ k ⋯) snd✝\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst k ⋯) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst k ⋯) ⟨fst✝, snd✝⟩.snd","state_after":"no goals","tactic":"exact h","premises":[]}]}
{"url":"Mathlib/Data/Finset/Grade.lean","commit":"","full_name":"Finset.isAtom_iff","start":[134,0],"end":[135,56],"file_path":"Mathlib/Data/Finset/Grade.lean","tactics":[{"state_before":"α : Type u_1\ns t : Finset α\na : α\n⊢ IsAtom s ↔ ∃ a, s = {a}","state_after":"no goals","tactic":"simp [← bot_covBy_iff, covBy_iff_exists_cons, eq_comm]","premises":[{"full_name":"Finset.covBy_iff_exists_cons","def_path":"Mathlib/Data/Finset/Grade.lean","def_pos":[91,6],"def_end_pos":[91,27]},{"full_name":"bot_covBy_iff","def_path":"Mathlib/Order/Atoms.lean","def_pos":[98,8],"def_end_pos":[98,21]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/RingTheory/Filtration.lean","commit":"","full_name":"Ideal.Filtration.submodule_span_single","start":[282,0],"end":[285,50],"file_path":"Mathlib/RingTheory/Filtration.lean","tactics":[{"state_before":"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\n⊢ Submodule.span (↥(reesAlgebra I)) (⋃ i, ⇑(single R i) '' ↑(F.N i)) = F.submodule","state_after":"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\n⊢ Submodule.span ↥(reesAlgebra I) ↑F.submodule = F.submodule","tactic":"rw [← Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]","premises":[{"full_name":"Ideal.Filtration.submodule_closure_single","def_path":"Mathlib/RingTheory/Filtration.lean","def_pos":[267,8],"def_end_pos":[267,32]},{"full_name":"Submodule.coe_toAddSubmonoid","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[126,8],"def_end_pos":[126,26]},{"full_name":"Submodule.span_closure","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[151,8],"def_end_pos":[151,20]}]},{"state_before":"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\n⊢ Submodule.span ↥(reesAlgebra I) ↑F.submodule = F.submodule","state_after":"no goals","tactic":"exact Submodule.span_eq (Filtration.submodule F)","premises":[{"full_name":"Ideal.Filtration.submodule","def_path":"Mathlib/RingTheory/Filtration.lean","def_pos":[237,14],"def_end_pos":[237,23]},{"full_name":"Submodule.span_eq","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[85,8],"def_end_pos":[85,15]}]}]}
{"url":"Mathlib/Topology/LocalAtTarget.lean","commit":"","full_name":"closedEmbedding_iff_closedEmbedding_of_iSup_eq_top","start":[171,0],"end":[178,53],"file_path":"Mathlib/Topology/LocalAtTarget.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ ClosedEmbedding f ↔ ∀ (i : ι), ClosedEmbedding ((U i).carrier.restrictPreimage f)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ∧ IsClosed (range f) ↔\n    ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f) ∧ IsClosed (range ((U i).carrier.restrictPreimage f))","tactic":"simp_rw [closedEmbedding_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"closedEmbedding_iff","def_path":"Mathlib/Topology/Defs/Induced.lean","def_pos":[120,2],"def_end_pos":[120,8]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ∧ IsClosed (range f) ↔\n    ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f) ∧ IsClosed (range ((U i).carrier.restrictPreimage f))","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ∧ IsClosed (range f) ↔\n    (∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)) ∧\n      ∀ (x : ι), IsClosed (range ((U x).carrier.restrictPreimage f))","tactic":"rw [forall_and]","premises":[{"full_name":"forall_and","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[256,8],"def_end_pos":[256,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ∧ IsClosed (range f) ↔\n    (∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)) ∧\n      ∀ (x : ι), IsClosed (range ((U x).carrier.restrictPreimage f))","state_after":"case h₁\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ↔ ∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)\n\ncase h₂\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ IsClosed (range f) ↔ ∀ (x : ι), IsClosed (range ((U x).carrier.restrictPreimage f))","tactic":"apply and_congr","premises":[{"full_name":"and_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[43,8],"def_end_pos":[43,17]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","commit":"","full_name":"Real.hasStrictDerivAt_log_of_pos","start":[33,0],"end":[38,26],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","tactics":[{"state_before":"x : ℝ\nhx : 0 < x\n⊢ HasStrictDerivAt log x⁻¹ x","state_after":"x : ℝ\nhx : 0 < x\nthis : HasStrictDerivAt log (rexp (log x))⁻¹ x\n⊢ HasStrictDerivAt log x⁻¹ x","tactic":"have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=\n    (hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')\n        (ne_of_gt <| exp_pos _) <|\n      Eventually.mono (lt_mem_nhds hx) @exp_log","premises":[{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"HasStrictDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[127,4],"def_end_pos":[127,20]},{"full_name":"HasStrictDerivAt.of_local_left_inverse","def_path":"Mathlib/Analysis/Calculus/Deriv/Inverse.lean","def_pos":[57,8],"def_end_pos":[57,46]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Real.continuousAt_log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[316,8],"def_end_pos":[316,24]},{"full_name":"Real.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[102,11],"def_end_pos":[102,14]},{"full_name":"Real.exp_log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[53,8],"def_end_pos":[53,15]},{"full_name":"Real.exp_pos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[982,8],"def_end_pos":[982,15]},{"full_name":"Real.hasStrictDerivAt_exp","def_path":"Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean","def_pos":[167,8],"def_end_pos":[167,28]},{"full_name":"Real.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[39,18],"def_end_pos":[39,21]},{"full_name":"lt_mem_nhds","def_path":"Mathlib/Topology/Order/Basic.lean","def_pos":[97,8],"def_end_pos":[97,19]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]}]},{"state_before":"x : ℝ\nhx : 0 < x\nthis : HasStrictDerivAt log (rexp (log x))⁻¹ x\n⊢ HasStrictDerivAt log x⁻¹ x","state_after":"no goals","tactic":"rwa [exp_log hx] at this","premises":[{"full_name":"Real.exp_log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[53,8],"def_end_pos":[53,15]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Pullbacks.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.Pullback.cocycle_fst_snd","start":[158,0],"end":[161,52],"file_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ t' 𝒰 f g i j k ≫\n      t' 𝒰 f g j k i ≫\n        t' 𝒰 f g k i j ≫\n          pullback.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.snd (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j) =\n    pullback.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.snd (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)","state_after":"no goals","tactic":"simp only [t'_fst_snd, t'_snd_snd, t'_fst_fst_fst]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Pullback.t'_fst_fst_fst","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[101,8],"def_end_pos":[101,22]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t'_fst_snd","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[117,8],"def_end_pos":[117,18]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.t'_snd_snd","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[141,8],"def_end_pos":[141,18]}]}]}
{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Real.abs_exp_sub_one_le","start":[1189,0],"end":[1195,12],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x : ℝ\nhx : |x| ≤ 1\n⊢ |rexp x - 1| ≤ 2 * |x|","state_after":"x : ℝ\nhx this : |x| ≤ 1\n⊢ |rexp x - 1| ≤ 2 * |x|","tactic":"have : |x| ≤ 1 := mod_cast hx","premises":[{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]}]},{"state_before":"x : ℝ\nhx this : |x| ≤ 1\n⊢ |rexp x - 1| ≤ 2 * |x|","state_after":"x : ℝ\nhx this✝ : |x| ≤ 1\nthis : Complex.abs (cexp ↑x - 1) ≤ 2 * Complex.abs ↑x\n⊢ |rexp x - 1| ≤ 2 * |x|","tactic":"have := Complex.abs_exp_sub_one_le (x := x) (by simpa using this)","premises":[{"full_name":"Complex.abs_exp_sub_one_le","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1148,8],"def_end_pos":[1148,26]}]},{"state_before":"x : ℝ\nhx this✝ : |x| ≤ 1\nthis : Complex.abs (cexp ↑x - 1) ≤ 2 * Complex.abs ↑x\n⊢ |rexp x - 1| ≤ 2 * |x|","state_after":"x : ℝ\nhx this✝ : |x| ≤ 1\nthis : |rexp x - 1| ≤ 2 * |x|\n⊢ |rexp x - 1| ≤ 2 * |x|","tactic":"rw [← ofReal_exp, ← ofReal_one, ← ofReal_sub, abs_ofReal, abs_ofReal] at this","premises":[{"full_name":"Complex.abs_ofReal","def_path":"Mathlib/Data/Complex/Abs.lean","def_pos":[69,8],"def_end_pos":[69,18]},{"full_name":"Complex.ofReal_exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[240,8],"def_end_pos":[240,18]},{"full_name":"Complex.ofReal_one","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[154,8],"def_end_pos":[154,18]},{"full_name":"Complex.ofReal_sub","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[646,8],"def_end_pos":[646,18]}]},{"state_before":"x : ℝ\nhx this✝ : |x| ≤ 1\nthis : |rexp x - 1| ≤ 2 * |x|\n⊢ |rexp x - 1| ≤ 2 * |x|","state_after":"no goals","tactic":"exact this","premises":[]}]}
{"url":"Mathlib/Data/NNReal/Basic.lean","commit":"","full_name":"Real.cast_natAbs_eq_nnabs_cast","start":[1043,0],"end":[1045,72],"file_path":"Mathlib/Data/NNReal/Basic.lean","tactics":[{"state_before":"n : ℤ\n⊢ ↑n.natAbs = nnabs ↑n","state_after":"case a\nn : ℤ\n⊢ ↑↑n.natAbs = ↑(nnabs ↑n)","tactic":"ext","premises":[]},{"state_before":"case a\nn : ℤ\n⊢ ↑↑n.natAbs = ↑(nnabs ↑n)","state_after":"no goals","tactic":"rw [NNReal.coe_natCast, Int.cast_natAbs, Real.coe_nnabs, Int.cast_abs]","premises":[{"full_name":"Int.cast_abs","def_path":"Mathlib/Algebra/Order/Ring/Cast.lean","def_pos":[73,6],"def_end_pos":[73,14]},{"full_name":"Int.cast_natAbs","def_path":"Mathlib/Algebra/Order/Ring/Cast.lean","def_pos":[98,6],"def_end_pos":[98,17]},{"full_name":"NNReal.coe_natCast","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[303,18],"def_end_pos":[303,29]},{"full_name":"Real.coe_nnabs","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[1028,8],"def_end_pos":[1028,17]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/DotProduct.lean","commit":"","full_name":"Matrix.self_mul_conjTranspose_mulVec_eq_zero","start":[144,0],"end":[146,91],"file_path":"Mathlib/LinearAlgebra/Matrix/DotProduct.lean","tactics":[{"state_before":"m : Type u_1\nn : Type u_2\np : Type u_3\nR : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : Fintype p\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ (A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0","state_after":"no goals","tactic":"simpa only [conjTranspose_conjTranspose] using conjTranspose_mul_self_mulVec_eq_zero Aᴴ _","premises":[{"full_name":"Matrix.conjTranspose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[152,4],"def_end_pos":[152,17]},{"full_name":"Matrix.conjTranspose_conjTranspose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1988,8],"def_end_pos":[1988,35]},{"full_name":"Matrix.conjTranspose_mul_self_mulVec_eq_zero","def_path":"Mathlib/LinearAlgebra/Matrix/DotProduct.lean","def_pos":[139,6],"def_end_pos":[139,43]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordset.pos_size_of_mem","start":[1564,0],"end":[1567,51],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : x ∈ t\n⊢ 0 < t.size","state_after":"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : Ordnode.mem x ↑t = true\n⊢ 0 < t.size","tactic":"simp? [Membership.mem, mem] at h_mem says\n    simp only [Membership.mem, mem, Bool.decide_eq_true] at h_mem","premises":[{"full_name":"Bool.decide_eq_true","def_path":".lake/packages/lean4/src/lean/Init/Data/Bool.lean","def_pos":[53,16],"def_end_pos":[53,30]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Ordset.mem","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[1550,4],"def_end_pos":[1550,7]}]},{"state_before":"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : Ordnode.mem x ↑t = true\n⊢ 0 < t.size","state_after":"no goals","tactic":"apply Ordnode.pos_size_of_mem t.property.sz h_mem","premises":[{"full_name":"Ordnode.Valid'.sz","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[884,2],"def_end_pos":[884,4]},{"full_name":"Ordnode.pos_size_of_mem","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[494,8],"def_end_pos":[494,23]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Coverage.lean","commit":"","full_name":"CategoryTheory.Presieve.le_of_factorsThru_sieve","start":[90,0],"end":[94,30],"file_path":"Mathlib/CategoryTheory/Sites/Coverage.lean","tactics":[{"state_before":"C : Type u_2\nD : Type ?u.23461\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.23469, ?u.23461} D\nX : C\nS : Presieve X\nT : Sieve X\nh : S.FactorsThru T.arrows\n⊢ S ≤ T.arrows","state_after":"C : Type u_2\nD : Type ?u.23461\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.23469, ?u.23461} D\nX : C\nS : Presieve X\nT : Sieve X\nh : S.FactorsThru T.arrows\nY : C\nf : Y ⟶ X\nhf : f ∈ S\n⊢ f ∈ T.arrows","tactic":"rintro Y f hf","premises":[]},{"state_before":"C : Type u_2\nD : Type ?u.23461\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.23469, ?u.23461} D\nX : C\nS : Presieve X\nT : Sieve X\nh : S.FactorsThru T.arrows\nY : C\nf : Y ⟶ X\nhf : f ∈ S\n⊢ f ∈ T.arrows","state_after":"case intro.intro.intro.intro\nC : Type u_2\nD : Type ?u.23461\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.23469, ?u.23461} D\nX : C\nS : Presieve X\nT : Sieve X\nh : S.FactorsThru T.arrows\nY W : C\ni : Y ⟶ W\ne : W ⟶ X\nh1 : T.arrows e\nhf : i ≫ e ∈ S\n⊢ i ≫ e ∈ T.arrows","tactic":"obtain ⟨W, i, e, h1, rfl⟩ := h hf","premises":[]},{"state_before":"case intro.intro.intro.intro\nC : Type u_2\nD : Type ?u.23461\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.23469, ?u.23461} D\nX : C\nS : Presieve X\nT : Sieve X\nh : S.FactorsThru T.arrows\nY W : C\ni : Y ⟶ W\ne : W ⟶ X\nh1 : T.arrows e\nhf : i ≫ e ∈ S\n⊢ i ≫ e ∈ T.arrows","state_after":"no goals","tactic":"exact T.downward_closed h1 _","premises":[{"full_name":"CategoryTheory.Sieve.downward_closed","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[249,2],"def_end_pos":[249,17]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","commit":"","full_name":"CochainComplex.mappingCone.mapHomologicalComplexXIso'_hom","start":[552,0],"end":[582,33],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ n + 1 = m","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ m + -1 = n","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ n + 1 = m","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ m + -1 = n","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (H.map ((↑(fst φ)).v n m ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫\n      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫ H.map ((inl φ).v m n ⋯) +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map ((inr φ).f n)) =\n    𝟙 (((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj (mappingCone φ)).X n)","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ H.map ((↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n) = 𝟙 (H.obj ((mappingCone φ).X n))","tactic":"simp only [Functor.mapHomologicalComplex_obj_X, comp_add, add_comp, assoc,\n      inl_v_fst_v_assoc, inr_f_fst_v_assoc, zero_comp, comp_zero, add_zero,\n      inl_v_snd_v_assoc, inr_f_snd_v_assoc, zero_add, ← Functor.map_comp, ← Functor.map_add]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.mapHomologicalComplex_obj_X","def_path":"Mathlib/Algebra/Homology/Additive.lean","def_pos":[107,2],"def_end_pos":[107,7]},{"full_name":"CategoryTheory.Functor.map_add","def_path":"Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean","def_pos":[52,8],"def_end_pos":[52,15]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.Limits.zero_comp","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ H.map ((↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n) = 𝟙 (H.obj ((mappingCone φ).X n))","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ H.map ((↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n) = H.map (𝟙 ((mappingCone φ).X n))","tactic":"rw [← H.map_id]","premises":[{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ H.map ((↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n) = H.map (𝟙 ((mappingCone φ).X n))","state_after":"case e_a\nC : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n = 𝟙 ((mappingCone φ).X n)","tactic":"congr 1","premises":[]},{"state_before":"case e_a\nC : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n = 𝟙 ((mappingCone φ).X n)","state_after":"no goals","tactic":"simp [ext_from_iff  _ _ _ hnm]","premises":[{"full_name":"CochainComplex.mappingCone.ext_from_iff","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[168,6],"def_end_pos":[168,18]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫ H.map ((inl φ).v m n ⋯) +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map ((inr φ).f n)) ≫\n      (H.map ((↑(fst φ)).v n m ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) =\n    𝟙 ((mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n)","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫\n          (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫\n          H.map 0 ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫\n          H.map 0 ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫\n          (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) =\n    𝟙 ((mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n)","tactic":"simp only [Functor.mapHomologicalComplex_obj_X, comp_add, add_comp, assoc,\n      ← H.map_comp_assoc, inl_v_fst_v, CategoryTheory.Functor.map_id, id_comp, inr_f_fst_v,\n      inl_v_snd_v, inr_f_snd_v]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.mapHomologicalComplex_obj_X","def_path":"Mathlib/Algebra/Homology/Additive.lean","def_pos":[107,2],"def_end_pos":[107,7]},{"full_name":"CategoryTheory.Functor.map_comp_assoc","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[57,6],"def_end_pos":[57,28]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]},{"full_name":"CochainComplex.mappingCone.inl_v_fst_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[72,6],"def_end_pos":[72,17]},{"full_name":"CochainComplex.mappingCone.inl_v_snd_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[78,6],"def_end_pos":[78,17]},{"full_name":"CochainComplex.mappingCone.inr_f_fst_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[83,6],"def_end_pos":[83,17]},{"full_name":"CochainComplex.mappingCone.inr_f_snd_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[88,6],"def_end_pos":[88,17]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫\n          (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫\n          H.map 0 ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫\n          H.map 0 ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫\n          (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) =\n    𝟙 ((mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n)","state_after":"no goals","tactic":"simp [ext_from_iff _ _ _ hnm]","premises":[{"full_name":"CochainComplex.mappingCone.ext_from_iff","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[168,6],"def_end_pos":[168,18]}]}]}
{"url":"Mathlib/Combinatorics/Enumerative/Composition.lean","commit":"","full_name":"Composition.boundary_zero","start":[223,0],"end":[224,75],"file_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","tactics":[{"state_before":"n : ℕ\nc : Composition n\n⊢ c.boundary 0 = 0","state_after":"no goals","tactic":"simp [boundary, Fin.ext_iff]","premises":[{"full_name":"Composition.boundary","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[219,4],"def_end_pos":[219,12]},{"full_name":"Fin.ext_iff","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[42,8],"def_end_pos":[42,15]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.rightShiftLinearEquiv_apply","start":[293,0],"end":[298,72],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn a n' : ℤ\nhn' : n' + a = n\nx : R\nγ : Cochain K L n\n⊢ (rightShiftAddEquiv K L n a n' hn') (x • γ) = x • (rightShiftAddEquiv K L n a n' hn') γ","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Multiset/Bind.lean","commit":"","full_name":"Multiset.bind_assoc","start":[159,0],"end":[161,80],"file_path":"Mathlib/Data/Multiset/Bind.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\ns✝ t : Multiset α\nf✝ g✝ : α → Multiset β\ns : Multiset α\nf : α → Multiset β\ng : β → Multiset γ\n⊢ (bind 0 f).bind g = bind 0 fun a => (f a).bind g","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\ns✝ t : Multiset α\nf✝ g✝ : α → Multiset β\ns : Multiset α\nf : α → Multiset β\ng : β → Multiset γ\n⊢ ∀ (a : α) (s : Multiset α),\n    ((s.bind f).bind g = s.bind fun a => (f a).bind g) →\n      ((a ::ₘ s).bind f).bind g = (a ::ₘ s).bind fun a => (f a).bind g","state_after":"no goals","tactic":"simp (config := { contextual := true })","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]}]}]}
{"url":"Mathlib/Algebra/Ring/Parity.lean","commit":"","full_name":"Even.add_odd","start":[99,0],"end":[100,80],"file_path":"Mathlib/Algebra/Ring/Parity.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝¹ : Semiring α\ninst✝ : Semiring β\na b : α\nm n : ℕ\n⊢ Even a → Odd b → Odd (a + b)","state_after":"case intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : ℕ\na b : α\n⊢ Odd (a + a + (2 * b + 1))","tactic":"rintro ⟨a, rfl⟩ ⟨b, rfl⟩","premises":[]},{"state_before":"case intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : ℕ\na b : α\n⊢ Odd (a + a + (2 * b + 1))","state_after":"no goals","tactic":"exact ⟨a + b, by rw [mul_add, ← two_mul, add_assoc]⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","commit":"","full_name":"ProbabilityTheory.measurable_preCDF'","start":[144,0],"end":[147,48],"file_path":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\n⊢ Measurable fun a r => (preCDF ρ r a).toReal","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\n⊢ ∀ (a : ℚ), Measurable fun x => (preCDF ρ a x).toReal","tactic":"rw [measurable_pi_iff]","premises":[{"full_name":"measurable_pi_iff","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[791,8],"def_end_pos":[791,25]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\n⊢ ∀ (a : ℚ), Measurable fun x => (preCDF ρ a x).toReal","state_after":"no goals","tactic":"exact fun _ ↦ measurable_preCDF.ennreal_toReal","premises":[{"full_name":"Measurable.ennreal_toReal","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean","def_pos":[347,8],"def_end_pos":[347,33]},{"full_name":"ProbabilityTheory.measurable_preCDF","def_path":"Mathlib/Probability/Kernel/Disintegration/CondCdf.lean","def_pos":[141,8],"def_end_pos":[141,25]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Sieves.lean","commit":"","full_name":"CategoryTheory.Sieve.sieveOfSubfunctor_functorInclusion","start":[793,0],"end":[800,24],"file_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\n⊢ sieveOfSubfunctor S.functorInclusion = S","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (sieveOfSubfunctor S.functorInclusion).arrows f✝ ↔ S.arrows f✝","tactic":"ext","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (sieveOfSubfunctor S.functorInclusion).arrows f✝ ↔ S.arrows f✝","state_after":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) ↔ S.arrows f✝","tactic":"simp only [functorInclusion_app, sieveOfSubfunctor_apply]","premises":[{"full_name":"CategoryTheory.Sieve.functorInclusion_app","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[767,2],"def_end_pos":[767,7]},{"full_name":"CategoryTheory.Sieve.sieveOfSubfunctor_apply","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[784,2],"def_end_pos":[784,7]}]},{"state_before":"case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) ↔ S.arrows f✝","state_after":"case h.mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) → S.arrows f✝\n\ncase h.mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ S.arrows f✝ → ∃ t, ↑t = f✝","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/Group/Nat.lean","commit":"","full_name":"Nat.even_mul_succ_self","start":[130,0],"end":[130,99],"file_path":"Mathlib/Algebra/Group/Nat.lean","tactics":[{"state_before":"m n✝ n : ℕ\n⊢ Even (n * (n + 1))","state_after":"m n✝ n : ℕ\n⊢ Even n ∨ ¬Even n","tactic":"rw [even_mul, even_add_one]","premises":[{"full_name":"Nat.even_add_one","def_path":"Mathlib/Algebra/Group/Nat.lean","def_pos":[102,22],"def_end_pos":[102,34]},{"full_name":"Nat.even_mul","def_path":"Mathlib/Algebra/Group/Nat.lean","def_pos":[119,22],"def_end_pos":[119,30]}]},{"state_before":"m n✝ n : ℕ\n⊢ Even n ∨ ¬Even n","state_after":"no goals","tactic":"exact em _","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","commit":"","full_name":"MeasurableEmbedding.memℒp_map_measure_iff","start":[1067,0],"end":[1069,74],"file_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nβ : Type u_5\nmβ : MeasurableSpace β\nf : α → β\ng✝ : β → E\ng : β → F\nhf : MeasurableEmbedding f\n⊢ Memℒp g p (Measure.map f μ) ↔ Memℒp (g ∘ f) p μ","state_after":"no goals","tactic":"simp_rw [Memℒp, hf.aestronglyMeasurable_map_iff, hf.eLpNorm_map_measure]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasurableEmbedding.aestronglyMeasurable_map_iff","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1492,8],"def_end_pos":[1492,63]},{"full_name":"MeasurableEmbedding.eLpNorm_map_measure","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[1053,8],"def_end_pos":[1053,54]},{"full_name":"MeasureTheory.Memℒp","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[122,4],"def_end_pos":[122,9]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean","commit":"","full_name":"Real.volume_interval","start":[124,0],"end":[126,52],"file_path":"Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝ : Fintype ι\na b : ℝ\n⊢ volume (uIcc a b) = ofReal |b - a|","state_after":"no goals","tactic":"rw [← Icc_min_max, volume_Icc, max_sub_min_eq_abs]","premises":[{"full_name":"Real.volume_Icc","def_path":"Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean","def_pos":[79,8],"def_end_pos":[79,18]},{"full_name":"Set.Icc_min_max","def_path":"Mathlib/Order/Interval/Set/UnorderedInterval.lean","def_pos":[192,8],"def_end_pos":[192,19]},{"full_name":"max_sub_min_eq_abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[257,2],"def_end_pos":[257,13]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","commit":"","full_name":"Rat.divInt_mul_divInt","start":[299,0],"end":[304,20],"file_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","tactics":[{"state_before":"n₁ n₂ d₁ d₂ : Int\nz₁ : d₁ ≠ 0\nz₂ : d₂ ≠ 0\n⊢ n₁ /. d₁ * (n₂ /. d₂) = n₁ * n₂ /. (d₁ * d₂)","state_after":"no goals","tactic":"rcases Int.eq_nat_or_neg d₁ with ⟨_, rfl | rfl⟩ <;>\n  rcases Int.eq_nat_or_neg d₂ with ⟨_, rfl | rfl⟩ <;>\n  simp_all [-Int.natCast_mul, divInt_neg', Int.mul_neg, Int.ofNat_mul_ofNat,  Int.neg_mul,\n    mkRat_mul_mkRat]","premises":[{"full_name":"Int.eq_nat_or_neg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[1006,8],"def_end_pos":[1006,21]},{"full_name":"Int.mul_neg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[450,32],"def_end_pos":[450,39]},{"full_name":"Int.natCast_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[527,16],"def_end_pos":[527,27]},{"full_name":"Int.neg_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[447,32],"def_end_pos":[447,39]},{"full_name":"Int.ofNat_mul_ofNat","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[48,22],"def_end_pos":[48,37]},{"full_name":"Rat.divInt_neg'","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[154,8],"def_end_pos":[154,19]},{"full_name":"Rat.mkRat_mul_mkRat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[294,8],"def_end_pos":[294,23]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.Point.toAffine_neg","start":[1457,0],"end":[1465,58],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhP : W.Nonsingular P\n⊢ toAffine W (W.neg P) = -toAffine W P","state_after":"case pos\nR : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhP : W.Nonsingular P\nhPz : P z = 0\n⊢ toAffine W (W.neg P) = -toAffine W P\n\ncase neg\nR : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhP : W.Nonsingular P\nhPz : ¬P z = 0\n⊢ toAffine W (W.neg P) = -toAffine W P","tactic":"by_cases hPz : P z = 0","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Topology/Connected/TotallyDisconnected.lean","commit":"","full_name":"Embedding.isTotallyDisconnected_image","start":[148,0],"end":[154,29],"file_path":"Mathlib/Topology/Connected/TotallyDisconnected.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\n⊢ IsTotallyDisconnected (f '' s) ↔ IsTotallyDisconnected s","state_after":"α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nu : Set β\nhus : u ⊆ f '' s\nhu : IsPreconnected u\n⊢ u.Subsingleton","tactic":"refine ⟨hf.isTotallyDisconnected, fun hs u hus hu ↦ ?_⟩","premises":[{"full_name":"Embedding.isTotallyDisconnected","def_path":"Mathlib/Topology/Connected/TotallyDisconnected.lean","def_pos":[144,8],"def_end_pos":[144,39]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nu : Set β\nhus : u ⊆ f '' s\nhu : IsPreconnected u\n⊢ u.Subsingleton","state_after":"case intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nv : Set α\nhvs : v ⊆ s\nhus : f '' v ⊆ f '' s\nhu : IsPreconnected (f '' v)\n⊢ (f '' v).Subsingleton","tactic":"obtain ⟨v, hvs, rfl⟩ : ∃ v, v ⊆ s ∧ f '' v = u :=\n    ⟨f ⁻¹' u ∩ s, inter_subset_right, by rwa [image_preimage_inter, inter_eq_left]⟩","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.image_preimage_inter","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[446,8],"def_end_pos":[446,28]},{"full_name":"Set.inter_eq_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[773,14],"def_end_pos":[773,27]},{"full_name":"Set.inter_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[764,8],"def_end_pos":[764,26]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]}]},{"state_before":"case intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nv : Set α\nhvs : v ⊆ s\nhus : f '' v ⊆ f '' s\nhu : IsPreconnected (f '' v)\n⊢ (f '' v).Subsingleton","state_after":"case intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nv : Set α\nhvs : v ⊆ s\nhus : f '' v ⊆ f '' s\nhu : IsPreconnected v\n⊢ (f '' v).Subsingleton","tactic":"rw [hf.toInducing.isPreconnected_image] at hu","premises":[{"full_name":"Inducing.isPreconnected_image","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[325,8],"def_end_pos":[325,37]}]},{"state_before":"case intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nv : Set α\nhvs : v ⊆ s\nhus : f '' v ⊆ f '' s\nhu : IsPreconnected v\n⊢ (f '' v).Subsingleton","state_after":"no goals","tactic":"exact (hs v hvs hu).image _","premises":[{"full_name":"Set.Subsingleton.image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[1002,8],"def_end_pos":[1002,26]}]}]}
{"url":"Mathlib/Analysis/Convolution.lean","commit":"","full_name":"HasCompactSupport.hasFDerivAt_convolution_left","start":[978,0],"end":[982,57],"file_path":"Mathlib/Analysis/Convolution.lean","tactics":[{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedAddCommGroup E'\ninst✝¹⁵ : NormedAddCommGroup E''\ninst✝¹⁴ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹³ : RCLike 𝕜\ninst✝¹² : NormedSpace 𝕜 E\ninst✝¹¹ : NormedSpace 𝕜 E'\ninst✝¹⁰ : NormedSpace 𝕜 E''\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedSpace 𝕜 F\nn : ℕ∞\ninst✝⁷ : CompleteSpace F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : BorelSpace G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : SFinite μ\ninst✝¹ : μ.IsAddLeftInvariant\ninst✝ : μ.IsNegInvariant\nhcf : HasCompactSupport f\nhf : ContDiff 𝕜 1 f\nhg : LocallyIntegrable g μ\nx₀ : G\n⊢ HasFDerivAt (f ⋆[L, μ] g) ((fderiv 𝕜 f ⋆[precompL G L, μ] g) x₀) x₀","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedAddCommGroup E'\ninst✝¹⁵ : NormedAddCommGroup E''\ninst✝¹⁴ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹³ : RCLike 𝕜\ninst✝¹² : NormedSpace 𝕜 E\ninst✝¹¹ : NormedSpace 𝕜 E'\ninst✝¹⁰ : NormedSpace 𝕜 E''\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedSpace 𝕜 F\nn : ℕ∞\ninst✝⁷ : CompleteSpace F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : BorelSpace G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : SFinite μ\ninst✝¹ : μ.IsAddLeftInvariant\ninst✝ : μ.IsNegInvariant\nhcf : HasCompactSupport f\nhf : ContDiff 𝕜 1 f\nhg : LocallyIntegrable g μ\nx₀ : G\n⊢ HasFDerivAt (g ⋆[L.flip, μ] f) ((g ⋆[(precompL G L).flip, μ] fderiv 𝕜 f) x₀) x₀","tactic":"simp (config := { singlePass := true }) only [← convolution_flip]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"MeasureTheory.convolution_flip","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[644,8],"def_end_pos":[644,24]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedAddCommGroup E'\ninst✝¹⁵ : NormedAddCommGroup E''\ninst✝¹⁴ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹³ : RCLike 𝕜\ninst✝¹² : NormedSpace 𝕜 E\ninst✝¹¹ : NormedSpace 𝕜 E'\ninst✝¹⁰ : NormedSpace 𝕜 E''\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedSpace 𝕜 F\nn : ℕ∞\ninst✝⁷ : CompleteSpace F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : BorelSpace G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : SFinite μ\ninst✝¹ : μ.IsAddLeftInvariant\ninst✝ : μ.IsNegInvariant\nhcf : HasCompactSupport f\nhf : ContDiff 𝕜 1 f\nhg : LocallyIntegrable g μ\nx₀ : G\n⊢ HasFDerivAt (g ⋆[L.flip, μ] f) ((g ⋆[(precompL G L).flip, μ] fderiv 𝕜 f) x₀) x₀","state_after":"no goals","tactic":"exact hcf.hasFDerivAt_convolution_right L.flip hg hf x₀","premises":[{"full_name":"ContinuousLinearMap.flip","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean","def_pos":[143,4],"def_end_pos":[143,8]},{"full_name":"HasCompactSupport.hasFDerivAt_convolution_right","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[945,8],"def_end_pos":[945,62]}]}]}
{"url":"Mathlib/Probability/Kernel/Basic.lean","commit":"","full_name":"ProbabilityTheory.Kernel.apply_congr_of_mem_measurableAtom","start":[211,0],"end":[215,63],"file_path":"Mathlib/Probability/Kernel/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ κ : Kernel α β\ny' y : α\nhy' : y' ∈ measurableAtom y\n⊢ κ y' = κ y","state_after":"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ κ : Kernel α β\ny' y : α\nhy' : y' ∈ measurableAtom y\ns : Set β\nhs : MeasurableSet s\n⊢ (κ y') s = (κ y) s","tactic":"ext s hs","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ κ : Kernel α β\ny' y : α\nhy' : y' ∈ measurableAtom y\ns : Set β\nhs : MeasurableSet s\n⊢ (κ y') s = (κ y) s","state_after":"no goals","tactic":"exact mem_of_mem_measurableAtom hy'\n    (κ.measurable_coe hs (measurableSet_singleton (κ y s))) rfl","premises":[{"full_name":"MeasurableSingletonClass.measurableSet_singleton","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[238,2],"def_end_pos":[238,25]},{"full_name":"ProbabilityTheory.Kernel.measurable_coe","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[207,18],"def_end_pos":[207,32]},{"full_name":"mem_of_mem_measurableAtom","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[567,6],"def_end_pos":[567,31]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Pullbacks.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.Pullback.openCoverOfRight_obj","start":[483,0],"end":[493,33],"file_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullback f g).OpenCover","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (i : 𝒰.J),\n    pullback.map f (𝒰.map i ≫ g) f g (𝟙 X) (𝒰.map i) (𝟙 Z) ⋯ ⋯ =\n      (pullbackSymmetry f (𝒰.map i ≫ g)).hom ≫\n        ((openCoverOfLeft 𝒰 g f).pushforwardIso (pullbackSymmetry g f).hom).map ((Equiv.refl 𝒰.J) i)","tactic":"fapply\n    ((openCoverOfLeft 𝒰 g f).pushforwardIso (pullbackSymmetry _ _).hom).copy 𝒰.J\n      (fun i => pullback f (𝒰.map i ≫ g))\n      (fun i => pullback.map _ _ _ _ (𝟙 _) (𝒰.map i) (𝟙 _) (by simp) (Category.comp_id _))\n      (Equiv.refl _) fun i => pullbackSymmetry _ _","premises":[{"full_name":"AlgebraicGeometry.Scheme.OpenCover.J","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[44,2],"def_end_pos":[44,3]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.copy","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[130,4],"def_end_pos":[130,18]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.pushforwardIso","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[147,4],"def_end_pos":[147,28]},{"full_name":"AlgebraicGeometry.Scheme.Pullback.openCoverOfLeft","def_path":"Mathlib/AlgebraicGeometry/Pullbacks.lean","def_pos":[471,4],"def_end_pos":[471,19]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Limits.pullback","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[92,7],"def_end_pos":[92,15]},{"full_name":"CategoryTheory.Limits.pullback.map","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[246,7],"def_end_pos":[246,19]},{"full_name":"CategoryTheory.Limits.pullbackSymmetry","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[448,4],"def_end_pos":[448,20]},{"full_name":"Equiv.refl","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[140,22],"def_end_pos":[140,26]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (i : 𝒰.J),\n    pullback.map f (𝒰.map i ≫ g) f g (𝟙 X) (𝒰.map i) (𝟙 Z) ⋯ ⋯ =\n      (pullbackSymmetry f (𝒰.map i ≫ g)).hom ≫\n        ((openCoverOfLeft 𝒰 g f).pushforwardIso (pullbackSymmetry g f).hom).map ((Equiv.refl 𝒰.J) i)","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ni : 𝒰.J\n⊢ pullback.map f (𝒰.map i ≫ g) f g (𝟙 X) (𝒰.map i) (𝟙 Z) ⋯ ⋯ =\n    (pullbackSymmetry f (𝒰.map i ≫ g)).hom ≫\n      ((openCoverOfLeft 𝒰 g f).pushforwardIso (pullbackSymmetry g f).hom).map ((Equiv.refl 𝒰.J) i)","tactic":"intro i","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ni : 𝒰.J\n⊢ pullback.map f (𝒰.map i ≫ g) f g (𝟙 X) (𝒰.map i) (𝟙 Z) ⋯ ⋯ =\n    (pullbackSymmetry f (𝒰.map i ≫ g)).hom ≫\n      ((openCoverOfLeft 𝒰 g f).pushforwardIso (pullbackSymmetry g f).hom).map ((Equiv.refl 𝒰.J) i)","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ni : 𝒰.J\n⊢ pullback.map f (𝒰.map i ≫ g) f g (𝟙 X) (𝒰.map i) (𝟙 Z) ⋯ ⋯ =\n    (pullbackSymmetry f (𝒰.map i ≫ g)).hom ≫\n      pullback.map (𝒰.map i ≫ g) f g f (𝒰.map i) (𝟙 X) (𝟙 Z) ⋯ ⋯ ≫ (pullbackSymmetry g f).hom","tactic":"dsimp [OpenCover.bind]","premises":[{"full_name":"AlgebraicGeometry.Scheme.OpenCover.bind","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[97,4],"def_end_pos":[97,18]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.J), HasPullback (𝒰✝.map i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Y.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ni : 𝒰.J\n⊢ pullback.map f (𝒰.map i ≫ g) f g (𝟙 X) (𝒰.map i) (𝟙 Z) ⋯ ⋯ =\n    (pullbackSymmetry f (𝒰.map i ≫ g)).hom ≫\n      pullback.map (𝒰.map i ≫ g) f g f (𝒰.map i) (𝟙 X) (𝟙 Z) ⋯ ⋯ ≫ (pullbackSymmetry g f).hom","state_after":"no goals","tactic":"apply pullback.hom_ext <;> simp","premises":[{"full_name":"CategoryTheory.Limits.pullback.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[212,8],"def_end_pos":[212,24]}]}]}
{"url":"Mathlib/Analysis/Matrix.lean","commit":"","full_name":"Matrix.nnnorm_lt_iff","start":[94,0],"end":[96,43],"file_path":"Mathlib/Analysis/Matrix.lean","tactics":[{"state_before":"R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁵ : Fintype l\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Unique ι\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nr : ℝ≥0\nhr : 0 < r\nA : Matrix m n α\n⊢ ‖A‖₊ < r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ < r","state_after":"no goals","tactic":"simp_rw [nnnorm_def, pi_nnnorm_lt_iff hr]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Matrix.nnnorm_def","def_path":"Mathlib/Analysis/Matrix.lean","def_pos":[83,8],"def_end_pos":[83,18]},{"full_name":"pi_nnnorm_lt_iff","def_path":"Mathlib/Analysis/Normed/Group/Constructions.lean","def_pos":[327,14],"def_end_pos":[327,30]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/Linearity.lean","commit":"","full_name":"LSeriesSummable.neg","start":[61,0],"end":[63,62],"file_path":"Mathlib/NumberTheory/LSeries/Linearity.lean","tactics":[{"state_before":"f : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\n⊢ LSeriesSummable (-f) s","state_after":"no goals","tactic":"simpa only [LSeriesSummable, term_neg] using Summable.neg hf","premises":[{"full_name":"LSeries.term_neg","def_path":"Mathlib/NumberTheory/LSeries/Linearity.lean","def_pos":[49,6],"def_end_pos":[49,22]},{"full_name":"LSeriesSummable","def_path":"Mathlib/NumberTheory/LSeries/Basic.lean","def_pos":[128,4],"def_end_pos":[128,19]},{"full_name":"Summable.neg","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Group.lean","def_pos":[33,2],"def_end_pos":[33,13]}]}]}
{"url":"Mathlib/Data/Matrix/RowCol.lean","commit":"","full_name":"Matrix.row_smul","start":[94,0],"end":[97,5],"file_path":"Mathlib/Data/Matrix/RowCol.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\ninst✝ : SMul R α\nx : R\nv : m → α\n⊢ row ι (x • v) = x • row ι v","state_after":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\ninst✝ : SMul R α\nx : R\nv : m → α\ni✝ : ι\nj✝ : m\n⊢ row ι (x • v) i✝ j✝ = (x • row ι v) i✝ j✝","tactic":"ext","premises":[]},{"state_before":"case a\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nα : Type v\nβ : Type w\nι : Type u_6\ninst✝ : SMul R α\nx : R\nv : m → α\ni✝ : ι\nj✝ : m\n⊢ row ι (x • v) i✝ j✝ = (x • row ι v) i✝ j✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/GroupTheory/FreeGroup/Basic.lean","commit":"","full_name":"FreeGroup.reduce.not","start":[939,0],"end":[965,22],"file_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","tactics":[{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\np : Prop\nL2 L3 : List (α × Bool)\nx✝¹ : α\nx✝ : Bool\nh : reduce [] = L2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: L3\n⊢ p","state_after":"no goals","tactic":"cases L2 <;> injections","premises":[]},{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\np : Prop\nx : α\nb : Bool\nL1 L2 L3 : List (α × Bool)\nx' : α\nb' : Bool\n⊢ reduce ((x, b) :: L1) = L2 ++ (x', b') :: (x', !b') :: L3 → p","state_after":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\np : Prop\nx : α\nb : Bool\nL1 L2 L3 : List (α × Bool)\nx' : α\nb' : Bool\n⊢ List.rec [(x, b)] (fun head tail tail_ih => if x = head.1 ∧ b = !head.2 then tail else (x, b) :: head :: tail)\n        (reduce L1) =\n      L2 ++ (x', b') :: (x', !b') :: L3 →\n    p","tactic":"dsimp","premises":[]}]}
{"url":"Mathlib/Combinatorics/SetFamily/CauchyDavenport.lean","commit":"","full_name":"_private.Mathlib.Combinatorics.SetFamily.CauchyDavenport.0.wellFoundedOn_devosMulRel","start":[98,0],"end":[107,52],"file_path":"Mathlib/Combinatorics/SetFamily/CauchyDavenport.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Group α\ninst✝ : DecidableEq α\nx y : Finset α × Finset α\ns t : Finset α\n⊢ {x | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn DevosMulRel","state_after":"α : Type u_1\ninst✝¹ : Group α\ninst✝ : DecidableEq α\nx y : Finset α × Finset α\ns t : Finset α\nn : ℕ\n⊢ ({x | x.1.Nonempty ∧ x.2.Nonempty} ∩ (fun x => (x.1 * x.2).card) ⁻¹' {n}).WellFoundedOn\n    ((fun x x_1 => x > x_1) on fun x => x.1.card + x.2.card)","tactic":"refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦\n    Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦\n      wellFounded_lt.onFun.wellFoundedOn","premises":[{"full_name":"Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[839,8],"def_end_pos":[839,57]},{"full_name":"WellFounded.onFun","def_path":"Mathlib/Order/WellFounded.lean","def_pos":[39,8],"def_end_pos":[39,13]},{"full_name":"WellFounded.wellFoundedOn","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[91,8],"def_end_pos":[91,40]},{"full_name":"wellFounded_lt","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[298,6],"def_end_pos":[298,20]}]},{"state_before":"α : Type u_1\ninst✝¹ : Group α\ninst✝ : DecidableEq α\nx y : Finset α × Finset α\ns t : Finset α\nn : ℕ\n⊢ ({x | x.1.Nonempty ∧ x.2.Nonempty} ∩ (fun x => (x.1 * x.2).card) ⁻¹' {n}).WellFoundedOn\n    ((fun x x_1 => x > x_1) on fun x => x.1.card + x.2.card)","state_after":"no goals","tactic":"exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|\n    add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|\n      (card_le_card_mul_left _ hx.1.1).trans_eq hx.2","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Finset.card_le_card_mul_left","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[1567,8],"def_end_pos":[1567,29]},{"full_name":"Finset.card_le_card_mul_right","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[1585,8],"def_end_pos":[1585,30]},{"full_name":"Set.WellFoundedOn.mono'","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[122,8],"def_end_pos":[122,13]},{"full_name":"WellFounded.onFun","def_path":"Mathlib/Order/WellFounded.lean","def_pos":[39,8],"def_end_pos":[39,13]},{"full_name":"WellFounded.wellFoundedOn","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[91,8],"def_end_pos":[91,40]},{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"tsub_lt_tsub_left_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[222,8],"def_end_pos":[222,31]},{"full_name":"wellFounded_lt","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[298,6],"def_end_pos":[298,20]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean","commit":"","full_name":"card_linearIndependent","start":[34,0],"end":[52,46],"file_path":"Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean","tactics":[{"state_before":"K : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Fintype V\nk : ℕ\nhk : k ≤ n\n⊢ Nat.card { s // LinearIndependent K s } = ∏ i : Fin k, (q ^ n - q ^ ↑i)","state_after":"K : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Fintype V\nk : ℕ\nhk : k ≤ n\n⊢ card { s // LinearIndependent K s } = ∏ i : Fin k, (q ^ n - q ^ ↑i)","tactic":"rw [Nat.card_eq_fintype_card]","premises":[{"full_name":"Nat.card_eq_fintype_card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[37,8],"def_end_pos":[37,28]}]},{"state_before":"K : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Fintype V\nk : ℕ\nhk : k ≤ n\n⊢ card { s // LinearIndependent K s } = ∏ i : Fin k, (q ^ n - q ^ ↑i)","state_after":"no goals","tactic":"induction k with\n  | zero => simp only [LinearIndependent, Finsupp.total_fin_zero, ker_zero, card_ofSubsingleton,\n      Finset.univ_eq_empty, Finset.prod_empty]\n  | succ k ih =>\n      have (s : { s : Fin k → V // LinearIndependent K s }) :\n          card ((Submodule.span K (Set.range (s : Fin k → V)))ᶜ : Set (V)) =\n          (q) ^ n - (q) ^ k := by\n            rw [card_compl_set, card_eq_pow_finrank (K := K)\n            (V := ((Submodule.span K (Set.range (s : Fin k → V))) : Set (V)))]\n            simp only [SetLike.coe_sort_coe, finrank_span_eq_card s.2, card_fin]\n            rw [card_eq_pow_finrank (K := K)]\n      simp [card_congr (equiv_linearIndependent k), sum_congr _ _ this, ih (Nat.le_of_succ_le hk),\n        mul_comm, Fin.prod_univ_succAbove _ k]","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Fin.prod_univ_succAbove","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[59,8],"def_end_pos":[59,27]},{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Finset.prod_empty","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[314,8],"def_end_pos":[314,18]},{"full_name":"Finset.univ_eq_empty","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[112,8],"def_end_pos":[112,21]},{"full_name":"Finsupp.total_fin_zero","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[750,8],"def_end_pos":[750,22]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Fintype.card_compl_set","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[274,8],"def_end_pos":[274,30]},{"full_name":"Fintype.card_congr","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[140,8],"def_end_pos":[140,18]},{"full_name":"Fintype.card_fin","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[279,8],"def_end_pos":[279,24]},{"full_name":"Fintype.card_ofSubsingleton","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[200,8],"def_end_pos":[200,27]},{"full_name":"Fintype.sum_congr","def_path":"Mathlib/Data/Fintype/BigOperators.lean","def_pos":[63,2],"def_end_pos":[63,13]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"LinearIndependent","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[99,4],"def_end_pos":[99,21]},{"full_name":"LinearMap.ker_zero","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[120,8],"def_end_pos":[120,16]},{"full_name":"Nat.le_of_succ_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[341,8],"def_end_pos":[341,21]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"SetLike.coe_sort_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[134,8],"def_end_pos":[134,20]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"card_eq_pow_finrank","def_path":"Mathlib/FieldTheory/Finite/Basic.lean","def_pos":[457,8],"def_end_pos":[457,27]},{"full_name":"equiv_linearIndependent","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[1316,4],"def_end_pos":[1316,27]},{"full_name":"finrank_span_eq_card","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[441,8],"def_end_pos":[441,28]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/GroupTheory/Coxeter/Inversion.lean","commit":"","full_name":"CoxeterSystem.IsReduced.nodup_rightInvSeq","start":[390,0],"end":[435,7],"file_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","tactics":[{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\n⊢ (cs.rightInvSeq ω).Nodup","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\n⊢ ∀ (i j : ℕ), i < j → j < (cs.rightInvSeq ω).length → (cs.rightInvSeq ω).get? i ≠ (cs.rightInvSeq ω).get? j","tactic":"apply List.nodup_iff_get?_ne_get?.mpr","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"List.nodup_iff_get?_ne_get?","def_path":"Mathlib/Data/List/Nodup.lean","def_pos":[129,8],"def_end_pos":[129,30]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\n⊢ ∀ (i j : ℕ), i < j → j < (cs.rightInvSeq ω).length → (cs.rightInvSeq ω).get? i ≠ (cs.rightInvSeq ω).get? j","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < (cs.rightInvSeq ω).length\ndup : (cs.rightInvSeq ω).get? j = (cs.rightInvSeq ω).get? j'\n⊢ False","tactic":"intro j j' j_lt_j' j'_lt_length (dup : get? (rightInvSeq cs ω) j = get? (rightInvSeq cs ω) j')","premises":[{"full_name":"CoxeterSystem.rightInvSeq","def_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","def_pos":[180,4],"def_end_pos":[180,15]},{"full_name":"List.get?","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[202,4],"def_end_pos":[202,8]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < (cs.rightInvSeq ω).length\ndup : (cs.rightInvSeq ω).get? j = (cs.rightInvSeq ω).get? j'\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\ndup : (cs.rightInvSeq ω).get? j = (cs.rightInvSeq ω).get? j'\nj'_lt_length : j' < ω.length\n⊢ False","tactic":"replace j'_lt_length : j' < List.length ω := by simpa using j'_lt_length","premises":[{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\ndup : (cs.rightInvSeq ω).get? j = (cs.rightInvSeq ω).get? j'\nj'_lt_length : j' < ω.length\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : some ((cs.rightInvSeq ω).get ⟨j, ⋯⟩) = some ((cs.rightInvSeq ω).get ⟨j', ⋯⟩)\n⊢ False","tactic":"rw [get?_eq_get (by simp; omega), get?_eq_get (by simp; omega)] at dup","premises":[{"full_name":"List.get?_eq_get","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[143,8],"def_end_pos":[143,19]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : some ((cs.rightInvSeq ω).get ⟨j, ⋯⟩) = some ((cs.rightInvSeq ω).get ⟨j', ⋯⟩)\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).get ⟨j, ⋯⟩ = (cs.rightInvSeq ω).get ⟨j', ⋯⟩\n⊢ False","tactic":"apply Option.some_injective at dup","premises":[{"full_name":"Option.some_injective","def_path":"Mathlib/Data/Option/Basic.lean","def_pos":[68,8],"def_end_pos":[68,22]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).get ⟨j, ⋯⟩ = (cs.rightInvSeq ω).get ⟨j', ⋯⟩\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\n⊢ False","tactic":"rw [← getD_eq_get _ 1, ← getD_eq_get _ 1] at dup","premises":[{"full_name":"List.getD_eq_get","def_path":"Mathlib/Data/List/GetD.lean","def_pos":[34,8],"def_end_pos":[34,19]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\n⊢ False","tactic":"set! t := (ris ω).getD j 1 with h₁","premises":[{"full_name":"CoxeterSystem.rightInvSeq","def_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","def_pos":[180,4],"def_end_pos":[180,15]},{"full_name":"List.getD","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[230,4],"def_end_pos":[230,8]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\n⊢ False","tactic":"set! t' := (ris (ω.eraseIdx j)).getD (j' - 1) 1 with h₂","premises":[{"full_name":"CoxeterSystem.rightInvSeq","def_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","def_pos":[180,4],"def_end_pos":[180,15]},{"full_name":"List.eraseIdx","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[962,4],"def_end_pos":[962,12]},{"full_name":"List.getD","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[230,4],"def_end_pos":[230,8]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\n⊢ False","tactic":"have h₃ : t' = (ris ω).getD j' 1                    := by\n    rw [h₂, cs.getD_rightInvSeq, cs.getD_rightInvSeq,\n      (Nat.sub_add_cancel (by omega) : j' - 1 + 1 = j'), eraseIdx_eq_take_drop_succ,\n      drop_append_eq_append_drop, drop_length_le (by simp [j_lt_j'.le]), length_take,\n      drop_drop, nil_append, min_eq_left_of_lt (j_lt_j'.trans j'_lt_length), ← add_assoc,\n      Nat.sub_add_cancel (by omega), mul_left_inj, mul_right_inj]\n    congr 2\n    show get? (take j ω ++ drop (j + 1) ω) (j' - 1) = get? ω j'\n    rw [get?_eq_getElem?, get?_eq_getElem?,\n      getElem?_append_right (by simp [Nat.le_sub_one_of_lt j_lt_j']), getElem?_drop]\n    congr\n    show j + 1 + (j' - 1 - List.length (take j ω)) = j'\n    rw [length_take]\n    omega","premises":[{"full_name":"CoxeterSystem.getD_rightInvSeq","def_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","def_pos":[243,8],"def_end_pos":[243,24]},{"full_name":"CoxeterSystem.rightInvSeq","def_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","def_pos":[180,4],"def_end_pos":[180,15]},{"full_name":"List.drop","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[732,4],"def_end_pos":[732,8]},{"full_name":"List.drop_append_eq_append_drop","def_path":".lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean","def_pos":[173,8],"def_end_pos":[173,34]},{"full_name":"List.drop_drop","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1659,16],"def_end_pos":[1659,25]},{"full_name":"List.drop_length_le","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1552,8],"def_end_pos":[1552,22]},{"full_name":"List.eraseIdx_eq_take_drop_succ","def_path":".lake/packages/batteries/Batteries/Data/List/EraseIdx.lean","def_pos":[22,8],"def_end_pos":[22,34]},{"full_name":"List.get?","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[202,4],"def_end_pos":[202,8]},{"full_name":"List.get?_eq_getElem?","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[156,16],"def_end_pos":[156,32]},{"full_name":"List.getD","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[230,4],"def_end_pos":[230,8]},{"full_name":"List.getElem?_append_right","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1021,8],"def_end_pos":[1021,29]},{"full_name":"List.getElem?_drop","def_path":".lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean","def_pos":[234,8],"def_end_pos":[234,21]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"List.length_take","def_path":".lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean","def_pos":[23,16],"def_end_pos":[23,27]},{"full_name":"List.nil_append","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[483,16],"def_end_pos":[483,26]},{"full_name":"List.take","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[715,4],"def_end_pos":[715,8]},{"full_name":"Nat.le_sub_one_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[682,8],"def_end_pos":[682,24]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"min_eq_left_of_lt","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[125,8],"def_end_pos":[125,25]},{"full_name":"mul_left_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[81,8],"def_end_pos":[81,20]},{"full_name":"mul_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[64,8],"def_end_pos":[64,21]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\n⊢ False","tactic":"have h₄ : t * t' = 1                                := by\n    rw [h₁, h₃, dup]\n    exact cs.getD_rightInvSeq_mul_self _ _","premises":[{"full_name":"CoxeterSystem.getD_rightInvSeq_mul_self","def_path":"Mathlib/GroupTheory/Coxeter/Inversion.lean","def_pos":[272,8],"def_end_pos":[272,33]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\n⊢ False","tactic":"have h₆ := calc\n    ω.length = ℓ (π ω)                                    := rω.symm\n    _        = ℓ (π ((ω.eraseIdx j).eraseIdx (j' - 1)))   := congrArg cs.length h₅\n    _        ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length  := cs.length_wordProd_le _","premises":[{"full_name":"CoxeterSystem.length","def_path":"Mathlib/GroupTheory/Coxeter/Length.lean","def_pos":[67,18],"def_end_pos":[67,24]},{"full_name":"CoxeterSystem.length_wordProd_le","def_path":"Mathlib/GroupTheory/Coxeter/Length.lean","def_pos":[75,8],"def_end_pos":[75,26]},{"full_name":"CoxeterSystem.wordProd","def_path":"Mathlib/GroupTheory/Coxeter/Basic.lean","def_pos":[352,4],"def_end_pos":[352,12]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"List.eraseIdx","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[962,4],"def_end_pos":[962,12]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"congrArg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[362,8],"def_end_pos":[362,16]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1\n⊢ False","tactic":"have h₇ := add_le_add_right (add_le_add_right h₆ 1) 1","premises":[{"full_name":"add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[74,31],"def_end_pos":[74,47]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False","tactic":"have h₈ : j' - 1 < List.length (eraseIdx ω j)           := by\n    apply (@Nat.add_lt_add_iff_right 1).mp\n    rw [Nat.sub_add_cancel (by omega)]\n    rw [length_eraseIdx_add_one (by omega)]\n    omega","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"List.eraseIdx","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[962,4],"def_end_pos":[962,12]},{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"List.length_eraseIdx_add_one","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2360,8],"def_end_pos":[2360,31]},{"full_name":"Nat.add_lt_add_iff_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[58,18],"def_end_pos":[58,38]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ (ω.eraseIdx j).length + 1\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False","tactic":"rw [length_eraseIdx_add_one h₈] at h₇","premises":[{"full_name":"List.length_eraseIdx_add_one","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2360,8],"def_end_pos":[2360,31]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ (ω.eraseIdx j).length + 1\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False","state_after":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ω.length\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False","tactic":"rw [length_eraseIdx_add_one (by omega)] at h₇","premises":[{"full_name":"List.length_eraseIdx_add_one","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2360,8],"def_end_pos":[2360,31]}]},{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ω.length\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False","state_after":"no goals","tactic":"omega","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/TorusIntegral.lean","commit":"","full_name":"torusIntegral_add","start":[155,0],"end":[158,62],"file_path":"Mathlib/MeasureTheory/Integral/TorusIntegral.lean","tactics":[{"state_before":"n : ℕ\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ (∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x","state_after":"no goals","tactic":"simpa only [torusIntegral, smul_add, Pi.add_apply] using\n    integral_add hf.function_integrable hg.function_integrable","premises":[{"full_name":"MeasureTheory.integral_add","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[784,8],"def_end_pos":[784,20]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"TorusIntegrable.function_integrable","def_path":"Mathlib/MeasureTheory/Integral/TorusIntegral.lean","def_pos":[128,8],"def_end_pos":[128,27]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]},{"full_name":"torusIntegral","def_path":"Mathlib/MeasureTheory/Integral/TorusIntegral.lean","def_pos":[141,4],"def_end_pos":[141,17]}]}]}
{"url":"Mathlib/FieldTheory/KrullTopology.lean","commit":"","full_name":"IntermediateField.fixingSubgroup.antimono","start":[112,0],"end":[116,22],"file_path":"Mathlib/FieldTheory/KrullTopology.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nh12 : E1 ≤ E2\n⊢ E2.fixingSubgroup ≤ E1.fixingSubgroup","state_after":"case mk\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nh12 : E1 ≤ E2\nσ : L ≃ₐ[K] L\nhσ : σ ∈ E2.fixingSubgroup\nx : L\nhx : x ∈ ↑E1\n⊢ σ • ↑⟨x, hx⟩ = ↑⟨x, hx⟩","tactic":"rintro σ hσ ⟨x, hx⟩","premises":[]},{"state_before":"case mk\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nh12 : E1 ≤ E2\nσ : L ≃ₐ[K] L\nhσ : σ ∈ E2.fixingSubgroup\nx : L\nhx : x ∈ ↑E1\n⊢ σ • ↑⟨x, hx⟩ = ↑⟨x, hx⟩","state_after":"no goals","tactic":"exact hσ ⟨x, h12 hx⟩","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/AmpleSet.lean","commit":"","full_name":"ampleSet_univ","start":[51,0],"end":[56,95],"file_path":"Mathlib/Analysis/Convex/AmpleSet.lean","tactics":[{"state_before":"F✝ : Type u_1\ninst✝⁴ : AddCommGroup F✝\ninst✝³ : Module ℝ F✝\ninst✝² : TopologicalSpace F✝\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\n⊢ AmpleSet univ","state_after":"F✝ : Type u_1\ninst✝⁴ : AddCommGroup F✝\ninst✝³ : Module ℝ F✝\ninst✝² : TopologicalSpace F✝\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\na✝ : x ∈ univ\n⊢ (convexHull ℝ) (connectedComponentIn univ x) = univ","tactic":"intro x _","premises":[]},{"state_before":"F✝ : Type u_1\ninst✝⁴ : AddCommGroup F✝\ninst✝³ : Module ℝ F✝\ninst✝² : TopologicalSpace F✝\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\na✝ : x ∈ univ\n⊢ (convexHull ℝ) (connectedComponentIn univ x) = univ","state_after":"no goals","tactic":"rw [connectedComponentIn_univ, PreconnectedSpace.connectedComponent_eq_univ, convexHull_univ]","premises":[{"full_name":"PreconnectedSpace.connectedComponent_eq_univ","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[673,8],"def_end_pos":[673,52]},{"full_name":"connectedComponentIn_univ","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[553,8],"def_end_pos":[553,33]},{"full_name":"convexHull_univ","def_path":"Mathlib/Analysis/Convex/Hull.lean","def_pos":[73,8],"def_end_pos":[73,23]}]}]}
{"url":"Mathlib/Algebra/CharP/Reduced.lean","commit":"","full_name":"ExpChar.pow_prime_pow_mul_eq_one_iff","start":[41,0],"end":[46,15],"file_path":"Mathlib/Algebra/CharP/Reduced.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ x ^ (p ^ k * m) = 1 ↔ x ^ m = 1","state_after":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ (x ^ m) ^ p ^ k = 1 ↔ x ^ m = 1","tactic":"rw [pow_mul']","premises":[{"full_name":"pow_mul'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[610,6],"def_end_pos":[610,14]}]},{"state_before":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ (x ^ m) ^ p ^ k = 1 ↔ x ^ m = 1","state_after":"case h.e'_1.h.e'_3\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ (iterateFrobenius R p k) 1 = 1","tactic":"convert ← (iterateFrobenius_inj R p k).eq_iff","premises":[{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"iterateFrobenius_inj","def_path":"Mathlib/Algebra/CharP/Reduced.lean","def_pos":[20,8],"def_end_pos":[20,28]}]},{"state_before":"case h.e'_1.h.e'_3\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ (iterateFrobenius R p k) 1 = 1","state_after":"no goals","tactic":"apply map_one","premises":[{"full_name":"map_one","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[190,8],"def_end_pos":[190,15]}]}]}
{"url":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","commit":"","full_name":"Submonoid.LocalizationMap.inv_inj","start":[476,0],"end":[485,62],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","tactics":[{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : M →* N\nhf : ∀ (y : ↥S), IsUnit (f ↑y)\ny z : ↥S\nh : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹\n⊢ f ↑y = f ↑z","state_after":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : M →* N\nhf : ∀ (y : ↥S), IsUnit (f ↑y)\ny z : ↥S\nh : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹\n⊢ f ↑z * ↑((IsUnit.liftRight (f.restrict S) hf) z)⁻¹ = 1","tactic":"rw [← mul_one (f y), eq_comm, ← mul_inv_left hf y (f z) 1, h]","premises":[{"full_name":"Submonoid.LocalizationMap.mul_inv_left","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[448,8],"def_end_pos":[448,20]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : M →* N\nhf : ∀ (y : ↥S), IsUnit (f ↑y)\ny z : ↥S\nh : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹\n⊢ f ↑z * ↑((IsUnit.liftRight (f.restrict S) hf) z)⁻¹ = 1","state_after":"no goals","tactic":"exact Units.inv_mul (IsUnit.liftRight (f.restrict S) hf z)⁻¹","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"IsUnit.liftRight","def_path":"Mathlib/Algebra/Group/Units/Hom.lean","def_pos":[180,18],"def_end_pos":[180,27]},{"full_name":"MonoidHom.restrict","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[806,4],"def_end_pos":[806,12]},{"full_name":"Units.inv_mul","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[212,8],"def_end_pos":[212,15]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.tendsto_lintegral_of_dominated_convergence'","start":[1139,0],"end":[1157,17],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))","tactic":"have : ∀ n, ∫⁻ a, F n a ∂μ = ∫⁻ a, (hF_meas n).mk (F n) a ∂μ := fun n =>\n    lintegral_congr_ae (hF_meas n).ae_eq_mk","premises":[{"full_name":"AEMeasurable.ae_eq_mk","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[395,8],"def_end_pos":[395,16]},{"full_name":"AEMeasurable.mk","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[388,4],"def_end_pos":[388,6]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.lintegral_congr_ae","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[309,8],"def_end_pos":[309,26]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))","tactic":"simp_rw [this]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) ⋯ a) atTop (𝓝 (f a))\n\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ ∀ (n : ℕ), AEMeasurable.mk (F n) ⋯ ≤ᶠ[ae μ] bound","tactic":"apply\n    tendsto_lintegral_of_dominated_convergence bound (fun n => (hF_meas n).measurable_mk) _ h_fin","premises":[{"full_name":"AEMeasurable.measurable_mk","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[392,8],"def_end_pos":[392,21]},{"full_name":"MeasureTheory.tendsto_lintegral_of_dominated_convergence","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1123,8],"def_end_pos":[1123,50]}]}]}
{"url":"Mathlib/Topology/ContinuousOn.lean","commit":"","full_name":"nhdsWithin_biUnion","start":[194,0],"end":[197,65],"file_path":"Mathlib/Topology/ContinuousOn.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : TopologicalSpace α\nι : Type u_5\nI : Set ι\nhI : I.Finite\ns : ι → Set α\na : α\n⊢ 𝓝[⋃ i ∈ ∅, s i] a = ⨆ i ∈ ∅, 𝓝[s i] a","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : TopologicalSpace α\nι : Type u_5\nI : Set ι\nhI : I.Finite\ns : ι → Set α\na : α\na✝ : ι\ns✝ : Set ι\nx✝¹ : a✝ ∉ s✝\nx✝ : s✝.Finite\nhT : 𝓝[⋃ i ∈ s✝, s i] a = ⨆ i ∈ s✝, 𝓝[s i] a\n⊢ 𝓝[⋃ i ∈ insert a✝ s✝, s i] a = ⨆ i ∈ insert a✝ s✝, 𝓝[s i] a","state_after":"no goals","tactic":"simp only [hT, nhdsWithin_union, iSup_insert, biUnion_insert]","premises":[{"full_name":"Set.biUnion_insert","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[809,8],"def_end_pos":[809,22]},{"full_name":"iSup_insert","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1198,8],"def_end_pos":[1198,19]},{"full_name":"nhdsWithin_union","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[190,8],"def_end_pos":[190,24]}]}]}
{"url":"Mathlib/SetTheory/Game/PGame.lean","commit":"","full_name":"SetTheory.PGame.le_zero","start":[587,0],"end":[590,6],"file_path":"Mathlib/SetTheory/Game/PGame.lean","tactics":[{"state_before":"xl xr : Type u\nx : PGame\n⊢ x ≤ 0 ↔ ∀ (i : x.LeftMoves), ∃ j, (x.moveLeft i).moveRight j ≤ 0","state_after":"xl xr : Type u\nx : PGame\n⊢ ((∀ (i : x.LeftMoves), (∃ i', x.moveLeft i ≤ moveLeft 0 i') ∨ ∃ j, (x.moveLeft i).moveRight j ≤ 0) ∧\n      ∀ (j : RightMoves 0), (∃ i, x ≤ (moveRight 0 j).moveLeft i) ∨ ∃ j', x.moveRight j' ≤ moveRight 0 j) ↔\n    ∀ (i : x.LeftMoves), ∃ j, (x.moveLeft i).moveRight j ≤ 0","tactic":"rw [le_def]","premises":[{"full_name":"SetTheory.PGame.le_def","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[543,8],"def_end_pos":[543,14]}]},{"state_before":"xl xr : Type u\nx : PGame\n⊢ ((∀ (i : x.LeftMoves), (∃ i', x.moveLeft i ≤ moveLeft 0 i') ∨ ∃ j, (x.moveLeft i).moveRight j ≤ 0) ∧\n      ∀ (j : RightMoves 0), (∃ i, x ≤ (moveRight 0 j).moveLeft i) ∨ ∃ j', x.moveRight j' ≤ moveRight 0 j) ↔\n    ∀ (i : x.LeftMoves), ∃ j, (x.moveLeft i).moveRight j ≤ 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Monovary.lean","commit":"","full_name":"monovary_neg_right","start":[55,0],"end":[56,46],"file_path":"Mathlib/Algebra/Order/Monovary.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : OrderedCommGroup α\ninst✝ : OrderedCommGroup β\ns : Set ι\nf f₁ f₂ : ι → α\ng g₁ g₂ : ι → β\n⊢ Monovary f g⁻¹ ↔ Antivary f g","state_after":"no goals","tactic":"simpa [Monovary, Antivary] using forall_swap","premises":[{"full_name":"Antivary","def_path":"Mathlib/Order/Monotone/Monovary.lean","def_pos":[42,4],"def_end_pos":[42,12]},{"full_name":"Monovary","def_path":"Mathlib/Order/Monotone/Monovary.lean","def_pos":[38,4],"def_end_pos":[38,12]},{"full_name":"forall_swap","def_path":"Mathlib/Logic/Basic.lean","def_pos":[463,8],"def_end_pos":[463,19]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"AddSubgroup.map_comap_eq_self","start":[2308,0],"end":[2311,34],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\nf✝ f : G →* N\nH : Subgroup N\nh : H ≤ f.range\n⊢ map f (comap f H) = H","state_after":"no goals","tactic":"rwa [map_comap_eq, inf_eq_right]","premises":[{"full_name":"Subgroup.map_comap_eq","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2296,8],"def_end_pos":[2296,20]},{"full_name":"inf_eq_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[341,8],"def_end_pos":[341,20]}]}]}
{"url":"Mathlib/Data/Int/ConditionallyCompleteOrder.lean","commit":"","full_name":"Int.csInf_mem","start":[90,0],"end":[92,24],"file_path":"Mathlib/Data/Int/ConditionallyCompleteOrder.lean","tactics":[{"state_before":"s : Set ℤ\nh1 : s.Nonempty\nh2 : BddBelow s\n⊢ sInf s ∈ s","state_after":"case h.e'_4\ns : Set ℤ\nh1 : s.Nonempty\nh2 : BddBelow s\n⊢ sInf s = ↑((Classical.choose h2).leastOfBdd ⋯ h1)","tactic":"convert (leastOfBdd _ (Classical.choose_spec h2) h1).2.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"Int.leastOfBdd","def_path":"Mathlib/Data/Int/LeastGreatest.lean","def_pos":[44,4],"def_end_pos":[44,14]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"case h.e'_4\ns : Set ℤ\nh1 : s.Nonempty\nh2 : BddBelow s\n⊢ sInf s = ↑((Classical.choose h2).leastOfBdd ⋯ h1)","state_after":"no goals","tactic":"exact dif_pos ⟨h1, h2⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.sum_range_induction","start":[1418,0],"end":[1431,58],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf s : ℕ → β\nbase : s 0 = 1\nstep : ∀ (n : ℕ), s (n + 1) = s n * f n\nn : ℕ\n⊢ ∏ k ∈ range n, f k = s n","state_after":"case zero\nι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf s : ℕ → β\nbase : s 0 = 1\nstep : ∀ (n : ℕ), s (n + 1) = s n * f n\n⊢ ∏ k ∈ range 0, f k = s 0\n\ncase succ\nι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf s : ℕ → β\nbase : s 0 = 1\nstep : ∀ (n : ℕ), s (n + 1) = s n * f n\nk : ℕ\nhk : ∏ k ∈ range k, f k = s k\n⊢ ∏ k ∈ range (k + 1), f k = s (k + 1)","tactic":"induction' n with k hk","premises":[]}]}
{"url":"Mathlib/Data/Matroid/Closure.lean","commit":"","full_name":"Matroid.Basis'.closure_eq_closure","start":[313,0],"end":[314,74],"file_path":"Mathlib/Data/Matroid/Closure.lean","tactics":[{"state_before":"ι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X Y : Set α\ne : α\nι : Sort u_3\nI J B : Set α\nf x y : α\nh : M.Basis' I X\n⊢ M.closure I = M.closure X","state_after":"no goals","tactic":"rw [← closure_inter_ground _ X, h.basis_inter_ground.closure_eq_closure]","premises":[{"full_name":"Matroid.Basis'.basis_inter_ground","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[780,8],"def_end_pos":[780,33]},{"full_name":"Matroid.Basis.closure_eq_closure","def_path":"Mathlib/Data/Matroid/Closure.lean","def_pos":[304,6],"def_end_pos":[304,30]},{"full_name":"Matroid.closure_inter_ground","def_path":"Mathlib/Data/Matroid/Closure.lean","def_pos":[142,14],"def_end_pos":[142,34]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Basic.lean","commit":"","full_name":"Cardinal.ord_le","start":[1172,0],"end":[1184,56],"file_path":"Mathlib/SetTheory/Ordinal/Basic.lean","tactics":[{"state_before":"α✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\n⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card","state_after":"α✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\nr : α → α → Prop\nw✝ : IsWellOrder α r\ne : (#α).ord = type r\n⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card","tactic":"let ⟨r, _, e⟩ := ord_eq α","premises":[{"full_name":"Cardinal.ord_eq","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[1165,8],"def_end_pos":[1165,14]}]},{"state_before":"α✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\nr : α → α → Prop\nw✝ : IsWellOrder α r\ne : (#α).ord = type r\n⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card","state_after":"α✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\nr : α → α → Prop\nw✝ : IsWellOrder α r\ne : (#α).ord = type r\n⊢ (#α).ord ≤ type s ↔ #α ≤ #β","tactic":"simp only [card_type]","premises":[{"full_name":"Ordinal.card_type","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[527,8],"def_end_pos":[527,17]}]},{"state_before":"α✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\nr : α → α → Prop\nw✝ : IsWellOrder α r\ne : (#α).ord = type r\n⊢ (#α).ord ≤ type s ↔ #α ≤ #β","state_after":"case mp\nα✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\nr : α → α → Prop\nw✝ : IsWellOrder α r\ne : (#α).ord = type r\nh : (#α).ord ≤ type s\n⊢ #α ≤ #β\n\ncase mpr\nα✝ : Type u\nβ✝ : Type u_1\nγ : Type u_2\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\nα β : Type u_3\ns : β → β → Prop\nx✝ : IsWellOrder β s\nr : α → α → Prop\nw✝ : IsWellOrder α r\ne : (#α).ord = type r\nh : #α ≤ #β\n⊢ (#α).ord ≤ type s","tactic":"constructor <;> intro h","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Function/AEEqFun.lean","commit":"","full_name":"MeasureTheory.AEEqFun.comp_compQuasiMeasurePreserving","start":[269,0],"end":[273,15],"file_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : MeasurableSpace β\nν : Measure β\ng : γ → δ\nhg : Continuous g\nf : β →ₘ[ν] γ\nφ : α → β\nhφ : Measure.QuasiMeasurePreserving φ μ ν\n⊢ (comp g hg f).compQuasiMeasurePreserving φ hφ = comp g hg (f.compQuasiMeasurePreserving φ hφ)","state_after":"case mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : MeasurableSpace β\nν : Measure β\ng : γ → δ\nhg : Continuous g\nf : β →ₘ[ν] γ\nφ : α → β\nhφ : Measure.QuasiMeasurePreserving φ μ ν\na✝ : { f // AEStronglyMeasurable f ν }\n⊢ (comp g hg (Quot.mk Setoid.r a✝)).compQuasiMeasurePreserving φ hφ =\n    comp g hg (compQuasiMeasurePreserving (Quot.mk Setoid.r a✝) φ hφ)","tactic":"rcases f","premises":[]},{"state_before":"case mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : MeasurableSpace α\nμ ν✝ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : MeasurableSpace β\nν : Measure β\ng : γ → δ\nhg : Continuous g\nf : β →ₘ[ν] γ\nφ : α → β\nhφ : Measure.QuasiMeasurePreserving φ μ ν\na✝ : { f // AEStronglyMeasurable f ν }\n⊢ (comp g hg (Quot.mk Setoid.r a✝)).compQuasiMeasurePreserving φ hφ =\n    comp g hg (compQuasiMeasurePreserving (Quot.mk Setoid.r a✝) φ hφ)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Side.lean","commit":"","full_name":"Function.Injective.wOppSide_map_iff","start":[97,0],"end":[107,9],"file_path":"Mathlib/Analysis/Convex/Side.lean","tactics":[{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\n⊢ (map f s).WOppSide (f x) (f y) ↔ s.WOppSide x y","state_after":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nh : (map f s).WOppSide (f x) (f y)\n⊢ s.WOppSide x y","tactic":"refine ⟨fun h => ?_, fun h => h.map _⟩","premises":[{"full_name":"AffineSubspace.WOppSide.map","def_path":"Mathlib/Analysis/Convex/Side.lean","def_pos":[90,8],"def_end_pos":[90,20]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nh : (map f s).WOppSide (f x) (f y)\n⊢ s.WOppSide x y","state_after":"case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₁ : P'\nhfp₁ : fp₁ ∈ map f s\nfp₂ : P'\nhfp₂ : fp₂ ∈ map f s\nh : SameRay R (f x -ᵥ fp₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y","tactic":"rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩","premises":[]},{"state_before":"case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₁ : P'\nhfp₁ : fp₁ ∈ map f s\nfp₂ : P'\nhfp₂ : fp₂ ∈ map f s\nh : SameRay R (f x -ᵥ fp₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y","state_after":"case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₁ : P'\nhfp₁ : ∃ y ∈ s, f y = fp₁\nfp₂ : P'\nhfp₂ : ∃ y ∈ s, f y = fp₂\nh : SameRay R (f x -ᵥ fp₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y","tactic":"rw [mem_map] at hfp₁ hfp₂","premises":[{"full_name":"AffineSubspace.mem_map","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[1392,8],"def_end_pos":[1392,15]}]},{"state_before":"case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₁ : P'\nhfp₁ : ∃ y ∈ s, f y = fp₁\nfp₂ : P'\nhfp₂ : ∃ y ∈ s, f y = fp₂\nh : SameRay R (f x -ᵥ fp₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y","state_after":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₂ : P'\nhfp₂ : ∃ y ∈ s, f y = fp₂\np₁ : P\nhp₁ : p₁ ∈ s\nh : SameRay R (f x -ᵥ f p₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y","tactic":"rcases hfp₁ with ⟨p₁, hp₁, rfl⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₂ : P'\nhfp₂ : ∃ y ∈ s, f y = fp₂\np₁ : P\nhp₁ : p₁ ∈ s\nh : SameRay R (f x -ᵥ f p₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)\n⊢ s.WOppSide x y","tactic":"rcases hfp₂ with ⟨p₂, hp₂, rfl⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)\n⊢ s.WOppSide x y","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)","tactic":"refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)","tactic":"simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h","premises":[{"full_name":"AffineMap.linearMap_vsub","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[125,8],"def_end_pos":[125,22]},{"full_name":"AffineMap.linear_injective_iff","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[414,8],"def_end_pos":[414,28]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.Injective.sameRay_map_iff","def_path":"Mathlib/LinearAlgebra/Ray.lean","def_pos":[148,8],"def_end_pos":[148,49]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)","state_after":"no goals","tactic":"exact h","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Iso.lean","commit":"","full_name":"CategoryTheory.IsIso.inv_hom_id_assoc","start":[262,0],"end":[264,25],"file_path":"Mathlib/CategoryTheory/Iso.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z✝ : C\nf : X ⟶ Y\nI : IsIso f\nZ : C\ng : Y ⟶ Z\n⊢ inv f ≫ f ≫ g = g","state_after":"no goals","tactic":"simp [← Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]}]}
{"url":"Mathlib/Data/Finsupp/Defs.lean","commit":"","full_name":"Finsupp.eq_single_iff","start":[313,0],"end":[318,79],"file_path":"Mathlib/Data/Finsupp/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb✝ : M\nf : α →₀ M\na : α\nb : M\n⊢ f = single a b ↔ f.support ⊆ {a} ∧ f a = b","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb✝ : M\nf : α →₀ M\na : α\nb : M\n⊢ f.support ⊆ {a} ∧ f a = b → f = single a b","tactic":"refine ⟨fun h => h.symm ▸ ⟨support_single_subset, single_eq_same⟩, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finsupp.single_eq_same","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[258,8],"def_end_pos":[258,22]},{"full_name":"Finsupp.support_single_subset","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[289,8],"def_end_pos":[289,29]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb✝ : M\nf : α →₀ M\na : α\nb : M\n⊢ f.support ⊆ {a} ∧ f a = b → f = single a b","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\n⊢ f = single a (f a)","tactic":"rintro ⟨h, rfl⟩","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\n⊢ f = single a (f a)","state_after":"case intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\nx : α\n⊢ f x = (single a (f a)) x","tactic":"ext x","premises":[]},{"state_before":"case intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\nx : α\n⊢ f x = (single a (f a)) x","state_after":"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\nx : α\nhx : ¬a = x\n⊢ f x = 0","tactic":"by_cases hx : a = x <;> simp only [hx, single_eq_same, single_eq_of_ne, Ne, not_false_iff]","premises":[{"full_name":"Finsupp.single_eq_of_ne","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[262,8],"def_end_pos":[262,23]},{"full_name":"Finsupp.single_eq_same","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[258,8],"def_end_pos":[258,22]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"not_false_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1371,8],"def_end_pos":[1371,21]}]},{"state_before":"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\nx : α\nhx : ¬a = x\n⊢ f x = 0","state_after":"no goals","tactic":"exact not_mem_support_iff.1 (mt (fun hx => (mem_singleton.1 (h hx)).symm) hx)","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset.mem_singleton","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[584,8],"def_end_pos":[584,21]},{"full_name":"Finsupp.not_mem_support_iff","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[157,8],"def_end_pos":[157,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
{"url":"Mathlib/Topology/Sober.lean","commit":"","full_name":"ClosedEmbedding.quasiSober","start":[160,0],"end":[168,57],"file_path":"Mathlib/Topology/Sober.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\n⊢ ∃ x, IsGenericPoint x S✝","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝","tactic":"have hS'' := hS.image f hf.continuous.continuousOn","premises":[{"full_name":"ClosedEmbedding.continuous","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[566,8],"def_end_pos":[566,18]},{"full_name":"Continuous.continuousOn","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[810,8],"def_end_pos":[810,31]},{"full_name":"IsIrreducible.image","def_path":"Mathlib/Topology/Irreducible.lean","def_pos":[191,8],"def_end_pos":[191,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝","state_after":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\nx : β\nhx : IsGenericPoint x (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝","tactic":"obtain ⟨x, hx⟩ := QuasiSober.sober hS'' (hf.isClosedMap _ hS')","premises":[{"full_name":"ClosedEmbedding.isClosedMap","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[569,8],"def_end_pos":[569,19]},{"full_name":"QuasiSober.sober","def_path":"Mathlib/Topology/Sober.lean","def_pos":[104,2],"def_end_pos":[104,7]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\nx : β\nhx : IsGenericPoint x (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝","tactic":"obtain ⟨y, -, rfl⟩ := hx.mem","premises":[{"full_name":"IsGenericPoint.mem","def_path":"Mathlib/Topology/Sober.lean","def_pos":[59,18],"def_end_pos":[59,21]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝","state_after":"case h\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ IsGenericPoint y S✝","tactic":"use y","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ IsGenericPoint y S✝","state_after":"case h.a\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ f '' closure {y} = f '' S✝","tactic":"apply image_injective.mpr hf.inj","premises":[{"full_name":"Embedding.inj","def_path":"Mathlib/Topology/Defs/Induced.lean","def_pos":[111,2],"def_end_pos":[111,5]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.image_injective","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[1308,8],"def_end_pos":[1308,23]}]},{"state_before":"case h.a\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ f '' closure {y} = f '' S✝","state_after":"no goals","tactic":"rw [← hx.def, ← hf.closure_image_eq, image_singleton]","premises":[{"full_name":"ClosedEmbedding.closure_image_eq","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[605,8],"def_end_pos":[605,24]},{"full_name":"IsGenericPoint.def","def_path":"Mathlib/Topology/Sober.lean","def_pos":[39,8],"def_end_pos":[39,26]},{"full_name":"Set.image_singleton","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[286,8],"def_end_pos":[286,23]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Finsupp.lean","commit":"","full_name":"Finsupp.sum_ite_eq'","start":[109,0],"end":[114,29],"file_path":"Mathlib/Algebra/BigOperators/Finsupp.lean","tactics":[{"state_before":"α : Type u_1\nι : Type u_2\nγ : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst✝⁶ : AddCommMonoid A\ninst✝⁵ : AddCommMonoid B\ninst✝⁴ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst✝³ : Zero M\ninst✝² : Zero M'\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (f.prod fun x v => if x = a then b x v else 1) = if a ∈ f.support then b a (f a) else 1","state_after":"α : Type u_1\nι : Type u_2\nγ : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst✝⁶ : AddCommMonoid A\ninst✝⁵ : AddCommMonoid B\ninst✝⁴ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst✝³ : Zero M\ninst✝² : Zero M'\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a_1 = a then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1","tactic":"dsimp [Finsupp.prod]","premises":[{"full_name":"Finsupp.prod","def_path":"Mathlib/Algebra/BigOperators/Finsupp.lean","def_pos":[43,4],"def_end_pos":[43,8]}]},{"state_before":"α : Type u_1\nι : Type u_2\nγ : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst✝⁶ : AddCommMonoid A\ninst✝⁵ : AddCommMonoid B\ninst✝⁴ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst✝³ : Zero M\ninst✝² : Zero M'\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 ∈ f.support, if a_1 = a then b a_1 (f a_1) else 1) = if a ∈ f.support then b a (f a) else 1","state_after":"no goals","tactic":"rw [f.support.prod_ite_eq']","premises":[{"full_name":"Finset.prod_ite_eq'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1144,8],"def_end_pos":[1144,20]},{"full_name":"Finsupp.support","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[93,2],"def_end_pos":[93,9]}]}]}
{"url":"Mathlib/Combinatorics/Additive/ETransform.lean","commit":"","full_name":"Finset.addDysonETransform_idem","start":[66,0],"end":[73,28],"file_path":"Mathlib/Combinatorics/Additive/ETransform.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ne : α\nx : Finset α × Finset α\n⊢ mulDysonETransform e (mulDysonETransform e x) = mulDysonETransform e x","state_after":"case a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ne : α\nx : Finset α × Finset α\n⊢ x.1 ∪ e • x.2 ∪ e • (x.2 ∩ e⁻¹ • x.1) = x.1 ∪ e • x.2\n\ncase a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ne : α\nx : Finset α × Finset α\n⊢ x.2 ∩ e⁻¹ • x.1 ∩ e⁻¹ • (x.1 ∪ e • x.2) = x.2 ∩ e⁻¹ • x.1","tactic":"ext : 1 <;> dsimp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/EqToHom.lean","commit":"","full_name":"CategoryTheory.eqToHom_comp_iff","start":[59,0],"end":[62,50],"file_path":"Mathlib/CategoryTheory/EqToHom.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX X' Y : C\np : X = X'\nf : X ⟶ Y\ng : X' ⟶ Y\nh : eqToHom p ≫ g = f\n⊢ g = eqToHom ⋯ ≫ eqToHom p ≫ g","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX X' Y : C\np : X = X'\nf : X ⟶ Y\ng : X' ⟶ Y\nh : g = eqToHom ⋯ ≫ f\n⊢ eqToHom p ≫ eqToHom ⋯ ≫ f = f","state_after":"no goals","tactic":"simp [whisker_eq _ h]","premises":[{"full_name":"CategoryTheory.whisker_eq","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[184,8],"def_end_pos":[184,18]}]}]}
{"url":"Mathlib/Data/DList/Defs.lean","commit":"","full_name":"Batteries.DList.toList_cons","start":[55,0],"end":[56,15],"file_path":"Mathlib/Data/DList/Defs.lean","tactics":[{"state_before":"α : Type u\nx : α\nl : DList α\n⊢ (cons x l).toList = x :: l.toList","state_after":"case mk\nα : Type u\nx : α\napply✝ : List α → List α\ninvariant✝ : ∀ (l : List α), apply✝ l = apply✝ [] ++ l\n⊢ (cons x { apply := apply✝, invariant := invariant✝ }).toList =\n    x :: { apply := apply✝, invariant := invariant✝ }.toList","tactic":"cases l","premises":[]},{"state_before":"case mk\nα : Type u\nx : α\napply✝ : List α → List α\ninvariant✝ : ∀ (l : List α), apply✝ l = apply✝ [] ++ l\n⊢ (cons x { apply := apply✝, invariant := invariant✝ }).toList =\n    x :: { apply := apply✝, invariant := invariant✝ }.toList","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Mul.lean","commit":"","full_name":"hasFDerivAt_ring_inverse","start":[783,0],"end":[794,59],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Mul.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nR : Type u_6\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nthis : (fun t => Ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] _root_.id\n⊢ HasFDerivAt Ring.inverse (-((mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x","state_after":"no goals","tactic":"simpa [hasFDerivAt_iff_isLittleO_nhds_zero] using this","premises":[{"full_name":"hasFDerivAt_iff_isLittleO_nhds_zero","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[306,8],"def_end_pos":[306,43]}]}]}
{"url":"Mathlib/Data/Set/Function.lean","commit":"","full_name":"Set.InjOn.image","start":[648,0],"end":[649,93],"file_path":"Mathlib/Data/Set/Function.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁✝ s₂✝ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\ns₁ : Set α\nhs₁ : s₁ ∈ 𝒫 s\ns₂ : Set α\nhs₂ : s₂ ∈ 𝒫 s\nh' : f '' s₁ = f '' s₂\n⊢ s₁ = s₂","state_after":"no goals","tactic":"rw [← h.preimage_image_inter hs₁, h', h.preimage_image_inter hs₂]","premises":[{"full_name":"Set.InjOn.preimage_image_inter","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[628,8],"def_end_pos":[628,34]}]}]}
{"url":"Mathlib/Data/List/Zip.lean","commit":"","full_name":"List.zipWith_congr","start":[121,0],"end":[126,29],"file_path":"Mathlib/Data/List/Zip.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\nε : Type u_4\nf g : α → β → γ\nla : List α\nlb : List β\nh : Forall₂ (fun a b => f a b = g a b) la lb\n⊢ zipWith f la lb = zipWith g la lb","state_after":"case nil\nα : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\nε : Type u_4\nf g : α → β → γ\nla : List α\nlb : List β\n⊢ zipWith f [] [] = zipWith g [] []\n\ncase cons\nα : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\nε : Type u_4\nf g : α → β → γ\nla : List α\nlb : List β\na : α\nb : β\nas : List α\nbs : List β\nhfg : f a b = g a b\na✝ : Forall₂ (fun a b => f a b = g a b) as bs\nih : zipWith f as bs = zipWith g as bs\n⊢ zipWith f (a :: as) (b :: bs) = zipWith g (a :: as) (b :: bs)","tactic":"induction' h with a b as bs hfg _ ih","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean","commit":"","full_name":"CochainComplex.mappingConeHomOfDegreewiseSplitIso_hom_f","start":[108,0],"end":[122,9],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean","tactics":[{"state_before":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\n⊢ ∀ (i j : ℤ),\n    (ComplexShape.up ℤ).Rel i j →\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) i).hom ≫\n          ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d i j =\n        (mappingCone (homOfDegreewiseSplit S σ)).d i j ≫\n          ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) j).hom","state_after":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","tactic":"rintro p _ rfl","premises":[]},{"state_before":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","state_after":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f :\n  (σ (p + 1 + 1)).r ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).f =\n    𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).X₂ -\n      (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).g ≫ (σ (p + 1 + 1)).s\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","tactic":"have r_f := (σ (p + 1 + 1)).r_f","premises":[{"full_name":"CategoryTheory.ShortComplex.Splitting.r_f","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[489,6],"def_end_pos":[489,9]}]},{"state_before":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f :\n  (σ (p + 1 + 1)).r ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).f =\n    𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).X₂ -\n      (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).g ≫ (σ (p + 1 + 1)).s\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","state_after":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f :\n  (σ (p + 1 + 1)).r ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).f =\n    𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).X₂ -\n      (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).g ≫ (σ (p + 1 + 1)).s\ns_g : (σ (p + 1)).s ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1))).g = 𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1))).X₃\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","tactic":"have s_g := (σ (p + 1)).s_g","premises":[{"full_name":"CategoryTheory.ShortComplex.Splitting.s_g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[478,2],"def_end_pos":[478,5]}]},{"state_before":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f :\n  (σ (p + 1 + 1)).r ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).f =\n    𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).X₂ -\n      (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).g ≫ (σ (p + 1 + 1)).s\ns_g : (σ (p + 1)).s ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1))).g = 𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1))).X₃\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","state_after":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f : (σ (p + 1 + 1)).r ≫ S.f.f (p + 1 + 1) = 𝟙 (S.X₂.X (p + 1 + 1)) - S.g.f (p + 1 + 1) ≫ (σ (p + 1 + 1)).s\ns_g : (σ (p + 1)).s ≫ S.g.f (p + 1) = 𝟙 (S.X₃.X (p + 1))\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","tactic":"dsimp at r_f s_g","premises":[]},{"state_before":"C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f : (σ (p + 1 + 1)).r ≫ S.f.f (p + 1 + 1) = 𝟙 (S.X₂.X (p + 1 + 1)) - S.g.f (p + 1 + 1) ≫ (σ (p + 1 + 1)).s\ns_g : (σ (p + 1)).s ≫ S.g.f (p + 1) = 𝟙 (S.X₃.X (p + 1))\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom","state_after":"no goals","tactic":"set_option tactic.skipAssignedInstances false in\n    simp [mappingConeHomOfDegreewiseSplitXIso, mappingCone.ext_from_iff _ _ _ rfl,\n      mappingCone.inl_v_d_assoc _ (p + 1) _ (p + 1 + 1) (by linarith) (by linarith),\n      cocycleOfDegreewiseSplit, r_f]\n    rw [← S.g.comm_assoc, reassoc_of% s_g]\n    abel","premises":[{"full_name":"CategoryTheory.ShortComplex.g","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[39,2],"def_end_pos":[39,3]},{"full_name":"CochainComplex.cocycleOfDegreewiseSplit","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean","def_pos":[30,4],"def_end_pos":[30,28]},{"full_name":"CochainComplex.mappingCone.ext_from_iff","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","def_pos":[168,6],"def_end_pos":[168,18]},{"full_name":"CochainComplex.mappingConeHomOfDegreewiseSplitXIso","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean","def_pos":[76,18],"def_end_pos":[76,53]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Data/Finset/Lattice.lean","commit":"","full_name":"Finset.sup'_congr","start":[798,0],"end":[803,64],"file_path":"Mathlib/Data/Finset/Lattice.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : SemilatticeSup α\ns : Finset β\nH : s.Nonempty\nf✝ : β → α\nt : Finset β\nf g : β → α\nh₁ : s = t\nh₂ : ∀ x ∈ s, f x = g x\n⊢ s.sup' H f = t.sup' ⋯ g","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : t.Nonempty\nh₂ : ∀ x ∈ t, f x = g x\n⊢ t.sup' H f = t.sup' ⋯ g","tactic":"subst s","premises":[]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : t.Nonempty\nh₂ : ∀ x ∈ t, f x = g x\n⊢ t.sup' H f = t.sup' ⋯ g","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : t.Nonempty\nh₂ : ∀ x ∈ t, f x = g x\nc : α\n⊢ t.sup' H f ≤ c ↔ t.sup' ⋯ g ≤ c","tactic":"refine eq_of_forall_ge_iff fun c => ?_","premises":[{"full_name":"eq_of_forall_ge_iff","def_path":"Mathlib/Order/Basic.lean","def_pos":[457,8],"def_end_pos":[457,27]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : t.Nonempty\nh₂ : ∀ x ∈ t, f x = g x\nc : α\n⊢ t.sup' H f ≤ c ↔ t.sup' ⋯ g ≤ c","state_after":"no goals","tactic":"simp (config := { contextual := true }) only [sup'_le_iff, h₂]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Finset.sup'_le_iff","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[723,8],"def_end_pos":[723,19]}]}]}
{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.exists_subset_image_finite_and","start":[893,0],"end":[898,7],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\np : Set β → Prop\n⊢ (∃ t ⊆ f '' s, t.Finite ∧ p t) ↔ ∃ t ⊆ s, t.Finite ∧ p (f '' t)","state_after":"no goals","tactic":"classical\n  simp_rw [@and_comm (_ ⊆ _), and_assoc, exists_finite_iff_finset, @and_comm (p _),\n    Finset.subset_image_iff]\n  aesop","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.subset_image_iff","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[697,8],"def_end_pos":[697,24]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.exists_finite_iff_finset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[878,8],"def_end_pos":[878,32]},{"full_name":"and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[148,8],"def_end_pos":[148,17]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
{"url":"Mathlib/Algebra/Group/Ext.lean","commit":"","full_name":"DivInvMonoid.ext","start":[112,0],"end":[132,7],"file_path":"Mathlib/Algebra/Group/Ext.lean","tactics":[{"state_before":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\n⊢ m₁ = m₂","state_after":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\n⊢ m₁ = m₂","tactic":"have h_mon := Monoid.ext h_mul","premises":[{"full_name":"Monoid.ext","def_path":"Mathlib/Algebra/Group/Ext.lean","def_pos":[36,8],"def_end_pos":[36,18]}]},{"state_before":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\n⊢ m₁ = m₂","state_after":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\n⊢ m₁ = m₂","tactic":"let f : @MonoidHom M M m₁.toMulOneClass m₂.toMulOneClass :=\n    @MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)\n      (fun x y => congr_fun (congr_fun h_mul x) y)","premises":[{"full_name":"MonoidHom","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[321,10],"def_end_pos":[321,19]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis : Monoid.npow = Monoid.npow\n⊢ m₁ = m₂","state_after":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝ : Monoid.npow = Monoid.npow\nthis : DivInvMonoid.zpow = DivInvMonoid.zpow\n⊢ m₁ = m₂","tactic":"have : m₁.zpow = m₂.zpow := by\n    ext m x\n    exact @MonoidHom.map_zpow' M M m₁ m₂ f (congr_fun h_inv) x m","premises":[{"full_name":"DivInvMonoid.zpow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[803,12],"def_end_pos":[803,16]},{"full_name":"MonoidHom.map_zpow'","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[813,18],"def_end_pos":[813,37]}]},{"state_before":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝ : Monoid.npow = Monoid.npow\nthis : DivInvMonoid.zpow = DivInvMonoid.zpow\n⊢ m₁ = m₂","state_after":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝¹ : Monoid.npow = Monoid.npow\nthis✝ : DivInvMonoid.zpow = DivInvMonoid.zpow\nthis : Div.div = Div.div\n⊢ m₁ = m₂","tactic":"have : m₁.div = m₂.div := by\n    ext a b\n    exact @map_div' _ _\n      (F := @MonoidHom _ _ (_) _) _ (id _) _\n      (@MonoidHom.instMonoidHomClass _ _ (_) _) f (congr_fun h_inv) a b","premises":[{"full_name":"Div.div","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1334,2],"def_end_pos":[1334,5]},{"full_name":"MonoidHom","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[321,10],"def_end_pos":[321,19]},{"full_name":"MonoidHom.instMonoidHomClass","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[351,9],"def_end_pos":[351,37]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"map_div'","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[385,8],"def_end_pos":[385,16]}]},{"state_before":"M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝¹ : Monoid.npow = Monoid.npow\nthis✝ : DivInvMonoid.zpow = DivInvMonoid.zpow\nthis : Div.div = Div.div\n⊢ m₁ = m₂","state_after":"case mk.mk.mk\nM : Type u_1\nm₂ : DivInvMonoid M\ntoMonoid✝ : Monoid M\nzpow✝ : ℤ → M → M\nzpow_zero'✝ : ∀ (a : M), zpow✝ 0 a = 1\nzpow_succ'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.ofNat n.succ) a = zpow✝ (Int.ofNat n) a * a\ninv✝ : M → M\nzpow_neg'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.negSucc n) a = (zpow✝ (↑n.succ) a)⁻¹\ndiv✝ : M → M → M\ndiv_eq_mul_inv✝ : ∀ (a b : M), a / b = a * b⁻¹\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝¹ : Monoid.npow = Monoid.npow\nthis✝ : DivInvMonoid.zpow = DivInvMonoid.zpow\nthis : Div.div = Div.div\n⊢ mk div_eq_mul_inv✝ zpow✝ zpow_zero'✝ zpow_succ'✝ zpow_neg'✝ = m₂","tactic":"rcases m₁ with @⟨_, ⟨_⟩, ⟨_⟩⟩","premises":[]},{"state_before":"case mk.mk.mk\nM : Type u_1\nm₂ : DivInvMonoid M\ntoMonoid✝ : Monoid M\nzpow✝ : ℤ → M → M\nzpow_zero'✝ : ∀ (a : M), zpow✝ 0 a = 1\nzpow_succ'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.ofNat n.succ) a = zpow✝ (Int.ofNat n) a * a\ninv✝ : M → M\nzpow_neg'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.negSucc n) a = (zpow✝ (↑n.succ) a)⁻¹\ndiv✝ : M → M → M\ndiv_eq_mul_inv✝ : ∀ (a b : M), a / b = a * b⁻¹\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝¹ : Monoid.npow = Monoid.npow\nthis✝ : DivInvMonoid.zpow = DivInvMonoid.zpow\nthis : Div.div = Div.div\n⊢ mk div_eq_mul_inv✝ zpow✝ zpow_zero'✝ zpow_succ'✝ zpow_neg'✝ = m₂","state_after":"case mk.mk.mk.mk.mk.mk\nM : Type u_1\ntoMonoid✝¹ : Monoid M\nzpow✝¹ : ℤ → M → M\nzpow_zero'✝¹ : ∀ (a : M), zpow✝¹ 0 a = 1\nzpow_succ'✝¹ : ∀ (n : ℕ) (a : M), zpow✝¹ (Int.ofNat n.succ) a = zpow✝¹ (Int.ofNat n) a * a\ninv✝¹ : M → M\nzpow_neg'✝¹ : ∀ (n : ℕ) (a : M), zpow✝¹ (Int.negSucc n) a = (zpow✝¹ (↑n.succ) a)⁻¹\ndiv✝¹ : M → M → M\ndiv_eq_mul_inv✝¹ : ∀ (a b : M), a / b = a * b⁻¹\ntoMonoid✝ : Monoid M\nzpow✝ : ℤ → M → M\nzpow_zero'✝ : ∀ (a : M), zpow✝ 0 a = 1\nzpow_succ'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.ofNat n.succ) a = zpow✝ (Int.ofNat n) a * a\ninv✝ : M → M\nzpow_neg'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.negSucc n) a = (zpow✝ (↑n.succ) a)⁻¹\ndiv✝ : M → M → M\ndiv_eq_mul_inv✝ : ∀ (a b : M), a / b = a * b⁻¹\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝¹ : Monoid.npow = Monoid.npow\nthis✝ : DivInvMonoid.zpow = DivInvMonoid.zpow\nthis : Div.div = Div.div\n⊢ mk div_eq_mul_inv✝¹ zpow✝¹ zpow_zero'✝¹ zpow_succ'✝¹ zpow_neg'✝¹ =\n    mk div_eq_mul_inv✝ zpow✝ zpow_zero'✝ zpow_succ'✝ zpow_neg'✝","tactic":"rcases m₂ with @⟨_, ⟨_⟩, ⟨_⟩⟩","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.mk\nM : Type u_1\ntoMonoid✝¹ : Monoid M\nzpow✝¹ : ℤ → M → M\nzpow_zero'✝¹ : ∀ (a : M), zpow✝¹ 0 a = 1\nzpow_succ'✝¹ : ∀ (n : ℕ) (a : M), zpow✝¹ (Int.ofNat n.succ) a = zpow✝¹ (Int.ofNat n) a * a\ninv✝¹ : M → M\nzpow_neg'✝¹ : ∀ (n : ℕ) (a : M), zpow✝¹ (Int.negSucc n) a = (zpow✝¹ (↑n.succ) a)⁻¹\ndiv✝¹ : M → M → M\ndiv_eq_mul_inv✝¹ : ∀ (a b : M), a / b = a * b⁻¹\ntoMonoid✝ : Monoid M\nzpow✝ : ℤ → M → M\nzpow_zero'✝ : ∀ (a : M), zpow✝ 0 a = 1\nzpow_succ'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.ofNat n.succ) a = zpow✝ (Int.ofNat n) a * a\ninv✝ : M → M\nzpow_neg'✝ : ∀ (n : ℕ) (a : M), zpow✝ (Int.negSucc n) a = (zpow✝ (↑n.succ) a)⁻¹\ndiv✝ : M → M → M\ndiv_eq_mul_inv✝ : ∀ (a b : M), a / b = a * b⁻¹\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝¹ : Monoid.npow = Monoid.npow\nthis✝ : DivInvMonoid.zpow = DivInvMonoid.zpow\nthis : Div.div = Div.div\n⊢ mk div_eq_mul_inv✝¹ zpow✝¹ zpow_zero'✝¹ zpow_succ'✝¹ zpow_neg'✝¹ =\n    mk div_eq_mul_inv✝ zpow✝ zpow_zero'✝ zpow_succ'✝ zpow_neg'✝","state_after":"no goals","tactic":"congr","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Dual.lean","commit":"","full_name":"Submodule.mem_dualAnnihilator","start":[812,0],"end":[816,55],"file_path":"Mathlib/LinearAlgebra/Dual.lean","tactics":[{"state_before":"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nφ : Module.Dual R M\n⊢ φ ∈ W.dualAnnihilator ↔ ∀ w ∈ W, φ w = 0","state_after":"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nφ : Module.Dual R M\n⊢ W.dualRestrict φ = 0 ↔ ∀ w ∈ W, φ w = 0","tactic":"refine LinearMap.mem_ker.trans ?_","premises":[{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"LinearMap.mem_ker","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[62,8],"def_end_pos":[62,15]}]},{"state_before":"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nφ : Module.Dual R M\n⊢ W.dualRestrict φ = 0 ↔ ∀ w ∈ W, φ w = 0","state_after":"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nφ : Module.Dual R M\n⊢ (∀ (x : ↥W), φ ↑x = 0 x) ↔ ∀ w ∈ W, φ w = 0","tactic":"simp_rw [LinearMap.ext_iff, dualRestrict_apply]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearMap.ext_iff","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[325,18],"def_end_pos":[325,25]},{"full_name":"Submodule.dualRestrict_apply","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[801,8],"def_end_pos":[801,26]}]},{"state_before":"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nφ : Module.Dual R M\n⊢ (∀ (x : ↥W), φ ↑x = 0 x) ↔ ∀ w ∈ W, φ w = 0","state_after":"no goals","tactic":"exact ⟨fun h w hw => h ⟨w, hw⟩, fun h w => h w.1 w.2⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","commit":"","full_name":"AffineSubspace.direction_affineSpan_insert","start":[1323,0],"end":[1332,6],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ (affineSpan k (insert p2 ↑s)).direction = Submodule.span k {p2 -ᵥ p1} ⊔ s.direction","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ (affineSpan k (↑s ∪ ↑(affineSpan k {p2}))).direction = s.direction ⊔ Submodule.span k {p2 -ᵥ p1}","tactic":"rw [sup_comm, ← Set.union_singleton, ← coe_affineSpan_singleton k V p2]","premises":[{"full_name":"AffineSubspace.coe_affineSpan_singleton","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[636,8],"def_end_pos":[636,32]},{"full_name":"Set.union_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1060,8],"def_end_pos":[1060,23]},{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ (affineSpan k (↑s ∪ ↑(affineSpan k {p2}))).direction = s.direction ⊔ Submodule.span k {p2 -ᵥ p1}","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ (s ⊔ affineSpan k {p2}).direction = s.direction ⊔ Submodule.span k {p2 -ᵥ p1}","tactic":"change (s ⊔ affineSpan k {p2}).direction = _","premises":[{"full_name":"AffineSubspace.direction","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[176,4],"def_end_pos":[176,13]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Sup.sup","def_path":"Mathlib/Order/Notation.lean","def_pos":[47,2],"def_end_pos":[47,5]},{"full_name":"affineSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[483,4],"def_end_pos":[483,14]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ (s ⊔ affineSpan k {p2}).direction = s.direction ⊔ Submodule.span k {p2 -ᵥ p1}","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ s.direction ⊔ vectorSpan k {p2} ⊔ Submodule.span k {p2 -ᵥ p1} = s.direction ⊔ Submodule.span k {p2 -ᵥ p1}","tactic":"rw [direction_sup hp1 (mem_affineSpan k (Set.mem_singleton _)), direction_affineSpan]","premises":[{"full_name":"AffineSubspace.direction_sup","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[1295,8],"def_end_pos":[1295,21]},{"full_name":"Set.mem_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1015,8],"def_end_pos":[1015,21]},{"full_name":"direction_affineSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[500,8],"def_end_pos":[500,28]},{"full_name":"mem_affineSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[511,8],"def_end_pos":[511,22]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ s.direction ⊔ vectorSpan k {p2} ⊔ Submodule.span k {p2 -ᵥ p1} = s.direction ⊔ Submodule.span k {p2 -ᵥ p1}","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean","commit":"","full_name":"CategoryTheory.ShortComplex.quasiIso_iff_isIso_leftHomologyMap'","start":[121,0],"end":[125,44],"file_path":"Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean","tactics":[{"state_before":"C : Type u_2\ninst✝⁵ : Category.{u_1, u_2} C\ninst✝⁴ : HasZeroMorphisms C\nS₁ S₂ S₃ S₄ : ShortComplex C\ninst✝³ : S₁.HasHomology\ninst✝² : S₂.HasHomology\ninst✝¹ : S₃.HasHomology\ninst✝ : S₄.HasHomology\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\n⊢ QuasiIso φ ↔ IsIso (leftHomologyMap' φ h₁ h₂)","state_after":"C : Type u_2\ninst✝⁵ : Category.{u_1, u_2} C\ninst✝⁴ : HasZeroMorphisms C\nS₁ S₂ S₃ S₄ : ShortComplex C\ninst✝³ : S₁.HasHomology\ninst✝² : S₂.HasHomology\ninst✝¹ : S₃.HasHomology\ninst✝ : S₄.HasHomology\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nγ : LeftHomologyMapData φ h₁ h₂\n⊢ QuasiIso φ ↔ IsIso (leftHomologyMap' φ h₁ h₂)","tactic":"have γ : LeftHomologyMapData φ h₁ h₂ := default","premises":[{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[254,10],"def_end_pos":[254,29]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]}]},{"state_before":"C : Type u_2\ninst✝⁵ : Category.{u_1, u_2} C\ninst✝⁴ : HasZeroMorphisms C\nS₁ S₂ S₃ S₄ : ShortComplex C\ninst✝³ : S₁.HasHomology\ninst✝² : S₂.HasHomology\ninst✝¹ : S₃.HasHomology\ninst✝ : S₄.HasHomology\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nγ : LeftHomologyMapData φ h₁ h₂\n⊢ QuasiIso φ ↔ IsIso (leftHomologyMap' φ h₁ h₂)","state_after":"no goals","tactic":"rw [γ.quasiIso_iff, γ.leftHomologyMap'_eq]","premises":[{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData.leftHomologyMap'_eq","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[543,6],"def_end_pos":[543,25]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff","def_path":"Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean","def_pos":[97,6],"def_end_pos":[97,38]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Algebra/Lie/Nilpotent.lean","commit":"","full_name":"LieAlgebra.isNilpotent_range_ad_iff","start":[688,0],"end":[700,36],"file_path":"Mathlib/Algebra/Lie/Nilpotent.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\n⊢ IsNilpotent R ↥(ad R L).range ↔ IsNilpotent R L","state_after":"case refine_1\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nh : IsNilpotent R ↥(ad R L).range\n⊢ IsNilpotent R L\n\ncase refine_2\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\n⊢ IsNilpotent R L → IsNilpotent R ↥(ad R L).range","tactic":"refine ⟨fun h => ?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.ncard_le_ncard","start":[505,0],"end":[508,23],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nht : autoParam t.Finite _auto✝\n⊢ s.ncard ≤ t.ncard","state_after":"α : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nht : autoParam t.Finite _auto✝\n⊢ s.encard ≤ t.encard","tactic":"rw [← Nat.cast_le (α := ℕ∞), ht.cast_ncard_eq, (ht.subset hst).cast_ncard_eq]","premises":[{"full_name":"ENat","def_path":"Mathlib/Data/ENat/Basic.lean","def_pos":[28,4],"def_end_pos":[28,8]},{"full_name":"Nat.cast_le","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[78,8],"def_end_pos":[78,15]},{"full_name":"Set.Finite.cast_ncard_eq","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[477,8],"def_end_pos":[477,28]},{"full_name":"Set.Finite.subset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[633,8],"def_end_pos":[633,21]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nhst : s ⊆ t\nht : autoParam t.Finite _auto✝\n⊢ s.encard ≤ t.encard","state_after":"no goals","tactic":"exact encard_mono hst","premises":[{"full_name":"Set.encard_mono","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[154,8],"def_end_pos":[154,19]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean","commit":"","full_name":"AlgebraicGeometry.topologically_isStableUnderComposition","start":[191,0],"end":[199,22],"file_path":"Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean","tactics":[{"state_before":"P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop\nhP :\n  ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)\n    (g : β → γ), P f → P g → P (g ∘ f)\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) f\nhg : topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) g\n⊢ topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) (f ≫ g)","state_after":"P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop\nhP :\n  ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)\n    (g : β → γ), P f → P g → P (g ∘ f)\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) f\nhg : topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) g\n⊢ P (⇑g.val.base ∘ ⇑f.val.base)","tactic":"simp only [topologically, Scheme.comp_coeBase, TopCat.coe_comp]","premises":[{"full_name":"AlgebraicGeometry.Scheme.comp_coeBase","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[219,8],"def_end_pos":[219,20]},{"full_name":"AlgebraicGeometry.topologically","def_path":"Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean","def_pos":[185,4],"def_end_pos":[185,17]},{"full_name":"TopCat.coe_comp","def_path":"Mathlib/Topology/Category/TopCat/Basic.lean","def_pos":[71,16],"def_end_pos":[71,24]}]},{"state_before":"P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop\nhP :\n  ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)\n    (g : β → γ), P f → P g → P (g ∘ f)\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) f\nhg : topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) g\n⊢ P (⇑g.val.base ∘ ⇑f.val.base)","state_after":"no goals","tactic":"exact hP _ _ hf hg","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","commit":"","full_name":"CategoryTheory.Limits.isoZeroOfMonoZero_hom","start":[379,0],"end":[384,54],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nh : Mono 0\n⊢ (0 ≫ 0) ≫ 0 = 𝟙 X ≫ 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/BoundedVariation.lean","commit":"","full_name":"eVariationOn.comp_eq_of_antitoneOn","start":[541,0],"end":[554,40],"file_path":"Mathlib/Analysis/BoundedVariation.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\n⊢ eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)","state_after":"α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"apply le_antisymm (comp_le_of_antitoneOn f φ hφ (mapsTo_image φ t))","premises":[{"full_name":"Set.mapsTo_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[234,8],"def_end_pos":[234,20]},{"full_name":"eVariationOn.comp_le_of_antitoneOn","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[492,8],"def_end_pos":[492,29]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inl\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : IsEmpty β\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t\n\ncase inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"cases isEmpty_or_nonempty β","premises":[{"full_name":"isEmpty_or_nonempty","def_path":"Mathlib/Logic/IsEmpty.lean","def_pos":[195,8],"def_end_pos":[195,27]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"let ψ := φ.invFunOn t","premises":[{"full_name":"Function.invFunOn","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[1141,18],"def_end_pos":[1141,26]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"have ψφs : EqOn (φ ∘ ψ) id (φ '' t) := (surjOn_image φ t).rightInvOn_invFunOn","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Set.EqOn","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[229,4],"def_end_pos":[229,8]},{"full_name":"Set.SurjOn.rightInvOn_invFunOn","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[1199,8],"def_end_pos":[1199,34]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.surjOn_image","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[753,8],"def_end_pos":[753,20]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"have ψts := (surjOn_image φ t).mapsTo_invFunOn","premises":[{"full_name":"Set.SurjOn.mapsTo_invFunOn","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[1211,8],"def_end_pos":[1211,30]},{"full_name":"Set.surjOn_image","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[753,8],"def_end_pos":[753,20]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\nhψ : AntitoneOn ψ (φ '' t)\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"have hψ : AntitoneOn ψ (φ '' t) := Function.antitoneOn_of_rightInvOn_of_mapsTo hφ ψφs ψts","premises":[{"full_name":"AntitoneOn","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[88,4],"def_end_pos":[88,14]},{"full_name":"Function.antitoneOn_of_rightInvOn_of_mapsTo","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[1632,8],"def_end_pos":[1632,42]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\nhψ : AntitoneOn ψ (φ '' t)\n⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\nhψ : AntitoneOn ψ (φ '' t)\n⊢ eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"change eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"eVariationOn","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[61,18],"def_end_pos":[61,30]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\nhψ : AntitoneOn ψ (φ '' t)\n⊢ eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\nhψ : AntitoneOn ψ (φ '' t)\n⊢ eVariationOn (f ∘ φ ∘ ψ) (φ '' t) ≤ eVariationOn (f ∘ φ) t","tactic":"rw [← eq_of_eqOn (ψφs.comp_left : EqOn (f ∘ φ ∘ ψ) (f ∘ id) (φ '' t))]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Set.EqOn","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[229,4],"def_end_pos":[229,8]},{"full_name":"Set.EqOn.comp_left","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[208,8],"def_end_pos":[208,22]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"eVariationOn.eq_of_eqOn","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[90,8],"def_end_pos":[90,18]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"case inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : AntitoneOn φ t\nh✝ : Nonempty β\nψ : α → β := Function.invFunOn φ t\nψφs : EqOn (φ ∘ ψ) id (φ '' t)\nψts : MapsTo (Function.invFunOn φ t) (φ '' t) t\nhψ : AntitoneOn ψ (φ '' t)\n⊢ eVariationOn (f ∘ φ ∘ ψ) (φ '' t) ≤ eVariationOn (f ∘ φ) t","state_after":"no goals","tactic":"exact comp_le_of_antitoneOn _ ψ hψ ψts","premises":[{"full_name":"eVariationOn.comp_le_of_antitoneOn","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[492,8],"def_end_pos":[492,29]}]}]}
{"url":"Mathlib/Data/Finset/Lattice.lean","commit":"","full_name":"Finset.min'_image","start":[1486,0],"end":[1491,64],"file_path":"Mathlib/Data/Finset/Lattice.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : s✝.Nonempty\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : (image f s).Nonempty\n⊢ (image f s).min' h = f (s.min' ⋯)","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : s✝.Nonempty\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : (image f s).Nonempty\n⊢ s.inf' ⋯ (id ∘ f) = f (s.inf' ⋯ id)","tactic":"simp only [min', inf'_image]","premises":[{"full_name":"Finset.inf'_image","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[981,8],"def_end_pos":[981,18]},{"full_name":"Finset.min'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[1324,4],"def_end_pos":[1324,8]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : s✝.Nonempty\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : (image f s).Nonempty\n⊢ s.inf' ⋯ (id ∘ f) = f (s.inf' ⋯ id)","state_after":"no goals","tactic":"exact .symm <| comp_inf'_eq_inf'_comp _ _ fun _ _ ↦ hf.map_min","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset.comp_inf'_eq_inf'_comp","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[950,8],"def_end_pos":[950,30]},{"full_name":"Monotone.map_min","def_path":"Mathlib/Order/MinMax.lean","def_pos":[217,8],"def_end_pos":[217,24]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Roots.lean","commit":"","full_name":"Polynomial.rootSet_neg","start":[461,0],"end":[464,35],"file_path":"Mathlib/Algebra/Polynomial/Roots.lean","tactics":[{"state_before":"R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\np✝ q : R[X]\ninst✝³ : CommRing T\np : T[X]\nS : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra T S\n⊢ (-p).rootSet S = p.rootSet S","state_after":"no goals","tactic":"rw [rootSet, aroots_neg, rootSet]","premises":[{"full_name":"Polynomial.aroots_neg","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[397,8],"def_end_pos":[397,18]},{"full_name":"Polynomial.rootSet","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[439,4],"def_end_pos":[439,11]}]}]}
{"url":"Mathlib/Topology/Instances/NNReal.lean","commit":"","full_name":"NNReal.tendsto_of_antitone","start":[269,0],"end":[280,44],"file_path":"Mathlib/Topology/Instances/NNReal.lean","tactics":[{"state_before":"f : ℕ → ℝ≥0\nh_ant : Antitone f\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","state_after":"f : ℕ → ℝ≥0\nh_ant : Antitone f\nh_bdd_0 : 0 ∈ lowerBounds (range fun n => ↑(f n))\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","tactic":"have h_bdd_0 : (0 : ℝ) ∈ lowerBounds (Set.range fun n : ℕ => (f n : ℝ)) := by\n    rintro r ⟨n, hn⟩\n    simp_rw [← hn]\n    exact NNReal.coe_nonneg _","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"NNReal.coe_nonneg","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[122,8],"def_end_pos":[122,18]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"lowerBounds","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[48,4],"def_end_pos":[48,15]}]},{"state_before":"f : ℕ → ℝ≥0\nh_ant : Antitone f\nh_bdd_0 : 0 ∈ lowerBounds (range fun n => ↑(f n))\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","state_after":"case intro\nf : ℕ → ℝ≥0\nh_ant : Antitone f\nh_bdd_0 : 0 ∈ lowerBounds (range fun n => ↑(f n))\nL : ℝ\nhL : Tendsto (fun n => ↑(f n)) atTop (𝓝 L)\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","tactic":"obtain ⟨L, hL⟩ := Real.tendsto_of_bddBelow_antitone ⟨0, h_bdd_0⟩ h_ant","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Real.tendsto_of_bddBelow_antitone","def_path":"Mathlib/Topology/Instances/NNReal.lean","def_pos":[264,8],"def_end_pos":[264,48]}]},{"state_before":"case intro\nf : ℕ → ℝ≥0\nh_ant : Antitone f\nh_bdd_0 : 0 ∈ lowerBounds (range fun n => ↑(f n))\nL : ℝ\nhL : Tendsto (fun n => ↑(f n)) atTop (𝓝 L)\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","state_after":"case intro\nf : ℕ → ℝ≥0\nh_ant : Antitone f\nh_bdd_0 : 0 ∈ lowerBounds (range fun n => ↑(f n))\nL : ℝ\nhL : Tendsto (fun n => ↑(f n)) atTop (𝓝 L)\nhL0 : 0 ≤ L\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","tactic":"have hL0 : 0 ≤ L :=\n    haveI h_glb : IsGLB (Set.range fun n => (f n : ℝ)) L := isGLB_of_tendsto_atTop h_ant hL\n    (le_isGLB_iff h_glb).mpr h_bdd_0","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsGLB","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[72,4],"def_end_pos":[72,9]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"isGLB_of_tendsto_atTop","def_path":"Mathlib/Topology/Order/MonotoneConvergence.lean","def_pos":[276,8],"def_end_pos":[276,30]},{"full_name":"le_isGLB_iff","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[268,8],"def_end_pos":[268,20]}]},{"state_before":"case intro\nf : ℕ → ℝ≥0\nh_ant : Antitone f\nh_bdd_0 : 0 ∈ lowerBounds (range fun n => ↑(f n))\nL : ℝ\nhL : Tendsto (fun n => ↑(f n)) atTop (𝓝 L)\nhL0 : 0 ≤ L\n⊢ ∃ r, Tendsto f atTop (𝓝 r)","state_after":"no goals","tactic":"exact ⟨⟨L, hL0⟩, NNReal.tendsto_coe.mp hL⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"NNReal.tendsto_coe","def_path":"Mathlib/Topology/Instances/NNReal.lean","def_pos":[105,8],"def_end_pos":[105,19]}]}]}
{"url":"Mathlib/Topology/Sheaves/LocallySurjective.lean","commit":"","full_name":"TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks","start":[70,0],"end":[112,42],"file_path":"Mathlib/Topology/Sheaves/LocallySurjective.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ConcreteCategory C\nX : TopCat\nℱ 𝒢 : Presheaf C X\ninst✝¹ : Limits.HasColimits C\ninst✝ : Limits.PreservesFilteredColimits (forget C)\nT : ℱ ⟶ 𝒢\n⊢ IsLocallySurjective T ↔ ∀ (x : ↑X), Function.Surjective ⇑((stalkFunctor C x).map T)","state_after":"case mp\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ConcreteCategory C\nX : TopCat\nℱ 𝒢 : Presheaf C X\ninst✝¹ : Limits.HasColimits C\ninst✝ : Limits.PreservesFilteredColimits (forget C)\nT : ℱ ⟶ 𝒢\nhT : IsLocallySurjective T\n⊢ ∀ (x : ↑X), Function.Surjective ⇑((stalkFunctor C x).map T)\n\ncase mpr\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ConcreteCategory C\nX : TopCat\nℱ 𝒢 : Presheaf C X\ninst✝¹ : Limits.HasColimits C\ninst✝ : Limits.PreservesFilteredColimits (forget C)\nT : ℱ ⟶ 𝒢\nhT : ∀ (x : ↑X), Function.Surjective ⇑((stalkFunctor C x).map T)\n⊢ IsLocallySurjective T","tactic":"constructor <;> intro hT","premises":[]}]}
{"url":"Mathlib/Topology/Order/LawsonTopology.lean","commit":"","full_name":"Topology.scottHausdorff_le_isLawson","start":[191,0],"end":[193,32],"file_path":"Mathlib/Topology/Order/LawsonTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\nL S : TopologicalSpace α\ninst✝¹ : IsLawson α\ninst✝ : IsScott α\n⊢ scottHausdorff α ≤ L","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\nL S : TopologicalSpace α\ninst✝¹ : IsLawson α\ninst✝ : IsScott α\n⊢ scottHausdorff α ≤ lawson α","tactic":"rw [@IsLawson.topology_eq_lawson α _ L _]","premises":[{"full_name":"Topology.IsLawson.topology_eq_lawson","def_path":"Mathlib/Topology/Order/LawsonTopology.lean","def_pos":[73,2],"def_end_pos":[73,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\nL S : TopologicalSpace α\ninst✝¹ : IsLawson α\ninst✝ : IsScott α\n⊢ scottHausdorff α ≤ lawson α","state_after":"no goals","tactic":"exact scottHausdorff_le_lawson","premises":[{"full_name":"Topology.scottHausdorff_le_lawson","def_path":"Mathlib/Topology/Order/LawsonTopology.lean","def_pos":[169,6],"def_end_pos":[169,30]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","commit":"","full_name":"CategoryTheory.BicartesianSq.of_has_biproduct₂","start":[1243,0],"end":[1256,58],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproduct X Y\n⊢ BicartesianSq 0 0 biprod.inl biprod.inr","state_after":"no goals","tactic":"convert of_is_biproduct₂ (BinaryBiproduct.isBilimit X Y)","premises":[{"full_name":"CategoryTheory.BicartesianSq.of_is_biproduct₂","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","def_pos":[1224,8],"def_end_pos":[1224,24]},{"full_name":"CategoryTheory.Limits.BinaryBiproduct.isBilimit","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[1349,4],"def_end_pos":[1349,29]}]}]}
{"url":"Mathlib/CategoryTheory/Elements.lean","commit":"","full_name":"CategoryTheory.CategoryOfElements.to_fromCostructuredArrow_eq","start":[230,0],"end":[243,9],"file_path":"Mathlib/CategoryTheory/Elements.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nF✝ : C ⥤ Type w\nF : Cᵒᵖ ⥤ Type v\n⊢ (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 (CostructuredArrow yoneda F)","state_after":"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nF✝ : C ⥤ Type w\nF : Cᵒᵖ ⥤ Type v\n⊢ ∀ (X : CostructuredArrow yoneda F),\n    ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj X = (𝟭 (CostructuredArrow yoneda F)).obj X\n\ncase refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nF✝ : C ⥤ Type w\nF : Cᵒᵖ ⥤ Type v\n⊢ ∀ (X Y : CostructuredArrow yoneda F) (f : X ⟶ Y),\n    ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).map f =\n      eqToHom ⋯ ≫ (𝟭 (CostructuredArrow yoneda F)).map f ≫ eqToHom ⋯","tactic":"refine Functor.ext ?_ ?_","premises":[{"full_name":"CategoryTheory.Functor.ext","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[179,8],"def_end_pos":[179,11]}]}]}
{"url":"Mathlib/Topology/VectorBundle/Basic.lean","commit":"","full_name":"VectorBundleCore.localTriv_coordChange_eq","start":[629,0],"end":[633,49],"file_path":"Mathlib/Topology/VectorBundle/Basic.lean","tactics":[{"state_before":"R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ (Trivialization.coordChangeL R (Z.localTriv i) (Z.localTriv j) b) v = (Z.coordChange i j b) v","state_after":"case a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj ∈\n    Z.baseSet i ∩\n        Z.baseSet\n          (Z.indexAt { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj) ∩\n      Z.baseSet j\n\ncase hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ b ∈ (Z.localTriv i).baseSet ∩ (Z.localTriv j).baseSet","tactic":"rw [Trivialization.coordChangeL_apply', localTriv_symm_fst, localTriv_apply, coordChange_comp]","premises":[{"full_name":"Trivialization.coordChangeL_apply'","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[309,8],"def_end_pos":[309,27]},{"full_name":"VectorBundleCore.coordChange_comp","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[489,2],"def_end_pos":[489,18]},{"full_name":"VectorBundleCore.localTriv_apply","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[594,8],"def_end_pos":[594,23]},{"full_name":"VectorBundleCore.localTriv_symm_fst","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[620,8],"def_end_pos":[620,26]}]},{"state_before":"case a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj ∈\n    Z.baseSet i ∩\n        Z.baseSet\n          (Z.indexAt { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj) ∩\n      Z.baseSet j\n\ncase hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ b ∈ (Z.localTriv i).baseSet ∩ (Z.localTriv j).baseSet","state_after":"no goals","tactic":"exacts [⟨⟨hb.1, Z.mem_baseSet_at b⟩, hb.2⟩, hb]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"VectorBundleCore.mem_baseSet_at","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[485,2],"def_end_pos":[485,16]}]}]}
{"url":"Mathlib/LinearAlgebra/SymplecticGroup.lean","commit":"","full_name":"SymplecticGroup.J_mem","start":[97,0],"end":[99,6],"file_path":"Mathlib/LinearAlgebra/SymplecticGroup.lean","tactics":[{"state_before":"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\n⊢ J l R ∈ symplecticGroup l R","state_after":"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\n⊢ fromBlocks ((0 * 0 + -1 * 1) * 0ᵀ + (0 * -1 + -1 * 0) * (-1)ᵀ) ((0 * 0 + -1 * 1) * 1ᵀ + (0 * -1 + -1 * 0) * 0ᵀ)\n      ((1 * 0 + 0 * 1) * 0ᵀ + (1 * -1 + 0 * 0) * (-1)ᵀ) ((1 * 0 + 0 * 1) * 1ᵀ + (1 * -1 + 0 * 0) * 0ᵀ) =\n    fromBlocks 0 (-1) 1 0","tactic":"rw [mem_iff, J, fromBlocks_multiply, fromBlocks_transpose, fromBlocks_multiply]","premises":[{"full_name":"Matrix.J","def_path":"Mathlib/LinearAlgebra/SymplecticGroup.lean","def_pos":[35,4],"def_end_pos":[35,5]},{"full_name":"Matrix.fromBlocks_multiply","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[211,8],"def_end_pos":[211,27]},{"full_name":"Matrix.fromBlocks_transpose","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[129,8],"def_end_pos":[129,28]},{"full_name":"SymplecticGroup.mem_iff","def_path":"Mathlib/LinearAlgebra/SymplecticGroup.lean","def_pos":[86,8],"def_end_pos":[86,15]}]},{"state_before":"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\n⊢ fromBlocks ((0 * 0 + -1 * 1) * 0ᵀ + (0 * -1 + -1 * 0) * (-1)ᵀ) ((0 * 0 + -1 * 1) * 1ᵀ + (0 * -1 + -1 * 0) * 0ᵀ)\n      ((1 * 0 + 0 * 1) * 0ᵀ + (1 * -1 + 0 * 0) * (-1)ᵀ) ((1 * 0 + 0 * 1) * 1ᵀ + (1 * -1 + 0 * 0) * 0ᵀ) =\n    fromBlocks 0 (-1) 1 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Combinatorics/Colex.lean","commit":"","full_name":"Finset.Colex.toColex_le_toColex_iff_max'_mem","start":[314,0],"end":[325,75],"file_path":"Mathlib/Combinatorics/Colex.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\n𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)\ns t u : Finset α\na b : α\nr : ℕ\n⊢ { ofColex := s } ≤ { ofColex := t } ↔ ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\n𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)\ns t u : Finset α\na b : α\nr : ℕ\nh : { ofColex := s } ≤ { ofColex := t }\nhst : s ≠ t\n⊢ (s ∆ t).max' ⋯ ∈ t\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\n𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)\ns t u : Finset α\na✝ b : α\nr : ℕ\nh : ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t\na : α\nhas : a ∈ { ofColex := s }.ofColex\nhat : a ∉ { ofColex := t }.ofColex\n⊢ ∃ b ∈ { ofColex := t }.ofColex, b ∉ { ofColex := s }.ofColex ∧ a ≤ b","tactic":"refine ⟨fun h hst ↦ ?_, fun h a has hat ↦ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Algebra/Homology/HomologicalComplex.lean","commit":"","full_name":"HomologicalComplex.image_to_eq_image","start":[482,0],"end":[485,31],"file_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni j : ι\nr : c.Rel i j\n⊢ imageSubobject (C.dTo j) = imageSubobject (C.d i j)","state_after":"ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni j : ι\nr : c.Rel i j\n⊢ imageSubobject ((C.xPrevIso r).hom ≫ C.d i j) = imageSubobject (C.d i j)","tactic":"rw [C.dTo_eq r]","premises":[{"full_name":"HomologicalComplex.dTo_eq","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[440,8],"def_end_pos":[440,14]}]},{"state_before":"ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni j : ι\nr : c.Rel i j\n⊢ imageSubobject ((C.xPrevIso r).hom ≫ C.d i j) = imageSubobject (C.d i j)","state_after":"no goals","tactic":"apply imageSubobject_iso_comp","premises":[{"full_name":"CategoryTheory.Limits.imageSubobject_iso_comp","def_path":"Mathlib/CategoryTheory/Subobject/Limits.lean","def_pos":[378,8],"def_end_pos":[378,31]}]}]}
{"url":"Mathlib/Computability/Ackermann.lean","commit":"","full_name":"not_nat_primrec_ack_self","start":[342,0],"end":[344,20],"file_path":"Mathlib/Computability/Ackermann.lean","tactics":[{"state_before":"h : Nat.Primrec fun n => ack n n\n⊢ False","state_after":"case intro\nh : Nat.Primrec fun n => ack n n\nm : ℕ\nhm : ∀ (n : ℕ), ack n n < ack m n\n⊢ False","tactic":"cases' exists_lt_ack_of_nat_primrec h with m hm","premises":[{"full_name":"exists_lt_ack_of_nat_primrec","def_path":"Mathlib/Computability/Ackermann.lean","def_pos":[275,8],"def_end_pos":[275,36]}]},{"state_before":"case intro\nh : Nat.Primrec fun n => ack n n\nm : ℕ\nhm : ∀ (n : ℕ), ack n n < ack m n\n⊢ False","state_after":"no goals","tactic":"exact (hm m).false","premises":[{"full_name":"LT.lt.false","def_path":"Mathlib/Order/Basic.lean","def_pos":[264,18],"def_end_pos":[264,23]}]}]}
{"url":"Mathlib/RingTheory/RootsOfUnity/Lemmas.lean","commit":"","full_name":"IsPrimitiveRoot.prod_pow_sub_one_eq_order","start":[39,0],"end":[44,92],"file_path":"Mathlib/RingTheory/RootsOfUnity/Lemmas.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\n⊢ (-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : (-1) ^ n = ∏ k ∈ range n, -1\n⊢ (-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1","tactic":"have : (-1 : R) ^ n = ∏ k ∈ range n, -1 := by rw [prod_const, card_range]","premises":[{"full_name":"Finset.card_range","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[179,8],"def_end_pos":[179,18]},{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Finset.prod_const","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1470,8],"def_end_pos":[1470,18]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : (-1) ^ n = ∏ k ∈ range n, -1\n⊢ (-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1","state_after":"no goals","tactic":"simp only [this, ← prod_mul_distrib, neg_one_mul, neg_sub, ← prod_one_sub_pow_eq_order hμ]","premises":[{"full_name":"Finset.prod_mul_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[769,8],"def_end_pos":[769,24]},{"full_name":"IsPrimitiveRoot.prod_one_sub_pow_eq_order","def_path":"Mathlib/RingTheory/RootsOfUnity/Lemmas.lean","def_pos":[30,6],"def_end_pos":[30,31]},{"full_name":"neg_one_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[299,8],"def_end_pos":[299,19]},{"full_name":"neg_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]}]}]}
{"url":"Mathlib/Data/List/Infix.lean","commit":"","full_name":"List.tail_suffix","start":[100,0],"end":[100,88],"file_path":"Mathlib/Data/List/Infix.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nl : List α\n⊢ l.tail <:+ l","state_after":"α : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nl : List α\n⊢ drop 1 l <:+ l","tactic":"rw [← drop_one]","premises":[{"full_name":"List.drop_one","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[23,8],"def_end_pos":[23,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nl : List α\n⊢ drop 1 l <:+ l","state_after":"no goals","tactic":"apply drop_suffix","premises":[{"full_name":"List.drop_suffix","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[1215,8],"def_end_pos":[1215,19]}]}]}
{"url":"Mathlib/Algebra/Group/Center.lean","commit":"","full_name":"Set.addCentralizer_addCentralizer_addCentralizer","start":[150,0],"end":[158,53],"file_path":"Mathlib/Algebra/Group/Center.lean","tactics":[{"state_before":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\n⊢ S.centralizer.centralizer.centralizer = S.centralizer","state_after":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\n⊢ S.centralizer.centralizer.centralizer ⊆ S.centralizer","tactic":"refine Set.Subset.antisymm ?_ Set.subset_centralizer_centralizer","premises":[{"full_name":"Set.Subset.antisymm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[302,8],"def_end_pos":[302,23]},{"full_name":"Set.subset_centralizer_centralizer","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[145,6],"def_end_pos":[145,36]}]},{"state_before":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\n⊢ S.centralizer.centralizer.centralizer ⊆ S.centralizer","state_after":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\n⊢ x ∈ S.centralizer","tactic":"intro x hx","premises":[]},{"state_before":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\n⊢ x ∈ S.centralizer","state_after":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\n⊢ ∀ m ∈ S, m * x = x * m","tactic":"rw [Set.mem_centralizer_iff]","premises":[{"full_name":"Set.mem_centralizer_iff","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[111,6],"def_end_pos":[111,25]}]},{"state_before":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\n⊢ ∀ m ∈ S, m * x = x * m","state_after":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\ny : M\nhy : y ∈ S\n⊢ y * x = x * y","tactic":"intro y hy","premises":[]},{"state_before":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\ny : M\nhy : y ∈ S\n⊢ y * x = x * y","state_after":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : ∀ m ∈ S.centralizer.centralizer, m * x = x * m\ny : M\nhy : y ∈ S\n⊢ y * x = x * y","tactic":"rw [Set.mem_centralizer_iff] at hx","premises":[{"full_name":"Set.mem_centralizer_iff","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[111,6],"def_end_pos":[111,25]}]},{"state_before":"M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : ∀ m ∈ S.centralizer.centralizer, m * x = x * m\ny : M\nhy : y ∈ S\n⊢ y * x = x * y","state_after":"no goals","tactic":"exact hx y <| Set.subset_centralizer_centralizer hy","premises":[{"full_name":"Set.subset_centralizer_centralizer","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[145,6],"def_end_pos":[145,36]}]}]}
{"url":"Mathlib/Order/UpperLower/Basic.lean","commit":"","full_name":"UpperSet.upperClosure_inf_sdiff","start":[1585,0],"end":[1586,86],"file_path":"Mathlib/Order/UpperLower/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : Preorder α\ns : UpperSet α\nt : Set α\na : α\nhts : t ⊆ ↑s\nhst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t\n⊢ upperClosure t ⊓ s.sdiff t = s","state_after":"no goals","tactic":"rw [inf_comm, sdiff_inf_upperClosure hts hst]","premises":[{"full_name":"UpperSet.sdiff_inf_upperClosure","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[1576,6],"def_end_pos":[1576,28]},{"full_name":"inf_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[385,8],"def_end_pos":[385,16]}]}]}
{"url":"Mathlib/Topology/Homeomorph.lean","commit":"","full_name":"Homeomorph.piEquivPiSubtypeProd_apply","start":[662,0],"end":[673,97],"file_path":"Mathlib/Topology/Homeomorph.lean","tactics":[{"state_before":"X : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\n⊢ Continuous (Equiv.piEquivPiSubtypeProd p Y).toFun","state_after":"no goals","tactic":"apply Continuous.prod_mk <;> exact continuous_pi fun j => continuous_apply j.1","premises":[{"full_name":"Continuous.prod_mk","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[370,8],"def_end_pos":[370,26]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]},{"full_name":"continuous_apply","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1141,8],"def_end_pos":[1141,24]},{"full_name":"continuous_pi","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1137,8],"def_end_pos":[1137,21]}]},{"state_before":"X : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\n⊢ Continuous fun a => (Equiv.piEquivPiSubtypeProd p Y).invFun a j","state_after":"X : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\n⊢ Continuous fun a => if h : p j then a.1 ⟨j, h⟩ else a.2 ⟨j, h⟩","tactic":"dsimp only [Equiv.piEquivPiSubtypeProd]","premises":[{"full_name":"Equiv.piEquivPiSubtypeProd","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1193,4],"def_end_pos":[1193,24]}]},{"state_before":"X : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\n⊢ Continuous fun a => if h : p j then a.1 ⟨j, h⟩ else a.2 ⟨j, h⟩","state_after":"case pos\nX : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\nh✝ : p j\n⊢ Continuous fun a => a.1 ⟨j, h✝⟩\n\ncase neg\nX : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\nh✝ : ¬p j\n⊢ Continuous fun a => a.2 ⟨j, h✝⟩","tactic":"split_ifs","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nX : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\nh✝ : p j\n⊢ Continuous fun a => a.1 ⟨j, h✝⟩\n\ncase neg\nX : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\nh✝ : ¬p j\n⊢ Continuous fun a => a.2 ⟨j, h✝⟩","state_after":"no goals","tactic":"exacts [(continuous_apply _).comp continuous_fst, (continuous_apply _).comp continuous_snd]","premises":[{"full_name":"Continuous.comp","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1389,8],"def_end_pos":[1389,23]},{"full_name":"continuous_apply","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1141,8],"def_end_pos":[1141,24]},{"full_name":"continuous_fst","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[298,8],"def_end_pos":[298,22]},{"full_name":"continuous_snd","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[334,8],"def_end_pos":[334,22]}]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.tendsto_atTop_add_right_of_le'","start":[716,0],"end":[719,33],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : Tendsto f l atTop\nhg : ∀ᶠ (x : α) in l, C ≤ g x\n⊢ ∀ᶠ (x : α) in l, (fun x => -g x) x ≤ -C","state_after":"no goals","tactic":"simp [hg]","premises":[]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : Tendsto f l atTop\nhg : ∀ᶠ (x : α) in l, C ≤ g x\n⊢ Tendsto (fun x => (fun x => f x + g x) x + (fun x => -g x) x) l atTop","state_after":"no goals","tactic":"simp [hf]","premises":[]}]}
{"url":"Mathlib/Algebra/Algebra/Unitization.lean","commit":"","full_name":"Unitization.lift_symm_apply","start":[689,0],"end":[696,70],"file_path":"Mathlib/Algebra/Algebra/Unitization.lean","tactics":[{"state_before":"S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type u_5\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ : A →ₙₐ[R] C\n⊢ (fun φ => (↑φ).comp (inrNonUnitalAlgHom R A)) φ.toAlgHom = φ","state_after":"case h\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type u_5\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ : A →ₙₐ[R] C\nx✝ : A\n⊢ ((fun φ => (↑φ).comp (inrNonUnitalAlgHom R A)) φ.toAlgHom) x✝ = φ x✝","tactic":"ext","premises":[]},{"state_before":"case h\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type u_5\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ : A →ₙₐ[R] C\nx✝ : A\n⊢ ((fun φ => (↑φ).comp (inrNonUnitalAlgHom R A)) φ.toAlgHom) x✝ = φ x✝","state_after":"no goals","tactic":"simp [NonUnitalAlgHomClass.toNonUnitalAlgHom]","premises":[{"full_name":"NonUnitalAlgHomClass.toNonUnitalAlgHom","def_path":"Mathlib/Algebra/Algebra/NonUnitalHom.lean","def_pos":[134,4],"def_end_pos":[134,21]}]},{"state_before":"S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type u_5\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ : Unitization R A →ₐ[R] C\n⊢ ((fun φ => (↑φ).comp (inrNonUnitalAlgHom R A)) φ).toAlgHom = φ","state_after":"case h.h\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type u_5\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ : Unitization R A →ₐ[R] C\nx✝ : A\n⊢ ((↑((fun φ => (↑φ).comp (inrNonUnitalAlgHom R A)) φ).toAlgHom).comp (inrNonUnitalAlgHom R A)) x✝ =\n    ((↑φ).comp (inrNonUnitalAlgHom R A)) x✝","tactic":"ext","premises":[]},{"state_before":"case h.h\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type u_5\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ : Unitization R A →ₐ[R] C\nx✝ : A\n⊢ ((↑((fun φ => (↑φ).comp (inrNonUnitalAlgHom R A)) φ).toAlgHom).comp (inrNonUnitalAlgHom R A)) x✝ =\n    ((↑φ).comp (inrNonUnitalAlgHom R A)) x✝","state_after":"no goals","tactic":"simp [NonUnitalAlgHomClass.toNonUnitalAlgHom]","premises":[{"full_name":"NonUnitalAlgHomClass.toNonUnitalAlgHom","def_path":"Mathlib/Algebra/Algebra/NonUnitalHom.lean","def_pos":[134,4],"def_end_pos":[134,21]}]}]}
{"url":"Mathlib/Topology/StoneCech.lean","commit":"","full_name":"ultrafilter_pure_injective","start":[129,0],"end":[133,54],"file_path":"Mathlib/Topology/StoneCech.lean","tactics":[{"state_before":"α : Type u\n⊢ Function.Injective pure","state_after":"α : Type u\nx y : α\nh : pure x = pure y\n⊢ x = y","tactic":"intro x y h","premises":[]},{"state_before":"α : Type u\nx y : α\nh : pure x = pure y\n⊢ x = y","state_after":"α : Type u\nx y : α\nh : pure x = pure y\nthis : {x} ∈ pure x\n⊢ x = y","tactic":"have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure","premises":[{"full_name":"Filter.singleton_mem_pure","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2405,8],"def_end_pos":[2405,26]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Pure.pure","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2734,2],"def_end_pos":[2734,6]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Ultrafilter","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[40,10],"def_end_pos":[40,21]}]},{"state_before":"α : Type u\nx y : α\nh : pure x = pure y\nthis : {x} ∈ pure x\n⊢ x = y","state_after":"α : Type u\nx y : α\nh : pure x = pure y\nthis : {x} ∈ pure y\n⊢ x = y","tactic":"rw [h] at this","premises":[]},{"state_before":"α : Type u\nx y : α\nh : pure x = pure y\nthis : {x} ∈ pure y\n⊢ x = y","state_after":"no goals","tactic":"exact (mem_singleton_iff.mp (mem_pure.mp this)).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Filter.mem_pure","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1831,8],"def_end_pos":[1831,16]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_singleton_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[999,8],"def_end_pos":[999,25]}]}]}
{"url":"Mathlib/Data/Fintype/Basic.lean","commit":"","full_name":"Set.toFinset_empty","start":[634,0],"end":[637,6],"file_path":"Mathlib/Data/Fintype/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns t : Set α\ninst✝ : Fintype ↑∅\n⊢ ∅.toFinset = ∅","state_after":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ns t : Set α\ninst✝ : Fintype ↑∅\na✝ : α\n⊢ a✝ ∈ ∅.toFinset ↔ a✝ ∈ ∅","tactic":"ext","premises":[]},{"state_before":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ns t : Set α\ninst✝ : Fintype ↑∅\na✝ : α\n⊢ a✝ ∈ ∅.toFinset ↔ a✝ ∈ ∅","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","commit":"","full_name":"MeasureTheory.Measure.haar.addPrehaar_pos","start":[268,0],"end":[273,25],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : (interior U).Nonempty\nK : Set G\nh1K : IsCompact K\nh2K : (interior K).Nonempty\n⊢ 0 < prehaar (↑K₀) U { carrier := K, isCompact' := h1K }","state_after":"case ha\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : (interior U).Nonempty\nK : Set G\nh1K : IsCompact K\nh2K : (interior K).Nonempty\n⊢ 0 < index (↑{ carrier := K, isCompact' := h1K }) U\n\ncase hb\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : (interior U).Nonempty\nK : Set G\nh1K : IsCompact K\nh2K : (interior K).Nonempty\n⊢ 0 < index (↑K₀) U","tactic":"apply div_pos <;> norm_cast","premises":[{"full_name":"div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[45,6],"def_end_pos":[45,13]}]}]}
{"url":"Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean","commit":"","full_name":"Filter.BoundedAtFilter.mul","start":[96,0],"end":[100,6],"file_path":"Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean","tactics":[{"state_before":"𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ l.BoundedAtFilter (f * g)","state_after":"𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ (fun x => 1 x * 1 x) =O[l] 1","tactic":"refine (hf.mul hg).trans ?_","premises":[{"full_name":"Asymptotics.IsBigO.mul","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1335,8],"def_end_pos":[1335,18]},{"full_name":"Asymptotics.IsBigO.trans","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[431,8],"def_end_pos":[431,20]}]},{"state_before":"𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ (fun x => 1 x * 1 x) =O[l] 1","state_after":"case h.e'_8.h\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\nx✝ : α\n⊢ 1 x✝ = 1 x✝ * 1 x✝","tactic":"convert Asymptotics.isBigO_refl (E := ℝ) _ l","premises":[{"full_name":"Asymptotics.isBigO_refl","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[515,8],"def_end_pos":[515,19]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case h.e'_8.h\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\nx✝ : α\n⊢ 1 x✝ = 1 x✝ * 1 x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Part.lean","commit":"","full_name":"Part.some_ne_none","start":[164,0],"end":[167,39],"file_path":"Mathlib/Data/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nx : α\n⊢ some x ≠ none","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nx : α\nh : some x = none\n⊢ False","tactic":"intro h","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nx : α\nh : some x = none\n⊢ False","state_after":"no goals","tactic":"exact true_ne_false (congr_arg Dom h)","premises":[{"full_name":"Part.Dom","def_path":"Mathlib/Data/Part.lean","def_pos":[52,2],"def_end_pos":[52,5]},{"full_name":"true_ne_false","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[718,8],"def_end_pos":[718,21]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","commit":"","full_name":"CochainComplex.HomComplex.δ_zero_cochain_comp","start":[505,0],"end":[510,55],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m n₂ : ℤ\nz₁ : Cochain F G 0\nz₂ : Cochain G K n₂\nm₂ : ℤ\nh₂ : n₂ + 1 = m₂\n⊢ 1 + n₂ = m₂","state_after":"no goals","tactic":"rw [add_comm, h₂]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
{"url":"Mathlib/Analysis/Normed/Module/FiniteDimension.lean","commit":"","full_name":"Basis.opNNNorm_le","start":[275,0],"end":[291,91],"file_path":"Mathlib/Analysis/Normed/Module/FiniteDimension.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M\ne : E\n⊢ ‖u e‖₊ ≤ Fintype.card ι • ‖↑v.equivFunL‖₊ * M * ‖e‖₊","state_after":"𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑v.equivFunL\n⊢ ‖u e‖₊ ≤ Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊","tactic":"set φ := v.equivFunL.toContinuousLinearMap","premises":[{"full_name":"Basis.equivFunL","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[428,4],"def_end_pos":[428,13]},{"full_name":"ContinuousLinearEquiv.toContinuousLinearMap","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1615,4],"def_end_pos":[1615,25]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑v.equivFunL\n⊢ ‖u e‖₊ ≤ Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊","state_after":"no goals","tactic":"calc\n      ‖u e‖₊ = ‖u (∑ i, v.equivFun e i • v i)‖₊ := by rw [v.sum_equivFun]\n      _ = ‖∑ i, v.equivFun e i • (u <| v i)‖₊ := by simp [map_sum, LinearMap.map_smul]\n      _ ≤ ∑ i, ‖v.equivFun e i • (u <| v i)‖₊ := nnnorm_sum_le _ _\n      _ = ∑ i, ‖v.equivFun e i‖₊ * ‖u (v i)‖₊ := by simp only [nnnorm_smul]\n      _ ≤ ∑ i, ‖v.equivFun e i‖₊ * M := by gcongr; apply hu\n      _ = (∑ i, ‖v.equivFun e i‖₊) * M := by rw [Finset.sum_mul]\n      _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) * M := by\n        gcongr\n        calc\n          ∑ i, ‖v.equivFun e i‖₊ ≤ Fintype.card ι • ‖φ e‖₊ := Pi.sum_nnnorm_apply_le_nnnorm _\n          _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) := nsmul_le_nsmul_right (φ.le_opNNNorm e) _\n      _ = Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊ := by simp only [smul_mul_assoc, mul_right_comm]","premises":[{"full_name":"Basis.equivFun","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[817,4],"def_end_pos":[817,18]},{"full_name":"Basis.sum_equivFun","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[853,8],"def_end_pos":[853,26]},{"full_name":"ContinuousLinearMap.le_opNNNorm","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean","def_pos":[104,8],"def_end_pos":[104,19]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_mul","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[41,6],"def_end_pos":[41,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"LinearMap.map_smul","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[348,18],"def_end_pos":[348,26]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"Pi.sum_nnnorm_apply_le_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Constructions.lean","def_pos":[362,14],"def_end_pos":[362,43]},{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]},{"full_name":"mul_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[156,8],"def_end_pos":[156,22]},{"full_name":"nnnorm_smul","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[88,8],"def_end_pos":[88,19]},{"full_name":"nnnorm_sum_le","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1115,2],"def_end_pos":[1115,13]},{"full_name":"nsmul_le_nsmul_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean","def_pos":[31,37],"def_end_pos":[31,57]},{"full_name":"smul_mul_assoc","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[344,6],"def_end_pos":[344,20]}]}]}
{"url":"Mathlib/Data/List/Sym.lean","commit":"","full_name":"List.sym2_eq_nil_iff","start":[51,0],"end":[53,31],"file_path":"Mathlib/Data/List/Sym.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nxs : List α\n⊢ xs.sym2 = [] ↔ xs = []","state_after":"no goals","tactic":"cases xs <;> simp [List.sym2]","premises":[{"full_name":"List.sym2","def_path":"Mathlib/Data/List/Sym.lean","def_pos":[36,14],"def_end_pos":[36,18]}]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.bidirectionalRec_singleton","start":[685,0],"end":[690,25],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nmotive : List α → Sort u_2\nnil : motive []\nsingleton : (a : α) → motive [a]\ncons_append : (a : α) → (l : List α) → (b : α) → motive l → motive (a :: (l ++ [b]))\na : α\n⊢ bidirectionalRec nil singleton cons_append [a] = singleton a","state_after":"no goals","tactic":"simp [bidirectionalRec]","premises":[{"full_name":"List.bidirectionalRec","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[666,4],"def_end_pos":[666,20]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/OneHypercover.lean","commit":"","full_name":"CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve₁","start":[242,0],"end":[247,32],"file_path":"Mathlib/CategoryTheory/Sites/OneHypercover.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{?u.31588, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf₁ f₂ : S.Arrow\nW : C\np₁ : W ⟶ f₁.Y\np₂ : W ⟶ f₂.Y\nw : p₁ ≫ f₁.f = p₂ ≫ f₂.f\n⊢ S.preOneHypercover.sieve₁ p₁ p₂ = ⊤","state_after":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{?u.31588, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf₁ f₂ : S.Arrow\nW : C\np₁ : W ⟶ f₁.Y\np₂ : W ⟶ f₂.Y\nw : p₁ ≫ f₁.f = p₂ ≫ f₂.f\nY : C\nf : Y ⟶ W\n⊢ (S.preOneHypercover.sieve₁ p₁ p₂).arrows f ↔ ⊤.arrows f","tactic":"ext Y f","premises":[]},{"state_before":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{?u.31588, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf₁ f₂ : S.Arrow\nW : C\np₁ : W ⟶ f₁.Y\np₂ : W ⟶ f₂.Y\nw : p₁ ≫ f₁.f = p₂ ≫ f₂.f\nY : C\nf : Y ⟶ W\n⊢ (S.preOneHypercover.sieve₁ p₁ p₂).arrows f ↔ ⊤.arrows f","state_after":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{?u.31588, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf₁ f₂ : S.Arrow\nW : C\np₁ : W ⟶ f₁.Y\np₂ : W ⟶ f₂.Y\nw : p₁ ≫ f₁.f = p₂ ≫ f₂.f\nY : C\nf : Y ⟶ W\n⊢ (S.preOneHypercover.sieve₁ p₁ p₂).arrows f","tactic":"simp only [Sieve.top_apply, iff_true]","premises":[{"full_name":"CategoryTheory.Sieve.top_apply","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[356,8],"def_end_pos":[356,17]},{"full_name":"iff_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[127,16],"def_end_pos":[127,24]}]},{"state_before":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{?u.31588, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf₁ f₂ : S.Arrow\nW : C\np₁ : W ⟶ f₁.Y\np₂ : W ⟶ f₂.Y\nw : p₁ ≫ f₁.f = p₂ ≫ f₂.f\nY : C\nf : Y ⟶ W\n⊢ (S.preOneHypercover.sieve₁ p₁ p₂).arrows f","state_after":"no goals","tactic":"exact ⟨{ w := w}, f, rfl, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/GroupTheory/PGroup.lean","commit":"","full_name":"IsPGroup.comap_of_ker_isPGroup","start":[250,0],"end":[257,62],"file_path":"Mathlib/GroupTheory/PGroup.lean","tactics":[{"state_before":"p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\n⊢ IsPGroup p ↥(Subgroup.comap ϕ H)","state_after":"p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\n⊢ ∃ k, g ^ p ^ k = 1","tactic":"intro g","premises":[]},{"state_before":"p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\n⊢ ∃ k, g ^ p ^ k = 1","state_after":"case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ⟨ϕ ↑g, ⋯⟩ ^ p ^ j = 1\n⊢ ∃ k, g ^ p ^ k = 1","tactic":"obtain ⟨j, hj⟩ := hH ⟨ϕ g.1, g.2⟩","premises":[{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]},{"state_before":"case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ⟨ϕ ↑g, ⋯⟩ ^ p ^ j = 1\n⊢ ∃ k, g ^ p ^ k = 1","state_after":"case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ϕ (↑g ^ p ^ j) = ↑1\n⊢ ∃ k, g ^ p ^ k = 1","tactic":"rw [Subtype.ext_iff, H.coe_pow, Subtype.coe_mk, ← ϕ.map_pow] at hj","premises":[{"full_name":"MonoidHom.map_pow","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[809,18],"def_end_pos":[809,35]},{"full_name":"Subgroup.coe_pow","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[577,8],"def_end_pos":[577,15]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]},{"state_before":"case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ϕ (↑g ^ p ^ j) = ↑1\n⊢ ∃ k, g ^ p ^ k = 1","state_after":"case intro.intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ϕ (↑g ^ p ^ j) = ↑1\nk : ℕ\nhk : ⟨↑g ^ p ^ j, hj⟩ ^ p ^ k = 1\n⊢ ∃ k, g ^ p ^ k = 1","tactic":"obtain ⟨k, hk⟩ := hϕ ⟨g.1 ^ p ^ j, hj⟩","premises":[{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]},{"state_before":"case intro.intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ϕ (↑g ^ p ^ j) = ↑1\nk : ℕ\nhk : ⟨↑g ^ p ^ j, hj⟩ ^ p ^ k = 1\n⊢ ∃ k, g ^ p ^ k = 1","state_after":"case intro.intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ϕ (↑g ^ p ^ j) = ↑1\nk : ℕ\nhk : ↑g ^ p ^ (j + k) = ↑1\n⊢ ∃ k, g ^ p ^ k = 1","tactic":"rw [Subtype.ext_iff, ϕ.ker.coe_pow, Subtype.coe_mk, ← pow_mul, ← pow_add] at hk","premises":[{"full_name":"MonoidHom.ker","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2090,4],"def_end_pos":[2090,7]},{"full_name":"Subgroup.coe_pow","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[577,8],"def_end_pos":[577,15]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]}]},{"state_before":"case intro.intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nhH : IsPGroup p ↥H\nK : Type u_2\ninst✝ : Group K\nϕ : K →* G\nhϕ : IsPGroup p ↥ϕ.ker\ng : ↥(Subgroup.comap ϕ H)\nj : ℕ\nhj : ϕ (↑g ^ p ^ j) = ↑1\nk : ℕ\nhk : ↑g ^ p ^ (j + k) = ↑1\n⊢ ∃ k, g ^ p ^ k = 1","state_after":"no goals","tactic":"exact ⟨j + k, by rwa [Subtype.ext_iff, (H.comap ϕ).coe_pow]⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Subgroup.coe_pow","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[577,8],"def_end_pos":[577,15]},{"full_name":"Subgroup.comap","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1051,4],"def_end_pos":[1051,9]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]}]}
{"url":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","commit":"","full_name":"FractionalIdeal.absNorm_eq'","start":[78,0],"end":[82,19],"file_path":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : Module.Free ℤ R\ninst✝³ : Module.Finite ℤ R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : FractionalIdeal R⁰ K\na : ↥R⁰\nI₀ : Ideal R\nh : a • ↑I = Submodule.map (Algebra.linearMap R K) I₀\n⊢ absNorm I = ↑(Ideal.absNorm I₀) / ↑|(Algebra.norm ℤ) ↑a|","state_after":"no goals","tactic":"rw [absNorm, ← absNorm_div_norm_eq_absNorm_div_norm a I₀ h, MonoidWithZeroHom.coe_mk,\n    ZeroHom.coe_mk]","premises":[{"full_name":"FractionalIdeal.absNorm","def_path":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","def_pos":[55,18],"def_end_pos":[55,25]},{"full_name":"FractionalIdeal.absNorm_div_norm_eq_absNorm_div_norm","def_path":"Mathlib/RingTheory/FractionalIdeal/Norm.lean","def_pos":[36,8],"def_end_pos":[36,44]},{"full_name":"MonoidWithZeroHom.coe_mk","def_path":"Mathlib/Algebra/GroupWithZero/Hom.lean","def_pos":[126,14],"def_end_pos":[126,20]},{"full_name":"ZeroHom.coe_mk","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[484,2],"def_end_pos":[484,13]}]}]}
{"url":"Mathlib/Logic/Function/Basic.lean","commit":"","full_name":"Function.FactorsThrough.extend_apply","start":[617,0],"end":[620,62],"file_path":"Mathlib/Logic/Function/Basic.lean","tactics":[{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\ng : α → γ\nhf : FactorsThrough g f\ne' : β → γ\na : α\n⊢ extend f g e' (f a) = g a","state_after":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\ng : α → γ\nhf : FactorsThrough g f\ne' : β → γ\na : α\n⊢ g (Classical.choose ⋯) = g a","tactic":"simp only [extend_def, dif_pos, exists_apply_eq_apply]","premises":[{"full_name":"Function.extend_def","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[609,8],"def_end_pos":[609,18]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]},{"full_name":"exists_apply_eq_apply","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[310,16],"def_end_pos":[310,37]}]},{"state_before":"α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\ng : α → γ\nhf : FactorsThrough g f\ne' : β → γ\na : α\n⊢ g (Classical.choose ⋯) = g a","state_after":"no goals","tactic":"exact hf (Classical.choose_spec (exists_apply_eq_apply f a))","premises":[{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"exists_apply_eq_apply","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[310,16],"def_end_pos":[310,37]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Linear.lean","commit":"","full_name":"CategoryTheory.ShortComplex.rightHomologyMap'_smul","start":[124,0],"end":[128,99],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Linear.lean","tactics":[{"state_before":"R : Type u_1\nC : Type u_2\ninst✝³ : Semiring R\ninst✝² : Category.{u_3, u_2} C\ninst✝¹ : Preadditive C\ninst✝ : Linear R C\nS₁ S₂ : ShortComplex C\nφ φ' : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\na : R\n⊢ rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂","state_after":"R : Type u_1\nC : Type u_2\ninst✝³ : Semiring R\ninst✝² : Category.{u_3, u_2} C\ninst✝¹ : Preadditive C\ninst✝ : Linear R C\nS₁ S₂ : ShortComplex C\nφ φ' : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\na : R\nγ : RightHomologyMapData φ h₁ h₂\n⊢ rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂","tactic":"have γ : RightHomologyMapData φ h₁ h₂ := default","premises":[{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[344,10],"def_end_pos":[344,30]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]}]},{"state_before":"R : Type u_1\nC : Type u_2\ninst✝³ : Semiring R\ninst✝² : Category.{u_3, u_2} C\ninst✝¹ : Preadditive C\ninst✝ : Linear R C\nS₁ S₂ : ShortComplex C\nφ φ' : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\na : R\nγ : RightHomologyMapData φ h₁ h₂\n⊢ rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂","state_after":"no goals","tactic":"simp only [(γ.smul a).rightHomologyMap'_eq, RightHomologyMapData.smul_φH, γ.rightHomologyMap'_eq]","premises":[{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData.rightHomologyMap'_eq","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[637,6],"def_end_pos":[637,26]},{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData.smul","def_path":"Mathlib/Algebra/Homology/ShortComplex/Linear.lean","def_pos":[115,4],"def_end_pos":[115,8]},{"full_name":"CategoryTheory.ShortComplex.RightHomologyMapData.smul_φH","def_path":"Mathlib/Algebra/Homology/ShortComplex/Linear.lean","def_pos":[114,2],"def_end_pos":[114,7]}]}]}
{"url":"Mathlib/Order/Filter/IndicatorFunction.lean","commit":"","full_name":"Filter.EventuallyEq.mulIndicator_one","start":[106,0],"end":[109,52],"file_path":"Mathlib/Order/Filter/IndicatorFunction.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nM : Type u_3\nE : Type u_4\ninst✝ : One β\nl : Filter α\nf : α → β\ns : Set α\nhf : f =ᶠ[l] 1\n⊢ s.mulIndicator 1 =ᶠ[l] 1","state_after":"no goals","tactic":"rw [mulIndicator_one']","premises":[{"full_name":"Set.mulIndicator_one'","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[166,8],"def_end_pos":[166,25]}]}]}
{"url":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","commit":"","full_name":"MvQPF.wrepr_equiv","start":[125,0],"end":[131,28],"file_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","tactics":[{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\n⊢ WEquiv (wrepr x) x","state_after":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\n⊢ ∀ (a : (P F).A) (f' : (P F).drop.B a ⟹ α) (f : (P F).last.B a → (P F).W α),\n    (∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)) → WEquiv (wrepr ((P F).wMk a f' f)) ((P F).wMk a f' f)","tactic":"apply q.P.w_ind _ x","premises":[{"full_name":"MvPFunctor.w_ind","def_path":"Mathlib/Data/PFunctor/Multivariate/W.lean","def_pos":[177,8],"def_end_pos":[177,13]},{"full_name":"MvQPF.P","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[87,2],"def_end_pos":[87,3]}]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\n⊢ ∀ (a : (P F).A) (f' : (P F).drop.B a ⟹ α) (f : (P F).last.B a → (P F).W α),\n    (∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)) → WEquiv (wrepr ((P F).wMk a f' f)) ((P F).wMk a f' f)","state_after":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr ((P F).wMk a f' f)) ((P F).wMk a f' f)","tactic":"intro a f' f ih","premises":[]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr ((P F).wMk a f' f)) ((P F).wMk a f' f)","state_after":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr ((P F).wMk a f' f)) ((P F).wMk' ((TypeVec.id ::: wrepr) <$$> ⟨a, (P F).appendContents f' f⟩))\n\ncase a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv ((P F).wMk' ((TypeVec.id ::: wrepr) <$$> ⟨a, (P F).appendContents f' f⟩)) ((P F).wMk a f' f)","tactic":"apply WEquiv.trans _ (q.P.wMk' (appendFun id wrepr <$$> ⟨a, q.P.appendContents f' f⟩))","premises":[{"full_name":"MvFunctor.map","def_path":"Mathlib/Control/Functor/Multivariate.lean","def_pos":[30,2],"def_end_pos":[30,5]},{"full_name":"MvPFunctor.appendContents","def_path":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","def_pos":[219,7],"def_end_pos":[219,21]},{"full_name":"MvPFunctor.wMk'","def_path":"Mathlib/Data/PFunctor/Multivariate/W.lean","def_pos":[242,4],"def_end_pos":[242,8]},{"full_name":"MvQPF.P","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[87,2],"def_end_pos":[87,3]},{"full_name":"MvQPF.WEquiv.trans","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[80,4],"def_end_pos":[80,9]},{"full_name":"MvQPF.wrepr","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[116,4],"def_end_pos":[116,9]},{"full_name":"Sigma.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[173,2],"def_end_pos":[173,4]},{"full_name":"TypeVec.appendFun","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[129,4],"def_end_pos":[129,13]},{"full_name":"TypeVec.id","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[64,4],"def_end_pos":[64,6]}]},{"state_before":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv ((P F).wMk' ((TypeVec.id ::: wrepr) <$$> ⟨a, (P F).appendContents f' f⟩)) ((P F).wMk a f' f)","state_after":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv\n    (match ⟨a, splitFun f' (wrepr ∘ f)⟩ with\n    | ⟨a, f⟩ => (P F).wMk a (dropFun f) (lastFun f))\n    ((P F).wMk a f' f)","tactic":"rw [q.P.map_eq, MvPFunctor.wMk', appendFun_comp_splitFun, id_comp]","premises":[{"full_name":"MvPFunctor.map_eq","def_path":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","def_pos":[59,8],"def_end_pos":[59,14]},{"full_name":"MvPFunctor.wMk'","def_path":"Mathlib/Data/PFunctor/Multivariate/W.lean","def_pos":[242,4],"def_end_pos":[242,8]},{"full_name":"MvQPF.P","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[87,2],"def_end_pos":[87,3]},{"full_name":"TypeVec.appendFun_comp_splitFun","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[208,8],"def_end_pos":[208,31]},{"full_name":"TypeVec.id_comp","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[72,8],"def_end_pos":[72,15]}]},{"state_before":"case a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv\n    (match ⟨a, splitFun f' (wrepr ∘ f)⟩ with\n    | ⟨a, f⟩ => (P F).wMk a (dropFun f) (lastFun f))\n    ((P F).wMk a f' f)","state_after":"case a.a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ ∀ (x : (P F).last.B a), WEquiv (lastFun (splitFun f' (wrepr ∘ f)) x) (f x)","tactic":"apply WEquiv.ind","premises":[{"full_name":"MvQPF.WEquiv.ind","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[74,4],"def_end_pos":[74,7]}]},{"state_before":"case a.a\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx : (P F).W α\na : (P F).A\nf' : (P F).drop.B a ⟹ α\nf : (P F).last.B a → (P F).W α\nih : ∀ (i : (P F).last.B a), WEquiv (wrepr (f i)) (f i)\n⊢ ∀ (x : (P F).last.B a), WEquiv (lastFun (splitFun f' (wrepr ∘ f)) x) (f x)","state_after":"no goals","tactic":"exact ih","premises":[]}]}
{"url":"Mathlib/Data/Set/Image.lean","commit":"","full_name":"sigma_mk_preimage_image_eq_self","start":[1393,0],"end":[1394,14],"file_path":"Mathlib/Data/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : α → Type u_2\ni j : α\ns : Set (β i)\n⊢ Sigma.mk i ⁻¹' (Sigma.mk i '' s) = s","state_after":"no goals","tactic":"simp [image]","premises":[{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]}]}]}
{"url":"Mathlib/GroupTheory/Sylow.lean","commit":"","full_name":"Sylow.prime_dvd_card_quotient_normalizer","start":[482,0],"end":[496,92],"file_path":"Mathlib/GroupTheory/Sylow.lean","tactics":[{"state_before":"G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Finite G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Nat.card G\nH : Subgroup G\nhH : Nat.card ↥H = p ^ n\ns : ℕ\nhs : Nat.card G = s * p ^ (n + 1)\n⊢ Nat.card (G ⧸ H) * Nat.card ↥H = s * p * Nat.card ↥H","state_after":"no goals","tactic":"rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]","premises":[{"full_name":"Subgroup.card_eq_card_quotient_mul_card_subgroup","def_path":"Mathlib/GroupTheory/Coset.lean","def_pos":[616,8],"def_end_pos":[616,47]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Finite G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Nat.card G\nH : Subgroup G\nhH : Nat.card ↥H = p ^ n\ns : ℕ\nhs : Nat.card G = s * p ^ (n + 1)\nhcard : Nat.card (G ⧸ H) = s * p\nhm : s * p % p = Nat.card (↥H.normalizer ⧸ comap H.normalizer.subtype H) % p\n⊢ Nat.card (↥H.normalizer ⧸ comap H.normalizer.subtype H) % p = 0","state_after":"no goals","tactic":"rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm","premises":[{"full_name":"Nat.mod_eq_zero_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[69,8],"def_end_pos":[69,26]},{"full_name":"dvd_mul_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[176,8],"def_end_pos":[176,20]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/NumberTheory/MulChar/Basic.lean","commit":"","full_name":"MulChar.ringHomComp_pow","start":[448,0],"end":[452,58],"file_path":"Mathlib/NumberTheory/MulChar/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nχ : MulChar R R'\nf : R' →+* R''\nn : ℕ\n⊢ χ.ringHomComp f ^ n = (χ ^ n).ringHomComp f","state_after":"no goals","tactic":"induction n with\n  | zero => simp only [pow_zero, ringHomComp_one]\n  | succ n ih => simp only [pow_succ, ih, ringHomComp_mul]","premises":[{"full_name":"MulChar.ringHomComp_mul","def_path":"Mathlib/NumberTheory/MulChar/Basic.lean","def_pos":[443,6],"def_end_pos":[443,21]},{"full_name":"MulChar.ringHomComp_one","def_path":"Mathlib/NumberTheory/MulChar/Basic.lean","def_pos":[434,6],"def_end_pos":[434,21]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]}]}
{"url":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","commit":"","full_name":"LocallyLipschitz.continuous","start":[345,0],"end":[351,41],"file_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\n⊢ Continuous f","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\n⊢ ∀ (x : α), ContinuousAt f x","tactic":"rw [continuous_iff_continuousAt]","premises":[{"full_name":"continuous_iff_continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1427,8],"def_end_pos":[1427,35]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\n⊢ ∀ (x : α), ContinuousAt f x","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\nx : α\n⊢ ContinuousAt f x","tactic":"intro x","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\nx : α\n⊢ ContinuousAt f x","state_after":"case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\nx : α\nK : ℝ≥0\nt : Set α\nht : t ∈ 𝓝 x\nhK : LipschitzOnWith K f t\n⊢ ContinuousAt f x","tactic":"rcases (hf x) with ⟨K, t, ht, hK⟩","premises":[]},{"state_before":"case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\nx : α\nK : ℝ≥0\nt : Set α\nht : t ∈ 𝓝 x\nhK : LipschitzOnWith K f t\n⊢ ContinuousAt f x","state_after":"no goals","tactic":"exact (hK.continuousOn).continuousAt ht","premises":[{"full_name":"ContinuousOn.continuousAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[768,8],"def_end_pos":[768,33]},{"full_name":"LipschitzOnWith.continuousOn","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[293,18],"def_end_pos":[293,30]}]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Degrees.lean","commit":"","full_name":"MvPolynomial.degrees_prod","start":[126,0],"end":[128,60],"file_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_3\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ (∏ i ∈ s, f i).degrees ≤ ∑ i ∈ s, (f i).degrees","state_after":"no goals","tactic":"classical exact supDegree_prod_le (map_zero _) (map_add _)","premises":[{"full_name":"AddMonoidAlgebra.supDegree_prod_le","def_path":"Mathlib/Algebra/MonoidAlgebra/Degree.lean","def_pos":[267,8],"def_end_pos":[267,25]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"eq_of_one_div_eq_one_div","start":[470,0],"end":[472,46],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na b c d : α\nh : 1 / a = 1 / b\n⊢ a = b","state_after":"no goals","tactic":"rw [← one_div_one_div a, h, one_div_one_div]","premises":[{"full_name":"one_div_one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[406,8],"def_end_pos":[406,23]}]}]}
{"url":"Mathlib/Order/Interval/Set/Image.lean","commit":"","full_name":"_private.Mathlib.Order.Interval.Set.Image.0.Set.image_subtype_val_Ixx_Iix","start":[248,0],"end":[252,51],"file_path":"Mathlib/Order/Interval/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r x ↑c → q x b\n⊢ {x | p a x ∧ q x b} ∩ {y | r y ↑c} = {y | p a y ∧ r y ↑c}","state_after":"case h\nα : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r x ↑c → q x b\nx✝ : α\n⊢ x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r y ↑c} ↔ x✝ ∈ {y | p a y ∧ r y ↑c}","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r x ↑c → q x b\nx✝ : α\n⊢ x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r y ↑c} ↔ x✝ ∈ {y | p a y ∧ r y ↑c}","state_after":"no goals","tactic":"simp (config := { contextual := true}) [h]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.range_familyOfBFamily'","start":[1016,0],"end":[1023,46],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal.{u}\nho : type r = o\nf : (a : Ordinal.{u}) → a < o → α\n⊢ range (familyOfBFamily' r ho f) = o.brange f","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal.{u}\nho : type r = o\nf : (a : Ordinal.{u}) → a < o → α\na : α\n⊢ a ∈ range (familyOfBFamily' r ho f) → a ∈ o.brange f\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal.{u}\nho : type r = o\nf : (a : Ordinal.{u}) → a < o → α\na : α\n⊢ a ∈ o.brange f → a ∈ range (familyOfBFamily' r ho f)","tactic":"refine Set.ext fun a => ⟨?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Set.ext","def_path":"Mathlib/Init/Set.lean","def_pos":[68,8],"def_end_pos":[68,11]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Functor.lean","commit":"","full_name":"CategoryTheory.Functor.mapTriangleInvRotateIso_hom_app_hom₁","start":[107,0],"end":[114,66],"file_path":"Mathlib/CategoryTheory/Triangulated/Functor.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\nT : Triangle C\n⊢ ((F.mapTriangle ⋙ invRotate D).obj T).mor₁ ≫ (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂).hom =\n    ((F.commShiftIso (-1)).symm.app T.obj₃).hom ≫ ((invRotate C ⋙ F.mapTriangle).obj T).mor₁","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\nT : Triangle C\n⊢ ((F.mapTriangle ⋙ invRotate D).obj T).mor₂ ≫ (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃).hom =\n    (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂).hom ≫ ((invRotate C ⋙ F.mapTriangle).obj T).mor₂","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\nT : Triangle C\n⊢ ((F.mapTriangle ⋙ invRotate D).obj T).mor₃ ≫ (shiftFunctor D 1).map ((F.commShiftIso (-1)).symm.app T.obj₃).hom =\n    (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃).hom ≫ ((invRotate C ⋙ F.mapTriangle).obj T).mor₃","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹¹ : Category.{?u.222118, u_1} C\ninst✝¹⁰ : Category.{?u.222122, u_2} D\ninst✝⁹ : Category.{?u.222126, u_3} E\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : HasShift D ℤ\ninst✝⁶ : HasShift E ℤ\nF : C ⥤ D\ninst✝⁵ : F.CommShift ℤ\nG : D ⥤ E\ninst✝⁴ : G.CommShift ℤ\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\ninst✝¹ inst✝ : F.Additive\n⊢ ∀ {X Y : Triangle C} (f : X ⟶ Y),\n    (F.mapTriangle ⋙ invRotate D).map f ≫\n        ((fun T =>\n              ((F.mapTriangle ⋙ invRotate D).obj T).isoMk ((invRotate C ⋙ F.mapTriangle).obj T)\n                ((F.commShiftIso (-1)).symm.app T.obj₃) (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂)\n                (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃) ⋯ ⋯ ⋯)\n            Y).hom =\n      ((fun T =>\n              ((F.mapTriangle ⋙ invRotate D).obj T).isoMk ((invRotate C ⋙ F.mapTriangle).obj T)\n                ((F.commShiftIso (-1)).symm.app T.obj₃) (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₂)\n                (Iso.refl ((F.mapTriangle ⋙ invRotate D).obj T).obj₃) ⋯ ⋯ ⋯)\n            X).hom ≫\n        (invRotate C ⋙ F.mapTriangle).map f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Algebra/Homology/Embedding/Boundary.lean","commit":"","full_name":"ComplexShape.Embedding.not_boundaryLE_prev","start":[108,0],"end":[113,47],"file_path":"Mathlib/Algebra/Homology/Embedding/Boundary.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\n⊢ ¬e.BoundaryLE i","state_after":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\n⊢ ¬(c'.Rel (e.f i) (c'.next (e.f i)) ∧ ∀ (k : ι), ¬c'.Rel (e.f i) (e.f k))","tactic":"dsimp [BoundaryLE]","premises":[{"full_name":"ComplexShape.Embedding.BoundaryLE","def_path":"Mathlib/Algebra/Homology/Embedding/Boundary.lean","def_pos":[97,4],"def_end_pos":[97,14]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\n⊢ ¬(c'.Rel (e.f i) (c'.next (e.f i)) ∧ ∀ (k : ι), ¬c'.Rel (e.f i) (e.f k))","state_after":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\n⊢ c'.Rel (e.f i) (c'.next (e.f i)) → ∃ x, c'.Rel (e.f i) (e.f x)","tactic":"simp only [not_and, not_forall, not_not]","premises":[{"full_name":"Classical.not_forall","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[137,20],"def_end_pos":[137,30]},{"full_name":"Classical.not_not","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[135,16],"def_end_pos":[135,23]},{"full_name":"not_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[116,16],"def_end_pos":[116,23]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\n⊢ c'.Rel (e.f i) (c'.next (e.f i)) → ∃ x, c'.Rel (e.f i) (e.f x)","state_after":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\na✝ : c'.Rel (e.f i) (c'.next (e.f i))\n⊢ ∃ x, c'.Rel (e.f i) (e.f x)","tactic":"intro","premises":[]},{"state_before":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhi : c.Rel i j\na✝ : c'.Rel (e.f i) (c'.next (e.f i))\n⊢ ∃ x, c'.Rel (e.f i) (e.f x)","state_after":"no goals","tactic":"exact ⟨j, by simpa only [e.rel_iff] using hi⟩","premises":[{"full_name":"ComplexShape.Embedding.rel_iff","def_path":"Mathlib/Algebra/Homology/Embedding/Basic.lean","def_pos":[79,6],"def_end_pos":[79,13]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/GroupTheory/Coset.lean","commit":"","full_name":"QuotientAddGroup.leftRel_apply","start":[238,0],"end":[245,62],"file_path":"Mathlib/GroupTheory/Coset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∃ a, y * ↑a = x) ↔ ∃ a, x⁻¹ * y = ↑a⁻¹","state_after":"no goals","tactic":"simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]","premises":[{"full_name":"Subgroup.coe_inv","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[564,8],"def_end_pos":[564,15]},{"full_name":"eq_mul_inv_iff_mul_eq","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[654,8],"def_end_pos":[654,29]},{"full_name":"inv_mul_eq_iff_eq_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[662,8],"def_end_pos":[662,29]}]},{"state_before":"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∃ a, x⁻¹ * y = ↑a⁻¹) ↔ x⁻¹ * y ∈ s","state_after":"no goals","tactic":"simp [exists_inv_mem_iff_exists_mem]","premises":[{"full_name":"exists_inv_mem_iff_exists_mem","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[150,8],"def_end_pos":[150,37]}]}]}
{"url":"Mathlib/AlgebraicTopology/DoldKan/PInfty.lean","commit":"","full_name":"AlgebraicTopology.DoldKan.PInfty_idem","start":[85,0],"end":[88,23],"file_path":"Mathlib/AlgebraicTopology/DoldKan/PInfty.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ PInfty ≫ PInfty = PInfty","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (PInfty ≫ PInfty).f n = PInfty.f n","tactic":"ext n","premises":[]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (PInfty ≫ PInfty).f n = PInfty.f n","state_after":"no goals","tactic":"exact PInfty_f_idem n","premises":[{"full_name":"AlgebraicTopology.DoldKan.PInfty_f_idem","def_path":"Mathlib/AlgebraicTopology/DoldKan/PInfty.lean","def_pos":[82,8],"def_end_pos":[82,21]}]}]}
{"url":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","commit":"","full_name":"AlgebraicGeometry.StructureSheaf.isUnit_toStalk","start":[426,0],"end":[429,59],"file_path":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nx : ↑(PrimeSpectrum.Top R)\nf : ↥x.asIdeal.primeCompl\n⊢ IsUnit ((toStalk R x) ↑f)","state_after":"R : Type u\ninst✝ : CommRing R\nx : ↑(PrimeSpectrum.Top R)\nf : ↥x.asIdeal.primeCompl\n⊢ IsUnit (((structureSheaf R).presheaf.germ ⟨x, ⋯⟩) ((toOpen R (PrimeSpectrum.basicOpen ↑f)) ↑f))","tactic":"erw [← germ_toOpen R (PrimeSpectrum.basicOpen (f : R)) ⟨x, f.2⟩ (f : R)]","premises":[{"full_name":"AlgebraicGeometry.StructureSheaf.germ_toOpen","def_path":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","def_pos":[413,8],"def_end_pos":[413,19]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"PrimeSpectrum.basicOpen","def_path":"Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean","def_pos":[418,4],"def_end_pos":[418,13]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nx : ↑(PrimeSpectrum.Top R)\nf : ↥x.asIdeal.primeCompl\n⊢ IsUnit (((structureSheaf R).presheaf.germ ⟨x, ⋯⟩) ((toOpen R (PrimeSpectrum.basicOpen ↑f)) ↑f))","state_after":"no goals","tactic":"exact RingHom.isUnit_map _ (isUnit_to_basicOpen_self R f)","premises":[{"full_name":"AlgebraicGeometry.StructureSheaf.isUnit_to_basicOpen_self","def_path":"Mathlib/AlgebraicGeometry/StructureSheaf.lean","def_pos":[422,8],"def_end_pos":[422,32]},{"full_name":"RingHom.isUnit_map","def_path":"Mathlib/Algebra/Ring/Hom/Basic.lean","def_pos":[45,8],"def_end_pos":[45,18]}]}]}
{"url":"Mathlib/Logic/Relation.lean","commit":"","full_name":"Relation.TransGen.lift","start":[447,0],"end":[451,57],"file_path":"Mathlib/Logic/Relation.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nζ : Type u_6\nr : α → α → Prop\na✝ b✝ c d : α\np : β → β → Prop\na b : α\nf : α → β\nh : ∀ (a b : α), r a b → p (f a) (f b)\nhab : TransGen r a b\n⊢ TransGen p (f a) (f b)","state_after":"no goals","tactic":"induction hab with\n  | single hac => exact TransGen.single (h a _ hac)\n  | tail _ hcd hac => exact TransGen.tail hac (h _ _ hcd)","premises":[{"full_name":"Relation.TransGen.single","def_path":"Mathlib/Logic/Relation.lean","def_pos":[235,4],"def_end_pos":[235,10]},{"full_name":"Relation.TransGen.tail","def_path":"Mathlib/Logic/Relation.lean","def_pos":[236,4],"def_end_pos":[236,8]}]}]}
{"url":"Mathlib/Data/Finsupp/Fin.lean","commit":"","full_name":"Finsupp.cons_support","start":[82,0],"end":[87,42],"file_path":"Mathlib/Data/Finsupp/Fin.lean","tactics":[{"state_before":"n : ℕ\ni : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\n⊢ (cons y s).support ⊆ insert 0 (Finset.map (Fin.succEmb n) s.support)","state_after":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin (n + 1)\nhi : i ∈ (cons y s).support\n⊢ i ∈ insert 0 (Finset.map (Fin.succEmb n) s.support)","tactic":"intro i hi","premises":[]},{"state_before":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin (n + 1)\nhi : i ∈ (cons y s).support\n⊢ i ∈ insert 0 (Finset.map (Fin.succEmb n) s.support)","state_after":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin (n + 1)\nhi : i ∈ (cons y s).support\n⊢ i = 0 ∨ ∃ a, ¬s a = 0 ∧ a.succ = i","tactic":"suffices i = 0 ∨ ∃ a, ¬s a = 0 ∧ a.succ = i by simpa","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Fin.succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Basic.lean","def_pos":[34,4],"def_end_pos":[34,8]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Or","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[532,10],"def_end_pos":[532,12]}]},{"state_before":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin (n + 1)\nhi : i ∈ (cons y s).support\n⊢ i = 0 ∨ ∃ a, ¬s a = 0 ∧ a.succ = i","state_after":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin (n + 1)\nhi : i ∈ (cons y s).support\n⊢ ∀ (a : Fin n), i = a.succ → ¬s a = 0 ∧ a.succ = i","tactic":"apply (Fin.eq_zero_or_eq_succ i).imp id (Exists.imp _)","premises":[{"full_name":"Exists.imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[193,8],"def_end_pos":[193,18]},{"full_name":"Fin.eq_zero_or_eq_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[139,8],"def_end_pos":[139,26]},{"full_name":"Or.imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[172,8],"def_end_pos":[172,14]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin (n + 1)\nhi : i ∈ (cons y s).support\n⊢ ∀ (a : Fin n), i = a.succ → ¬s a = 0 ∧ a.succ = i","state_after":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin n\nhi : i.succ ∈ (cons y s).support\n⊢ ¬s i = 0 ∧ i.succ = i.succ","tactic":"rintro i rfl","premises":[]},{"state_before":"n : ℕ\ni✝ : Fin n\nM : Type u_1\ninst✝ : Zero M\ny : M\nt : Fin (n + 1) →₀ M\ns : Fin n →₀ M\ni : Fin n\nhi : i.succ ∈ (cons y s).support\n⊢ ¬s i = 0 ∧ i.succ = i.succ","state_after":"no goals","tactic":"simpa [Finsupp.mem_support_iff] using hi","premises":[{"full_name":"Finsupp.mem_support_iff","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[150,8],"def_end_pos":[150,23]}]}]}
{"url":"Mathlib/Data/Finsupp/BigOperators.lean","commit":"","full_name":"List.support_sum_eq","start":[73,0],"end":[88,22],"file_path":"Mathlib/Data/Finsupp/BigOperators.lean","tactics":[{"state_before":"ι : Type u_1\nM : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : AddMonoid M\nl : List (ι →₀ M)\nhl : Pairwise (_root_.Disjoint on Finsupp.support) l\n⊢ l.sum.support = foldr (fun x x_1 => x.support ⊔ x_1) ∅ l","state_after":"case nil\nι : Type u_1\nM : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : AddMonoid M\nhl : Pairwise (_root_.Disjoint on Finsupp.support) []\n⊢ [].sum.support = foldr (fun x x_1 => x.support ⊔ x_1) ∅ []\n\ncase cons\nι : Type u_1\nM : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : AddMonoid M\nhd : ι →₀ M\ntl : List (ι →₀ M)\nIH : Pairwise (_root_.Disjoint on Finsupp.support) tl → tl.sum.support = foldr (fun x x_1 => x.support ⊔ x_1) ∅ tl\nhl : Pairwise (_root_.Disjoint on Finsupp.support) (hd :: tl)\n⊢ (hd :: tl).sum.support = foldr (fun x x_1 => x.support ⊔ x_1) ∅ (hd :: tl)","tactic":"induction' l with hd tl IH","premises":[]}]}
{"url":"Mathlib/Topology/PartialHomeomorph.lean","commit":"","full_name":"PartialHomeomorph.restr_source'","start":[629,0],"end":[630,37],"file_path":"Mathlib/Topology/PartialHomeomorph.lean","tactics":[{"state_before":"X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne : PartialHomeomorph X Y\ns : Set X\nhs : IsOpen s\n⊢ (e.restr s).source = e.source ∩ s","state_after":"no goals","tactic":"rw [e.restr_source, hs.interior_eq]","premises":[{"full_name":"IsOpen.interior_eq","def_path":"Mathlib/Topology/Basic.lean","def_pos":[228,8],"def_end_pos":[228,26]},{"full_name":"PartialHomeomorph.restr_source","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[620,79],"def_end_pos":[620,85]}]}]}
{"url":"Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean","commit":"","full_name":"Multiset.le_sum_nonempty_of_subadditive_on_pred","start":[98,0],"end":[112,85],"file_path":"Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\ns : Multiset α\nhs_nonempty : s ≠ ∅\nhs : ∀ a ∈ s, p a\n⊢ f s.prod ≤ (map f s).prod","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\n⊢ ∀ (s : Multiset α), s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod","tactic":"revert s","premises":[]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\n⊢ ∀ (s : Multiset α), s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod","state_after":"case refine_1\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\n⊢ 0 ≠ ∅ → (∀ a ∈ 0, p a) → f (prod 0) ≤ (map f 0).prod\n\ncase refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\n⊢ ∀ (a : α) (s : Multiset α),\n    (s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod) →\n      a ::ₘ s ≠ ∅ → (∀ a_2 ∈ a ::ₘ s, p a_2) → f (a ::ₘ s).prod ≤ (map f (a ::ₘ s)).prod","tactic":"refine Multiset.induction ?_ ?_","premises":[{"full_name":"Multiset.induction","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[147,18],"def_end_pos":[147,27]}]},{"state_before":"case refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\n⊢ ∀ (a : α) (s : Multiset α),\n    (s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod) →\n      a ::ₘ s ≠ ∅ → (∀ a_2 ∈ a ::ₘ s, p a_2) → f (a ::ₘ s).prod ≤ (map f (a ::ₘ s)).prod","state_after":"case refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ f (a ::ₘ s).prod ≤ (map f (a ::ₘ s)).prod","tactic":"rintro a s hs - hsa_prop","premises":[]},{"state_before":"case refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ f (a ::ₘ s).prod ≤ (map f (a ::ₘ s)).prod","state_after":"case refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","tactic":"rw [prod_cons, map_cons, prod_cons]","premises":[{"full_name":"Multiset.map_cons","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1038,8],"def_end_pos":[1038,16]},{"full_name":"Multiset.prod_cons","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[65,8],"def_end_pos":[65,17]}]},{"state_before":"case refine_2\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","state_after":"case pos\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : s = ∅\n⊢ f (a * s.prod) ≤ f a * (map f s).prod\n\ncase neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","tactic":"by_cases hs_empty : s = ∅","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","state_after":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhsa_restrict : ∀ x ∈ s, p x\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","tactic":"have hsa_restrict : ∀ x, x ∈ s → p x := fun x hx => hsa_prop x (mem_cons_of_mem hx)","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Multiset.mem_cons_of_mem","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[221,8],"def_end_pos":[221,23]}]},{"state_before":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhsa_restrict : ∀ x ∈ s, p x\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","state_after":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhsa_restrict : ∀ x ∈ s, p x\nhp_sup : p s.prod\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","tactic":"have hp_sup : p s.prod := prod_induction_nonempty p hp_mul hs_empty hsa_restrict","premises":[{"full_name":"Multiset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[38,4],"def_end_pos":[38,8]},{"full_name":"Multiset.prod_induction_nonempty","def_path":"Mathlib/Algebra/BigOperators/Group/Multiset.lean","def_pos":[178,8],"def_end_pos":[178,31]}]},{"state_before":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhsa_restrict : ∀ x ∈ s, p x\nhp_sup : p s.prod\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","state_after":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhsa_restrict : ∀ x ∈ s, p x\nhp_sup : p s.prod\nhp_a : p a\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","tactic":"have hp_a : p a := hsa_prop a (mem_cons_self a s)","premises":[{"full_name":"Multiset.mem_cons_self","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[225,8],"def_end_pos":[225,21]}]},{"state_before":"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\na : α\ns : Multiset α\nhs : s ≠ ∅ → (∀ a ∈ s, p a) → f s.prod ≤ (map f s).prod\nhsa_prop : ∀ a_1 ∈ a ::ₘ s, p a_1\nhs_empty : ¬s = ∅\nhsa_restrict : ∀ x ∈ s, p x\nhp_sup : p s.prod\nhp_a : p a\n⊢ f (a * s.prod) ≤ f a * (map f s).prod","state_after":"no goals","tactic":"exact (h_mul a _ hp_a hp_sup).trans (mul_le_mul_left' (hs hs_empty hsa_restrict) _)","premises":[{"full_name":"mul_le_mul_left'","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[62,8],"def_end_pos":[62,24]}]}]}
{"url":"Mathlib/Algebra/Order/Module/Defs.lean","commit":"","full_name":"smul_le_of_le_one_left","start":[668,0],"end":[669,63],"file_path":"Mathlib/Algebra/Order/Module/Defs.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁶ : Monoid α\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nhb : 0 ≤ b\nh : a ≤ 1\n⊢ a • b ≤ b","state_after":"no goals","tactic":"simpa only [one_smul] using smul_le_smul_of_nonneg_right h hb","premises":[{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"smul_le_smul_of_nonneg_right","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[308,16],"def_end_pos":[308,44]}]}]}
{"url":"Mathlib/RingTheory/Jacobson.lean","commit":"","full_name":"_private.Mathlib.RingTheory.Jacobson.0.Ideal.MvPolynomial.aux_IH","start":[609,0],"end":[638,22],"file_path":"Mathlib/RingTheory/Jacobson.lean","tactics":[{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"let Q := P.comap v.toAlgHom.toRingHom","premises":[{"full_name":"AlgEquiv.toAlgHom","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[220,4],"def_end_pos":[220,12]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"have hw : Ideal.map v Q = P := map_comap_of_surjective v v.surjective P","premises":[{"full_name":"AlgEquiv.surjective","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[251,18],"def_end_pos":[251,28]},{"full_name":"Ideal.map","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[34,4],"def_end_pos":[34,7]},{"full_name":"Ideal.map_comap_of_surjective","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[247,8],"def_end_pos":[247,31]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"haveI hQ : IsMaximal Q := comap_isMaximal_of_surjective _ v.surjective","premises":[{"full_name":"AlgEquiv.surjective","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[251,18],"def_end_pos":[251,28]},{"full_name":"Ideal.IsMaximal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[246,6],"def_end_pos":[246,15]},{"full_name":"Ideal.comap_isMaximal_of_surjective","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[387,8],"def_end_pos":[387,37]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"let w : (S[X] ⧸ Q) ≃ₐ[R] (T ⧸ P) := Ideal.quotientEquivAlg Q P v hw.symm","premises":[{"full_name":"AlgEquiv","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[26,10],"def_end_pos":[26,18]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Ideal.quotientEquivAlg","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[573,4],"def_end_pos":[573,20]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"let Q' := Q.comap (Polynomial.C)","premises":[{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"let w' : (S ⧸ Q') →ₐ[R] (S[X] ⧸ Q) := Ideal.quotientMapₐ Q (Cₐ R S) le_rfl","premises":[{"full_name":"AlgHom","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[28,10],"def_end_pos":[28,16]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Ideal.quotientMapₐ","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[557,4],"def_end_pos":[557,16]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"_private.Mathlib.RingTheory.Jacobson.0.Ideal.MvPolynomial.Cₐ","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[605,26],"def_end_pos":[605,28]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","tactic":"have h_eq : algebraMap R (T ⧸ P) =\n    w.toRingEquiv.toRingHom.comp (w'.toRingHom.comp (algebraMap R (S ⧸ Q'))) := by\n    ext r\n    simp only [AlgEquiv.toAlgHom_eq_coe, AlgHom.toRingHom_eq_coe, AlgEquiv.toRingEquiv_eq_coe,\n      RingEquiv.toRingHom_eq_coe, AlgHom.comp_algebraMap_of_tower, coe_comp, coe_coe,\n      AlgEquiv.coe_ringEquiv, Function.comp_apply, AlgEquiv.commutes]","premises":[{"full_name":"AlgEquiv.coe_ringEquiv","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[170,8],"def_end_pos":[170,21]},{"full_name":"AlgEquiv.commutes","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[196,8],"def_end_pos":[196,16]},{"full_name":"AlgEquiv.toAlgHom_eq_coe","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[226,8],"def_end_pos":[226,23]},{"full_name":"AlgEquiv.toRingEquiv_eq_coe","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[162,8],"def_end_pos":[162,26]},{"full_name":"AlgHom.comp_algebraMap_of_tower","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[155,8],"def_end_pos":[155,46]},{"full_name":"AlgHom.toRingHom_eq_coe","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[151,8],"def_end_pos":[151,24]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"RingEquiv.toRingHom","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[594,4],"def_end_pos":[594,13]},{"full_name":"RingEquiv.toRingHom_eq_coe","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[600,16],"def_end_pos":[600,32]},{"full_name":"RingHom.coe_coe","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[392,8],"def_end_pos":[392,15]},{"full_name":"RingHom.coe_comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[572,8],"def_end_pos":[572,16]},{"full_name":"RingHom.comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[563,4],"def_end_pos":[563,8]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ (algebraMap R (T ⧸ P)).IsIntegral","state_after":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ (w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))).IsIntegral","tactic":"rw [h_eq]","premises":[]},{"state_before":"n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ (w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))).IsIntegral","state_after":"case hf\nn : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ (w'.comp (algebraMap R (S ⧸ Q'))).IsIntegral\n\ncase hg\nn : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ w.toRingEquiv.toRingHom.IsIntegral","tactic":"apply RingHom.IsIntegral.trans","premises":[{"full_name":"RingHom.IsIntegral.trans","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[584,18],"def_end_pos":[584,42]}]}]}
{"url":"Mathlib/CategoryTheory/Dialectica/Monoidal.lean","commit":"","full_name":"CategoryTheory.Dial.braiding_inv_f","start":[124,0],"end":[127,70],"file_path":"Mathlib/CategoryTheory/Dialectica/Monoidal.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\n⊢ (X.tensorObj Y).rel =\n    (Subobject.pullback (prod.map (prod.braiding X.src Y.src).hom (prod.braiding X.tgt Y.tgt).hom)).obj\n      (Y.tensorObj X).rel","state_after":"no goals","tactic":"simp [Subobject.inf_pullback, ← Subobject.pullback_comp, inf_comm]","premises":[{"full_name":"CategoryTheory.Subobject.inf_pullback","def_path":"Mathlib/CategoryTheory/Subobject/Lattice.lean","def_pos":[424,8],"def_end_pos":[424,20]},{"full_name":"CategoryTheory.Subobject.pullback_comp","def_path":"Mathlib/CategoryTheory/Subobject/Basic.lean","def_pos":[498,8],"def_end_pos":[498,21]},{"full_name":"inf_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[385,8],"def_end_pos":[385,16]}]}]}
{"url":"Mathlib/Algebra/Ring/Defs.lean","commit":"","full_name":"add_sq'","start":[244,0],"end":[245,55],"file_path":"Mathlib/Algebra/Ring/Defs.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : CommSemiring α\na✝ b✝ c a b : α\n⊢ (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b","state_after":"no goals","tactic":"rw [add_sq, add_assoc, add_comm _ (b ^ 2), add_assoc]","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"add_sq","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[241,6],"def_end_pos":[241,12]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.pred_succ","start":[159,0],"end":[162,85],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal.{u_4}\n⊢ (succ o).pred = o","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal.{u_4}\nh : ∃ a, succ o = succ a\n⊢ (succ o).pred = o","tactic":"have h : ∃ a, succ o = succ a := ⟨_, rfl⟩","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Order.succ","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[203,4],"def_end_pos":[203,8]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal.{u_4}\nh : ∃ a, succ o = succ a\n⊢ (succ o).pred = o","state_after":"no goals","tactic":"simpa only [pred, dif_pos h] using (succ_injective <| Classical.choose_spec h).symm","premises":[{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Order.succ_injective","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[433,8],"def_end_pos":[433,22]},{"full_name":"Ordinal.pred","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[156,4],"def_end_pos":[156,8]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]}]}
{"url":"Mathlib/Order/CompleteBooleanAlgebra.lean","commit":"","full_name":"CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq'","start":[231,0],"end":[243,52],"file_path":"Mathlib/Order/CompleteBooleanAlgebra.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\nminAx : MinimalAxioms α\nf : (a : ι) → κ a → α\n⊢ let x := minAx.toCompleteLattice;\n  ⨅ i, ⨆ j, f i j = ⨆ g, ⨅ i, f i (g i)","state_after":"α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\nminAx : MinimalAxioms α\nf : (a : ι) → κ a → α\nx✝ : CompleteLattice α := minAx.toCompleteLattice\n⊢ let x := minAx.toCompleteLattice;\n  ⨅ i, ⨆ j, f i j = ⨆ g, ⨅ i, f i (g i)","tactic":"let _ := minAx.toCompleteLattice","premises":[]},{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\nminAx : MinimalAxioms α\nf : (a : ι) → κ a → α\nx✝ : CompleteLattice α := minAx.toCompleteLattice\n⊢ let x := minAx.toCompleteLattice;\n  ⨅ i, ⨆ j, f i j = ⨆ g, ⨅ i, f i (g i)","state_after":"α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\nminAx : MinimalAxioms α\nf : (a : ι) → κ a → α\nx✝ : CompleteLattice α := minAx.toCompleteLattice\n⊢ ⨅ i, ⨆ j, f i j ≤ ⨆ g, ⨅ i, f i (g i)","tactic":"refine le_antisymm ?_ le_iInf_iSup","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_iInf_iSup","def_path":"Mathlib/Order/CompleteBooleanAlgebra.lean","def_pos":[122,8],"def_end_pos":[122,20]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","commit":"","full_name":"CochainComplex.mappingCone.mapHomologicalComplexXIso_eq","start":[592,0],"end":[595,5],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_4, u_1} C\ninst✝⁵ : Category.{u_3, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ mapHomologicalComplexXIso φ H n = mapHomologicalComplexXIso' φ H n m hnm","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_4, u_1} C\ninst✝⁵ : Category.{u_3, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn : ℤ\n⊢ mapHomologicalComplexXIso φ H n = mapHomologicalComplexXIso' φ H n (n + 1) ⋯","tactic":"subst hnm","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_4, u_1} C\ninst✝⁵ : Category.{u_3, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn : ℤ\n⊢ mapHomologicalComplexXIso φ H n = mapHomologicalComplexXIso' φ H n (n + 1) ⋯","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","commit":"","full_name":"Mon_.mul_associator","start":[383,0],"end":[396,28],"file_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫\n      (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","tactic":"simp only [tensor_obj, prodMonoidal_tensorObj, Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.prodMonoidal_tensorObj","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[936,9],"def_end_pos":[936,18]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_obj","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[775,2],"def_end_pos":[775,7]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","tactic":"slice_lhs 2 3 => rw [← Category.id_comp P.mul, tensor_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[166,2],"def_end_pos":[166,13]},{"full_name":"Mon_.mul","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[35,2],"def_end_pos":[35,5]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","tactic":"slice_lhs 3 4 => rw [associator_naturality]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_naturality","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[175,2],"def_end_pos":[175,23]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)","tactic":"slice_rhs 3 4 => rw [← Category.id_comp M.mul, tensor_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[166,2],"def_end_pos":[166,13]},{"full_name":"Mon_.mul","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[35,2],"def_end_pos":[35,5]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      tensor_μ C (M.X, N.X) (M.X, N.X) ▷ (P.X ⊗ P.X) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)","tactic":"simp only [tensorHom_id, id_tensorHom]","premises":[{"full_name":"CategoryTheory.MonoidalCategory.id_tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[222,8],"def_end_pos":[222,20]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[227,8],"def_end_pos":[227,20]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      tensor_μ C (M.X, N.X) (M.X, N.X) ▷ (P.X ⊗ P.X) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)","state_after":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n        tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫\n      (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)","tactic":"slice_lhs 1 3 => rw [associator_monoidal]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.associator_monoidal","def_path":"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean","def_pos":[644,8],"def_end_pos":[644,27]}]},{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n        tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫\n      (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)","state_after":"no goals","tactic":"simp only [Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]}]}
{"url":"Mathlib/CategoryTheory/Preadditive/Projective.lean","commit":"","full_name":"CategoryTheory.Adjunction.projective_of_map_projective","start":[200,0],"end":[206,36],"file_path":"Mathlib/CategoryTheory/Preadditive/Projective.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\n⊢ ∃ f', f' ≫ g = f","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\n⊢ ∃ f', f' ≫ g = f","tactic":"haveI := Adjunction.leftAdjointPreservesColimits.{0, 0} adj","premises":[{"full_name":"CategoryTheory.Adjunction.leftAdjointPreservesColimits","def_path":"Mathlib/CategoryTheory/Adjunction/Limits.lean","def_pos":[86,4],"def_end_pos":[86,32]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\n⊢ ∃ f', f' ≫ g = f","state_after":"case intro\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P ⟶ F.obj E✝\nhf' : f' ≫ F.map g = F.map f\n⊢ ∃ f', f' ≫ g = f","tactic":"rcases (@hP).1 (F.map f) (F.map g) with ⟨f', hf'⟩","premises":[{"full_name":"CategoryTheory.Projective.factors","def_path":"Mathlib/CategoryTheory/Preadditive/Projective.lean","def_pos":[46,2],"def_end_pos":[46,9]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"case intro\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P ⟶ F.obj E✝\nhf' : f' ≫ F.map g = F.map f\n⊢ ∃ f', f' ≫ g = f","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P ⟶ F.obj E✝\nhf' : f' ≫ F.map g = F.map f\n⊢ (adj.unit.app P ≫ G.map f' ≫ inv (adj.unit.app E✝)) ≫ g = f","tactic":"use adj.unit.app _ ≫ G.map f' ≫ (inv <| adj.unit.app _)","premises":[{"full_name":"CategoryTheory.Adjunction.unit","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[63,2],"def_end_pos":[63,6]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[235,18],"def_end_pos":[235,21]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P ⟶ F.obj E✝\nhf' : f' ≫ F.map g = F.map f\n⊢ (adj.unit.app P ≫ G.map f' ≫ inv (adj.unit.app E✝)) ≫ g = f","state_after":"no goals","tactic":"exact F.map_injective (by simpa)","premises":[{"full_name":"CategoryTheory.Functor.map_injective","def_path":"Mathlib/CategoryTheory/Functor/FullyFaithful.lean","def_pos":[57,8],"def_end_pos":[57,21]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.support_erase","start":[899,0],"end":[902,55],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn : ℕ\n⊢ (erase n p).support = p.support.erase n","state_after":"case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\ntoFinsupp✝ : R[ℕ]\n⊢ (erase n { toFinsupp := toFinsupp✝ }).support = { toFinsupp := toFinsupp✝ }.support.erase n","tactic":"rcases p with ⟨⟩","premises":[]},{"state_before":"case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\ntoFinsupp✝ : R[ℕ]\n⊢ (erase n { toFinsupp := toFinsupp✝ }).support = { toFinsupp := toFinsupp✝ }.support.erase n","state_after":"no goals","tactic":"simp only [support, erase_def, Finsupp.support_erase]","premises":[{"full_name":"Finsupp.support_erase","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[550,8],"def_end_pos":[550,21]},{"full_name":"Polynomial.erase_def","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[885,16],"def_end_pos":[885,21]},{"full_name":"Polynomial.support","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[348,4],"def_end_pos":[348,11]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean","commit":"","full_name":"InnerProductGeometry.angle_eq_zero_iff","start":[184,0],"end":[189,67],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean","tactics":[{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ angle x y = 0 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x","state_after":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) ≤ 1","tactic":"rw [angle, ← real_inner_div_norm_mul_norm_eq_one_iff, Real.arccos_eq_zero, LE.le.le_iff_eq,\n    eq_comm]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"InnerProductGeometry.angle","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean","def_pos":[43,4],"def_end_pos":[43,9]},{"full_name":"LE.le.le_iff_eq","def_path":"Mathlib/Order/Basic.lean","def_pos":[238,8],"def_end_pos":[238,17]},{"full_name":"Real.arccos_eq_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean","def_pos":[330,8],"def_end_pos":[330,22]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"real_inner_div_norm_mul_norm_eq_one_iff","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1460,8],"def_end_pos":[1460,47]}]},{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) ≤ 1","state_after":"no goals","tactic":"exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"abs_le","def_path":"Mathlib/Algebra/Order/Group/Abs.lean","def_pos":[70,8],"def_end_pos":[70,14]},{"full_name":"abs_real_inner_div_norm_mul_norm_le_one","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1321,8],"def_end_pos":[1321,47]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.RightHomologyData.isIso_p","start":[129,0],"end":[131,66],"file_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\n⊢ S.f ≫ 𝟙 S.X₂ = 0","state_after":"no goals","tactic":"rw [hf, comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\n⊢ h.descQ (𝟙 S.X₂) ⋯ ≫ h.p = 𝟙 h.Q","state_after":"no goals","tactic":"simp only [← cancel_epi h.p, p_descQ_assoc, id_comp, comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.ShortComplex.RightHomologyData.p","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[48,2],"def_end_pos":[48,3]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.mem_nsmul","start":[638,0],"end":[645,18],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\n⊢ a ∈ n • s ↔ n ≠ 0 ∧ a ∈ s","state_after":"case refine_1\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nha : a ∈ n • s\n⊢ n ≠ 0\n\ncase refine_2\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nh : n ≠ 0 ∧ a ∈ s\n⊢ a ∈ n • s","tactic":"refine ⟨fun ha ↦ ⟨?_, mem_of_mem_nsmul ha⟩, fun h => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Multiset.mem_of_mem_nsmul","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[631,8],"def_end_pos":[631,24]}]},{"state_before":"case refine_2\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nh : n ≠ 0 ∧ a ∈ s\n⊢ a ∈ n • s","state_after":"case refine_2.intro\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nh : n.succ ≠ 0 ∧ a ∈ s\n⊢ a ∈ n.succ • s","tactic":"obtain ⟨n, rfl⟩ := exists_eq_succ_of_ne_zero h.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Nat.exists_eq_succ_of_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[691,8],"def_end_pos":[691,33]}]},{"state_before":"case refine_2.intro\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nh : n.succ ≠ 0 ∧ a ∈ s\n⊢ a ∈ n.succ • s","state_after":"case refine_2.intro\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nh : n.succ ≠ 0 ∧ a ∈ s\n⊢ a ∈ n • s ∨ a ∈ s","tactic":"rw [succ_nsmul, mem_add]","premises":[{"full_name":"Multiset.mem_add","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[628,8],"def_end_pos":[628,15]},{"full_name":"succ_nsmul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[566,14],"def_end_pos":[566,24]}]},{"state_before":"case refine_2.intro\nα : Type u_1\nβ : Type v\nγ : Type u_2\na : α\ns : Multiset α\nn : ℕ\nh : n.succ ≠ 0 ∧ a ∈ s\n⊢ a ∈ n • s ∨ a ∈ s","state_after":"no goals","tactic":"exact Or.inr h.2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]}]}]}
{"url":"Mathlib/Combinatorics/Enumerative/Composition.lean","commit":"","full_name":"Composition.index_embedding","start":[369,0],"end":[373,26],"file_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","tactics":[{"state_before":"n : ℕ\nc : Composition n\ni : Fin c.length\nj : Fin (c.blocksFun i)\n⊢ c.index ((c.embedding i) j) = i","state_after":"n : ℕ\nc : Composition n\ni : Fin c.length\nj : Fin (c.blocksFun i)\n⊢ i = c.index ((c.embedding i) j)","tactic":"symm","premises":[]},{"state_before":"n : ℕ\nc : Composition n\ni : Fin c.length\nj : Fin (c.blocksFun i)\n⊢ i = c.index ((c.embedding i) j)","state_after":"n : ℕ\nc : Composition n\ni : Fin c.length\nj : Fin (c.blocksFun i)\n⊢ (c.embedding i) j ∈ Set.range ⇑(c.embedding i)","tactic":"rw [← mem_range_embedding_iff']","premises":[{"full_name":"Composition.mem_range_embedding_iff'","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[359,8],"def_end_pos":[359,32]}]},{"state_before":"n : ℕ\nc : Composition n\ni : Fin c.length\nj : Fin (c.blocksFun i)\n⊢ (c.embedding i) j ∈ Set.range ⇑(c.embedding i)","state_after":"no goals","tactic":"apply Set.mem_range_self","premises":[{"full_name":"Set.mem_range_self","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[148,22],"def_end_pos":[148,36]}]}]}
{"url":"Mathlib/MeasureTheory/Function/AEEqFun.lean","commit":"","full_name":"MeasureTheory.AEEqFun.ext_iff","start":[160,0],"end":[161,38],"file_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf g : α →ₘ[μ] β\nh : f = g\n⊢ ↑f =ᶠ[ae μ] ↑g","state_after":"no goals","tactic":"rw [h]","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Category.lean","commit":"","full_name":"CategoryTheory.MonoidalCategory.tensor_left_iff","start":[762,0],"end":[762,96],"file_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","tactics":[{"state_before":"C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z X Y : C\nf g : X ⟶ Y\n⊢ 𝟙 (𝟙_ C) ⊗ f = 𝟙 (𝟙_ C) ⊗ g ↔ f = g","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/Cover/Open.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.openCoverOfIsIso_map","start":[114,0],"end":[125,18],"file_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","tactics":[{"state_before":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ x ∈ Set.range ⇑((fun x => f) ((fun x => PUnit.unit) x)).val.base","state_after":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ x ∈ Set.univ\n\nX✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ Function.Surjective ⇑((fun x => f) ((fun x => PUnit.unit) x)).val.base","tactic":"rw [Set.range_iff_surjective.mpr]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.range_iff_surjective","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[586,8],"def_end_pos":[586,28]}]},{"state_before":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ x ∈ Set.univ\n\nX✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ Function.Surjective ⇑((fun x => f) ((fun x => PUnit.unit) x)).val.base","state_after":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ Function.Surjective ⇑((fun x => f) ((fun x => PUnit.unit) x)).val.base","tactic":"all_goals try trivial","premises":[{"full_name":"True.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[192,4],"def_end_pos":[192,9]}]},{"state_before":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ Function.Surjective ⇑((fun x => f) ((fun x => PUnit.unit) x)).val.base","state_after":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ Epi ((fun x => f) ((fun x => PUnit.unit) x)).val.base","tactic":"rw [← TopCat.epi_iff_surjective]","premises":[{"full_name":"TopCat.epi_iff_surjective","def_path":"Mathlib/Topology/Category/TopCat/EpiMono.lean","def_pos":[26,8],"def_end_pos":[26,26]}]},{"state_before":"X✝ Y✝ Z : Scheme\n𝒰 : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰.J), HasPullback (𝒰.map x ≫ f✝) g\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nx : ↑↑Y.toPresheafedSpace\n⊢ Epi ((fun x => f) ((fun x => PUnit.unit) x)).val.base","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean","commit":"","full_name":"MeasureTheory.Measure.setIntegral_comp_smul","start":[102,0],"end":[115,7],"file_path":"Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean","tactics":[{"state_before":"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns✝ : Set E\nf : E → F\nR : ℝ\ns : Set E\nhR : R ≠ 0\n⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ","state_after":"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns✝ : Set E\nf : E → F\nR : ℝ\ns : Set E\nhR : R ≠ 0\ne : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv\n⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ","tactic":"let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv","premises":[{"full_name":"Homeomorph.smul","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[206,4],"def_end_pos":[206,19]},{"full_name":"Homeomorph.toMeasurableEquiv","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[520,4],"def_end_pos":[520,32]},{"full_name":"MeasurableEquiv","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[130,10],"def_end_pos":[130,25]},{"full_name":"Units.mk0","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[161,4],"def_end_pos":[161,7]}]},{"state_before":"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns✝ : Set E\nf : E → F\nR : ℝ\ns : Set E\nhR : R ≠ 0\ne : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv\n⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ","state_after":"no goals","tactic":"calc\n  ∫ x in s, f (R • x) ∂μ\n    = ∫ x in e ⁻¹' (e.symm ⁻¹' s), f (e x) ∂μ := by simp [← preimage_comp]; rfl\n  _ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm\n  _ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.symm ⁻¹' s, f y ∂μ := by\n    simp [map_addHaar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]\n  _ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by\n    congr\n    ext y\n    rw [mem_smul_set_iff_inv_smul_mem₀ hR]\n    rfl","premises":[{"full_name":"ENNReal.toReal_ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[202,8],"def_end_pos":[202,21]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasurableEquiv.symm","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[184,4],"def_end_pos":[184,8]},{"full_name":"MeasureTheory.Measure.map","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1090,16],"def_end_pos":[1090,19]},{"full_name":"MeasureTheory.Measure.map_addHaar_smul","def_path":"Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean","def_pos":[336,8],"def_end_pos":[336,24]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"MeasureTheory.integral_smul_measure","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1509,8],"def_end_pos":[1509,29]},{"full_name":"MeasureTheory.setIntegral_map_equiv","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[564,8],"def_end_pos":[564,29]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.mem_smul_set_iff_inv_smul_mem₀","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[824,8],"def_end_pos":[824,38]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Set.preimage_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[130,8],"def_end_pos":[130,21]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"abs_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,29],"def_end_pos":[95,39]}]}]}
{"url":"Mathlib/RingTheory/FinitePresentation.lean","commit":"","full_name":"Algebra.FinitePresentation.ker_fG_of_surjective","start":[377,0],"end":[384,95],"file_path":"Mathlib/RingTheory/FinitePresentation.lean","tactics":[{"state_before":"R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\n⊢ (RingHom.ker f.toRingHom).FG","state_after":"case intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ (RingHom.ker f.toRingHom).FG","tactic":"obtain ⟨n, g, hg, _⟩ := FinitePresentation.out (R := R) (A := A)","premises":[{"full_name":"Algebra.FinitePresentation.out","def_path":"Mathlib/RingTheory/FinitePresentation.lean","def_pos":[40,2],"def_end_pos":[40,5]}]},{"state_before":"case intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ (RingHom.ker f.toRingHom).FG","state_after":"case h.e'_3\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ RingHom.ker f.toRingHom = Ideal.map g.toRingHom (RingHom.ker (f.comp g).toRingHom)","tactic":"convert (ker_fg_of_mvPolynomial (f.comp g) (hf.comp hg)).map g.toRingHom","premises":[{"full_name":"AlgHom.comp","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[274,4],"def_end_pos":[274,8]},{"full_name":"Algebra.FinitePresentation.ker_fg_of_mvPolynomial","def_path":"Mathlib/RingTheory/FinitePresentation.lean","def_pos":[305,8],"def_end_pos":[305,30]},{"full_name":"Function.Surjective.comp","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[112,8],"def_end_pos":[112,23]},{"full_name":"Ideal.FG.map","def_path":"Mathlib/RingTheory/Finiteness.lean","def_pos":[455,8],"def_end_pos":[455,14]}]},{"state_before":"case h.e'_3\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ RingHom.ker f.toRingHom = Ideal.map g.toRingHom (RingHom.ker (f.comp g).toRingHom)","state_after":"case h.e'_3\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ Ideal.comap ↑f ⊥ = Ideal.map (↑g) (Ideal.comap ((↑f).comp ↑g) ⊥)","tactic":"simp_rw [RingHom.ker_eq_comap_bot, AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom]","premises":[{"full_name":"AlgHom.comp_toRingHom","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[285,8],"def_end_pos":[285,22]},{"full_name":"AlgHom.toRingHom_eq_coe","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[151,8],"def_end_pos":[151,24]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"RingHom.ker_eq_comap_bot","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[529,8],"def_end_pos":[529,24]}]},{"state_before":"case h.e'_3\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ Ideal.comap ↑f ⊥ = Ideal.map (↑g) (Ideal.comap ((↑f).comp ↑g) ⊥)","state_after":"no goals","tactic":"rw [← Ideal.comap_comap, Ideal.map_comap_of_surjective (g : MvPolynomial (Fin n) R →+* A) hg]","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Ideal.comap_comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[118,8],"def_end_pos":[118,19]},{"full_name":"Ideal.map_comap_of_surjective","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[247,8],"def_end_pos":[247,31]},{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"RingHom","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[297,10],"def_end_pos":[297,17]}]}]}
{"url":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","commit":"","full_name":"Metric.infDist_zero_of_mem_closure","start":[529,0],"end":[533,30],"file_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhx : x ∈ closure s\n⊢ infDist x s = 0","state_after":"ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhx : x ∈ closure s\n⊢ infDist x (closure s) = 0","tactic":"rw [← infDist_closure]","premises":[{"full_name":"Metric.infDist_closure","def_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","def_pos":[526,8],"def_end_pos":[526,23]}]},{"state_before":"ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhx : x ∈ closure s\n⊢ infDist x (closure s) = 0","state_after":"no goals","tactic":"exact infDist_zero_of_mem hx","premises":[{"full_name":"Metric.infDist_zero_of_mem","def_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","def_pos":[459,8],"def_end_pos":[459,27]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","commit":"","full_name":"Bimod.TensorBimod.middle_assoc'","start":[311,0],"end":[327,6],"file_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ actLeft P Q ▷ T.X ≫ actRight P Q = (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (tensorLeft R.X ⋙ tensorRight T.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n      actLeft P Q ▷ T.X ≫ actRight P Q =\n    (tensorLeft R.X ⋙ tensorRight T.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n      (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q","tactic":"refine (cancel_epi ((tensorLeft _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 ?_","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Limits.coequalizer.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","def_pos":[841,21],"def_end_pos":[841,34]},{"full_name":"CategoryTheory.MonoidalCategory.tensorLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[819,4],"def_end_pos":[819,14]},{"full_name":"CategoryTheory.MonoidalCategory.tensorRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[823,4],"def_end_pos":[823,15]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (tensorLeft R.X ⋙ tensorRight T.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n      actLeft P Q ▷ T.X ≫ actRight P Q =\n    (tensorLeft R.X ⋙ tensorRight T.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n      (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      actLeft P Q ▷ T.X ≫ actRight P Q =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","tactic":"dsimp [X]","premises":[{"full_name":"Bimod.TensorBimod.X","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[185,18],"def_end_pos":[185,19]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      actLeft P Q ▷ T.X ≫ actRight P Q =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ ((α_ R.X P.X Q.X).inv ▷ T.X ≫\n        P.actLeft ▷ Q.X ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫\n      actRight P Q =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","tactic":"slice_lhs 1 2 => rw [← comp_whiskerRight, whiskerLeft_π_actLeft, comp_whiskerRight,\n    comp_whiskerRight]","premises":[{"full_name":"Bimod.TensorBimod.whiskerLeft_π_actLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[212,8],"def_end_pos":[212,29]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ ((α_ R.X P.X Q.X).inv ▷ T.X ≫\n        P.actLeft ▷ Q.X ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫\n      actRight P Q =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      P.actLeft ▷ Q.X ▷ T.X ≫\n        (α_ P.X Q.X T.X).hom ≫\n          P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","tactic":"slice_lhs 3 4 => rw [π_tensor_id_actRight]","premises":[{"full_name":"Bimod.TensorBimod.π_tensor_id_actRight","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[271,8],"def_end_pos":[271,28]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      P.actLeft ▷ Q.X ▷ T.X ≫\n        (α_ P.X Q.X T.X).hom ≫\n          P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","tactic":"slice_lhs 2 3 => rw [associator_naturality_left]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_naturality_left","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[452,8],"def_end_pos":[452,34]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (((α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n          R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫\n        R.X ◁ actRight P Q) ≫\n      actLeft P Q","tactic":"slice_rhs 1 2 => rw [associator_naturality_middle]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_naturality_middle","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[464,8],"def_end_pos":[464,36]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (((α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n          R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫\n        R.X ◁ actRight P Q) ≫\n      actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      (R.X ◁ (α_ P.X Q.X T.X).hom ≫\n          R.X ◁ P.X ◁ Q.actRight ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n        actLeft P Q","tactic":"slice_rhs 2 3 => rw [← MonoidalCategory.whiskerLeft_comp, π_tensor_id_actRight,\n    MonoidalCategory.whiskerLeft_comp, MonoidalCategory.whiskerLeft_comp]","premises":[{"full_name":"Bimod.TensorBimod.π_tensor_id_actRight","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[271,8],"def_end_pos":[271,28]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      (R.X ◁ (α_ P.X Q.X T.X).hom ≫\n          R.X ◁ P.X ◁ Q.actRight ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n        actLeft P Q","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        R.X ◁ P.X ◁ Q.actRight ≫\n          (α_ R.X P.X Q.X).inv ≫\n            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)","tactic":"slice_rhs 4 5 => rw [whiskerLeft_π_actLeft]","premises":[{"full_name":"Bimod.TensorBimod.whiskerLeft_π_actLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Bimod.lean","def_pos":[212,8],"def_end_pos":[212,29]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        R.X ◁ P.X ◁ Q.actRight ≫\n          (α_ R.X P.X Q.X).inv ≫\n            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        (((α_ R.X P.X (Q.X ⊗ T.X)).inv ≫ (R.X ⊗ P.X) ◁ Q.actRight) ≫ P.actLeft ▷ Q.X) ≫\n          coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)","tactic":"slice_rhs 3 4 => rw [associator_inv_naturality_right]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.associator_inv_naturality_right","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[480,8],"def_end_pos":[480,39]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        (((α_ R.X P.X (Q.X ⊗ T.X)).inv ≫ (R.X ⊗ P.X) ◁ Q.actRight) ≫ P.actLeft ▷ Q.X) ≫\n          coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        (α_ R.X P.X (Q.X ⊗ T.X)).inv ≫\n          (P.actLeft ▷ (Q.X ⊗ T.X) ≫ P.X ◁ Q.actRight) ≫\n            coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)","tactic":"slice_rhs 4 5 => rw [whisker_exchange]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.MonoidalCategory.whisker_exchange","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[273,8],"def_end_pos":[273,24]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        (α_ R.X P.X (Q.X ⊗ T.X)).inv ≫\n          (P.actLeft ▷ (Q.X ⊗ T.X) ≫ P.X ◁ Q.actRight) ≫\n            coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Bool/AllAny.lean","commit":"","full_name":"List.any_iff_exists_prop","start":[37,0],"end":[37,93],"file_path":"Mathlib/Data/Bool/AllAny.lean","tactics":[{"state_before":"α : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\nl : List α\na : α\n⊢ (l.any fun a => decide (p a)) = true ↔ ∃ a, a ∈ l ∧ p a","state_after":"no goals","tactic":"simp [any_iff_exists]","premises":[{"full_name":"List.any_iff_exists","def_path":"Mathlib/Data/Bool/AllAny.lean","def_pos":[32,8],"def_end_pos":[32,22]}]}]}
{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.filter_and_not","start":[2338,0],"end":[2340,87],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np✝ q✝ : α → Prop\ninst✝⁴ : DecidablePred p✝\ninst✝³ : DecidablePred q✝\ns✝ t : Finset α\ninst✝² : DecidableEq α\ns : Finset α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\n⊢ filter (fun a => p a ∧ ¬q a) s = filter p s \\ filter q s","state_after":"no goals","tactic":"rw [filter_and, filter_not, ← inter_sdiff_assoc, inter_eq_left.2 (filter_subset _ _)]","premises":[{"full_name":"Finset.filter_and","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2330,8],"def_end_pos":[2330,18]},{"full_name":"Finset.filter_not","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2333,8],"def_end_pos":[2333,18]},{"full_name":"Finset.filter_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2152,8],"def_end_pos":[2152,21]},{"full_name":"Finset.inter_eq_left","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1530,14],"def_end_pos":[1530,27]},{"full_name":"Finset.inter_sdiff_assoc","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1780,6],"def_end_pos":[1780,23]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
{"url":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","commit":"","full_name":"MeasureTheory.uniformIntegrable_of'","start":[793,0],"end":[832,27],"file_path":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ UniformIntegrable f p μ","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C","tactic":"refine ⟨fun i => (hf i).aestronglyMeasurable,\n    unifIntegrable_of hp hp' (fun i => (hf i).aestronglyMeasurable) h, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"MeasureTheory.StronglyMeasurable.aestronglyMeasurable","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[105,18],"def_end_pos":[105,57]},{"full_name":"MeasureTheory.unifIntegrable_of","def_path":"Mathlib/MeasureTheory/Function/UniformIntegrable.lean","def_pos":[708,8],"def_end_pos":[708,25]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1\n⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C","tactic":"obtain ⟨C, hC⟩ := h 1 one_pos","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1\n⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ eLpNorm (f i) p μ ≤ ↑(↑C * μ univ ^ p.toReal⁻¹ + 1).toNNReal","tactic":"refine ⟨((C : ℝ≥0∞) * μ Set.univ ^ p.toReal⁻¹ + 1).toNNReal, fun i => ?_⟩","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.toNNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[181,14],"def_end_pos":[181,22]},{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]}]}]}
{"url":"Mathlib/GroupTheory/Finiteness.lean","commit":"","full_name":"AddSubgroup.rank_closure_finite_le_nat_card","start":[367,0],"end":[372,46],"file_path":"Mathlib/GroupTheory/Finiteness.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_3\nH : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\n⊢ Group.rank ↥(closure s) ≤ Nat.card ↑s","state_after":"M : Type u_1\nN : Type u_2\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_3\nH : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank ↥(closure s) ≤ Nat.card ↑s","tactic":"haveI := Fintype.ofFinite s","premises":[{"full_name":"Fintype.ofFinite","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[395,18],"def_end_pos":[395,34]}]},{"state_before":"M : Type u_1\nN : Type u_2\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_3\nH : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank ↥(closure s) ≤ Nat.card ↑s","state_after":"M : Type u_1\nN : Type u_2\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_3\nH : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank ↥(closure ↑s.toFinset) ≤ s.toFinset.card","tactic":"rw [Nat.card_eq_fintype_card, ← s.toFinset_card, ← rank_congr (congr_arg _ s.coe_toFinset)]","premises":[{"full_name":"Nat.card_eq_fintype_card","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[37,8],"def_end_pos":[37,28]},{"full_name":"Set.coe_toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[557,8],"def_end_pos":[557,20]},{"full_name":"Set.toFinset_card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[222,8],"def_end_pos":[222,21]},{"full_name":"Subgroup.rank_congr","def_path":"Mathlib/GroupTheory/Finiteness.lean","def_pos":[351,8],"def_end_pos":[351,18]}]},{"state_before":"M : Type u_1\nN : Type u_2\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid N\nG : Type u_3\nH : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup H\ns : Set G\ninst✝ : Finite ↑s\nthis : Fintype ↑s\n⊢ Group.rank ↥(closure ↑s.toFinset) ≤ s.toFinset.card","state_after":"no goals","tactic":"exact rank_closure_finset_le_card s.toFinset","premises":[{"full_name":"Set.toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[532,4],"def_end_pos":[532,12]},{"full_name":"Subgroup.rank_closure_finset_le_card","def_path":"Mathlib/GroupTheory/Finiteness.lean","def_pos":[355,8],"def_end_pos":[355,35]}]}]}
{"url":"Mathlib/Data/List/Cycle.lean","commit":"","full_name":"List.pmap_next_eq_rotate_one","start":[325,0],"end":[329,49],"file_path":"Mathlib/Data/List/Cycle.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : l.Nodup\n⊢ pmap l.next l ⋯ = l.rotate 1","state_after":"case hl\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : l.Nodup\n⊢ (pmap l.next l ⋯).length = (l.rotate 1).length\n\ncase h\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : l.Nodup\n⊢ ∀ (n : ℕ) (h₁ : n < (pmap l.next l ⋯).length) (h₂ : n < (l.rotate 1).length),\n    (pmap l.next l ⋯).nthLe n h₁ = (l.rotate 1).nthLe n h₂","tactic":"apply List.ext_nthLe","premises":[{"full_name":"List.ext_nthLe","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[957,8],"def_end_pos":[957,17]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","commit":"","full_name":"Filter.HasBasis.uniformEquicontinuousOn_iff_right","start":[676,0],"end":[682,5],"file_path":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\nα : Type u_7\nα' : Type u_8\nβ : Type u_9\nβ' : Type u_10\nγ : Type u_11\n𝓕 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nuα : UniformSpace α\nuβ : UniformSpace β\nuγ : UniformSpace γ\np : κ → Prop\ns : κ → Set (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ UniformEquicontinuousOn F S ↔\n    ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i xy.2) ∈ s k","state_after":"ι : Type u_1\nκ : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\nα : Type u_7\nα' : Type u_8\nβ : Type u_9\nβ' : Type u_10\nγ : Type u_11\n𝓕 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nuα : UniformSpace α\nuβ : UniformSpace β\nuγ : UniformSpace γ\np : κ → Prop\ns : κ → Set (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ (∀ (i : κ),\n      p i →\n        ∀ᶠ (x : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S),\n          ((⇑UniformFun.ofFun ∘ swap F) x.1, (⇑UniformFun.ofFun ∘ swap F) x.2) ∈ (UniformFun.gen ι α ∘ s) i) ↔\n    ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i xy.2) ∈ s k","tactic":"rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,\n    (UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]","premises":[{"full_name":"Filter.HasBasis.tendsto_right_iff","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[766,8],"def_end_pos":[766,34]},{"full_name":"UniformContinuousOn","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[954,4],"def_end_pos":[954,23]},{"full_name":"UniformFun.hasBasis_uniformity_of_basis","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[303,18],"def_end_pos":[303,46]},{"full_name":"uniformEquicontinuousOn_iff_uniformContinuousOn","def_path":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","def_pos":[528,8],"def_end_pos":[528,55]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\nα : Type u_7\nα' : Type u_8\nβ : Type u_9\nβ' : Type u_10\nγ : Type u_11\n𝓕 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nuα : UniformSpace α\nuβ : UniformSpace β\nuγ : UniformSpace γ\np : κ → Prop\ns : κ → Set (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ (∀ (i : κ),\n      p i →\n        ∀ᶠ (x : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S),\n          ((⇑UniformFun.ofFun ∘ swap F) x.1, (⇑UniformFun.ofFun ∘ swap F) x.2) ∈ (UniformFun.gen ι α ∘ s) i) ↔\n    ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i xy.2) ∈ s k","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Ascoli.lean","commit":"","full_name":"Equicontinuous.tendsto_uniformFun_iff_pi","start":[160,0],"end":[199,70],"file_path":"Mathlib/Topology/UniformSpace/Ascoli.lean","tactics":[{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\nℱ : Filter ι\nf : X → α\n⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ℱ (𝓝 f)","state_after":"case inl\nι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\nf : X → α\n⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ⊥ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ⊥ (𝓝 f)\n\ncase inr\nι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\nℱ : Filter ι\nf : X → α\nℱ_ne : ℱ.NeBot\n⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ℱ (𝓝 f)","tactic":"rcases ℱ.eq_or_neBot with rfl | ℱ_ne","premises":[{"full_name":"Filter.eq_or_neBot","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[484,8],"def_end_pos":[484,19]}]},{"state_before":"case inr\nι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\nℱ : Filter ι\nf : X → α\nℱ_ne : ℱ.NeBot\n⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f)) ↔ Tendsto F ℱ (𝓝 f)","state_after":"case inr.mp\nι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\nℱ : Filter ι\nf : X → α\nℱ_ne : ℱ.NeBot\nH : Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))\n⊢ Tendsto F ℱ (𝓝 f)\n\ncase inr.mpr\nι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\nℱ : Filter ι\nf : X → α\nℱ_ne : ℱ.NeBot\nH : Tendsto F ℱ (𝓝 f)\n⊢ Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f))","tactic":"constructor <;> intro H","premises":[]}]}
{"url":"Mathlib/Data/List/Perm.lean","commit":"","full_name":"List.Perm.dedup","start":[256,0],"end":[258,94],"file_path":"Mathlib/Data/List/Perm.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl l₁✝ l₂✝ : List α\na✝ : α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\np : l₁ ~ l₂\na : α\nh : a ∈ l₁\n⊢ count a l₁.dedup = count a l₂.dedup","state_after":"no goals","tactic":"simp [nodup_dedup, h, p.subset h]","premises":[{"full_name":"List.Perm.subset","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[76,8],"def_end_pos":[76,19]},{"full_name":"List.nodup_dedup","def_path":"Mathlib/Data/List/Dedup.lean","def_pos":[62,8],"def_end_pos":[62,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl l₁✝ l₂✝ : List α\na✝ : α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\np : l₁ ~ l₂\na : α\nh : a ∉ l₁\n⊢ count a l₁.dedup = count a l₂.dedup","state_after":"no goals","tactic":"simp [h, mt p.mem_iff.2 h]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"List.Perm.mem_iff","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[69,8],"def_end_pos":[69,20]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
{"url":"Mathlib/Analysis/NormedSpace/AddTorsor.lean","commit":"","full_name":"DilationEquiv.smulTorsor_symm_apply","start":[271,0],"end":[281,25],"file_path":"Mathlib/Analysis/NormedSpace/AddTorsor.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx : E\n⊢ (k⁻¹ • fun x => x -ᵥ c) ((fun x => k • x +ᵥ c) x) = x","state_after":"no goals","tactic":"simp [inv_smul_smul₀ hk]","premises":[{"full_name":"inv_smul_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[35,8],"def_end_pos":[35,22]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\np : P\n⊢ (fun x => k • x +ᵥ c) ((k⁻¹ • fun x => x -ᵥ c) p) = p","state_after":"no goals","tactic":"simp [smul_inv_smul₀ hk]","premises":[{"full_name":"smul_inv_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[39,8],"def_end_pos":[39,22]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\n⊢ edist ({ toFun := fun x => k • x +ᵥ c, invFun := k⁻¹ • fun x => x -ᵥ c, left_inv := ⋯, right_inv := ⋯ }.toFun x)\n      ({ toFun := fun x => k • x +ᵥ c, invFun := k⁻¹ • fun x => x -ᵥ c, left_inv := ⋯, right_inv := ⋯ }.toFun y) =\n    ↑‖k‖₊ * edist x y","state_after":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\n⊢ edist (k • x) (k • y) = ↑‖k‖₊ * edist x y","tactic":"rw [show edist (k • x +ᵥ c) (k • y +ᵥ c) = _ from (IsometryEquiv.vaddConst c).isometry ..]","premises":[{"full_name":"EDist.edist","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[49,2],"def_end_pos":[49,7]},{"full_name":"HVAdd.hVAdd","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[61,2],"def_end_pos":[61,7]},{"full_name":"IsometryEquiv.isometry","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[278,18],"def_end_pos":[278,26]},{"full_name":"IsometryEquiv.vaddConst","def_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","def_pos":[119,4],"def_end_pos":[119,27]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\n⊢ edist (k • x) (k • y) = ↑‖k‖₊ * edist x y","state_after":"no goals","tactic":"exact edist_smul₀ ..","premises":[{"full_name":"edist_smul₀","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[104,8],"def_end_pos":[104,19]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/SetToL1.lean","commit":"","full_name":"MeasureTheory.continuous_L1_toL1","start":[1391,0],"end":[1435,32],"file_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\n⊢ Continuous fun f => Integrable.toL1 ↑↑f ⋯","state_after":"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : c' = 0\n⊢ Continuous fun f => Integrable.toL1 ↑↑f ⋯\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\n⊢ Continuous fun f => Integrable.toL1 ↑↑f ⋯","tactic":"by_cases hc'0 : c' = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\n⊢ Continuous fun f => Integrable.toL1 ↑↑f ⋯","state_after":"case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\n⊢ ∀ (b : ↥(Lp G 1 μ)),\n    ∀ ε > 0, ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a b < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑b ⋯) < ε","tactic":"rw [Metric.continuous_iff]","premises":[{"full_name":"Metric.continuous_iff","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[908,8],"def_end_pos":[908,22]}]},{"state_before":"case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\n⊢ ∀ (b : ↥(Lp G 1 μ)),\n    ∀ ε > 0, ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a b < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑b ⋯) < ε","state_after":"case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a f < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","tactic":"intro f ε hε_pos","premises":[]},{"state_before":"case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a f < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ε / 2 / c'.toReal > 0 ∧\n    ∀ (a : ↥(Lp G 1 μ)), dist a f < ε / 2 / c'.toReal → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","tactic":"use ε / 2 / c'.toReal","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ε / 2 / c'.toReal > 0 ∧\n    ∀ (a : ↥(Lp G 1 μ)), dist a f < ε / 2 / c'.toReal → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ∀ (a : ↥(Lp G 1 μ)), dist a f < ε / 2 / c'.toReal → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","tactic":"refine ⟨div_pos (half_pos hε_pos) (toReal_pos hc'0 hc'), ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"ENNReal.toReal_pos","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[155,8],"def_end_pos":[155,18]},{"full_name":"div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[45,6],"def_end_pos":[45,13]},{"full_name":"half_pos","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[351,8],"def_end_pos":[351,16]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ∀ (a : ↥(Lp G 1 μ)), dist a f < ε / 2 / c'.toReal → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : dist g f < ε / 2 / c'.toReal\n⊢ dist (Integrable.toL1 ↑↑g ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","tactic":"intro g hfg","premises":[]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : dist g f < ε / 2 / c'.toReal\n⊢ dist (Integrable.toL1 ↑↑g ⋯) (Integrable.toL1 ↑↑f ⋯) < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε","tactic":"rw [Lp.dist_def] at hfg ⊢","premises":[{"full_name":"MeasureTheory.Lp.dist_def","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[270,8],"def_end_pos":[270,16]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε","tactic":"let h_int := fun f' : α →₁[μ] G => (L1.integrable_coeFn f').of_measure_le_smul c' hc' hμ'_le","premises":[{"full_name":"MeasureTheory.Integrable.of_measure_le_smul","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[460,8],"def_end_pos":[460,37]},{"full_name":"MeasureTheory.L1.integrable_coeFn","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1250,8],"def_end_pos":[1250,24]},{"full_name":"MeasureTheory.Lp","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[104,4],"def_end_pos":[104,6]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε","tactic":"have :\n    eLpNorm (⇑(Integrable.toL1 g (h_int g)) - ⇑(Integrable.toL1 f (h_int f))) 1 μ' =\n      eLpNorm (⇑g - ⇑f) 1 μ' :=\n    eLpNorm_congr_ae ((Integrable.coeFn_toL1 _).sub (Integrable.coeFn_toL1 _))","premises":[{"full_name":"Filter.EventuallyEq.sub","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1359,2],"def_end_pos":[1359,13]},{"full_name":"MeasureTheory.Integrable.coeFn_toL1","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1322,8],"def_end_pos":[1322,18]},{"full_name":"MeasureTheory.Integrable.toL1","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1315,4],"def_end_pos":[1315,8]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]},{"full_name":"MeasureTheory.eLpNorm_congr_ae","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[683,8],"def_end_pos":[683,24]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε","tactic":"rw [this]","premises":[]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\nh_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ⊤\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε","tactic":"have h_eLpNorm_ne_top : eLpNorm (⇑g - ⇑f) 1 μ ≠ ∞ := by\n    rw [← eLpNorm_congr_ae (Lp.coeFn_sub _ _)]; exact Lp.eLpNorm_ne_top _","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"MeasureTheory.Lp.coeFn_sub","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[227,8],"def_end_pos":[227,17]},{"full_name":"MeasureTheory.Lp.eLpNorm_ne_top","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[197,8],"def_end_pos":[197,22]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]},{"full_name":"MeasureTheory.eLpNorm_congr_ae","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[683,8],"def_end_pos":[683,24]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\nh_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ⊤\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\nh_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ⊤\nh_eLpNorm_ne_top' : eLpNorm (↑↑g - ↑↑f) 1 μ' ≠ ⊤\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε","tactic":"have h_eLpNorm_ne_top' : eLpNorm (⇑g - ⇑f) 1 μ' ≠ ∞ := by\n    refine ((eLpNorm_mono_measure _ hμ'_le).trans_lt ?_).ne\n    rw [eLpNorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]\n    refine ENNReal.mul_lt_top ?_ h_eLpNorm_ne_top\n    simp [hc', hc'0]","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"ENNReal.mul_lt_top","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[189,8],"def_end_pos":[189,18]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]},{"full_name":"MeasureTheory.eLpNorm_mono_measure","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[743,8],"def_end_pos":[743,28]},{"full_name":"MeasureTheory.eLpNorm_smul_measure_of_ne_zero","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[817,8],"def_end_pos":[817,39]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\nh_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ⊤\nh_eLpNorm_ne_top' : eLpNorm (↑↑g - ↑↑f) 1 μ' ≠ ⊤\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε","state_after":"no goals","tactic":"calc\n    (eLpNorm (⇑g - ⇑f) 1 μ').toReal ≤ (c' * eLpNorm (⇑g - ⇑f) 1 μ).toReal := by\n      rw [toReal_le_toReal h_eLpNorm_ne_top' (ENNReal.mul_ne_top hc' h_eLpNorm_ne_top)]\n      refine (eLpNorm_mono_measure (⇑g - ⇑f) hμ'_le).trans ?_\n      rw [eLpNorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]\n      simp\n    _ = c'.toReal * (eLpNorm (⇑g - ⇑f) 1 μ).toReal := toReal_mul\n    _ ≤ c'.toReal * (ε / 2 / c'.toReal) :=\n      (mul_le_mul le_rfl hfg.le toReal_nonneg toReal_nonneg)\n    _ = ε / 2 := by\n      refine mul_div_cancel₀ (ε / 2) ?_; rw [Ne, toReal_eq_zero_iff]; simp [hc', hc'0]\n    _ < ε := half_lt_self hε_pos","premises":[{"full_name":"ENNReal.mul_ne_top","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[191,8],"def_end_pos":[191,18]},{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"ENNReal.toReal_eq_zero_iff","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[259,8],"def_end_pos":[259,26]},{"full_name":"ENNReal.toReal_le_toReal","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[68,8],"def_end_pos":[68,24]},{"full_name":"ENNReal.toReal_mul","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[378,8],"def_end_pos":[378,18]},{"full_name":"ENNReal.toReal_nonneg","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[224,16],"def_end_pos":[224,29]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]},{"full_name":"MeasureTheory.eLpNorm_mono_measure","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[743,8],"def_end_pos":[743,28]},{"full_name":"MeasureTheory.eLpNorm_smul_measure_of_ne_zero","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[817,8],"def_end_pos":[817,39]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"mul_div_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[424,6],"def_end_pos":[424,21]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
{"url":"Mathlib/AlgebraicTopology/SimplicialObject.lean","commit":"","full_name":"CategoryTheory.SimplicialObject.δ_comp_δ_self","start":[119,0],"end":[124,68],"file_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 2)\n⊢ X.δ i.castSucc ≫ X.δ i = X.δ i.succ ≫ X.δ i","state_after":"C : Type u\ninst✝ : Category.{v, u} C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 2)\n⊢ X.map (SimplexCategory.δ i.castSucc).op ≫ X.map (SimplexCategory.δ i).op =\n    X.map (SimplexCategory.δ i.succ).op ≫ X.map (SimplexCategory.δ i).op","tactic":"dsimp [δ]","premises":[{"full_name":"CategoryTheory.SimplicialObject.δ","def_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","def_pos":[82,4],"def_end_pos":[82,5]}]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 2)\n⊢ X.map (SimplexCategory.δ i.castSucc).op ≫ X.map (SimplexCategory.δ i).op =\n    X.map (SimplexCategory.δ i.succ).op ≫ X.map (SimplexCategory.δ i).op","state_after":"no goals","tactic":"simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ_self]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"SimplexCategory.δ_comp_δ_self","def_path":"Mathlib/AlgebraicTopology/SimplexCategory.lean","def_pos":[232,8],"def_end_pos":[232,21]}]}]}
{"url":"Mathlib/RingTheory/Algebraic.lean","commit":"","full_name":"isAlgebraic_int","start":[122,0],"end":[124,43],"file_path":"Mathlib/RingTheory/Algebraic.lean","tactics":[{"state_before":"R : Type u\nS : Type u_1\nA : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Nontrivial R\nn : ℤ\n⊢ IsAlgebraic R ↑n","state_after":"R : Type u\nS : Type u_1\nA : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Nontrivial R\nn : ℤ\n⊢ IsAlgebraic R ((algebraMap R A) ↑n)","tactic":"rw [← _root_.map_intCast (algebraMap R A)]","premises":[{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"map_intCast","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[359,8],"def_end_pos":[359,19]}]},{"state_before":"R : Type u\nS : Type u_1\nA : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Nontrivial R\nn : ℤ\n⊢ IsAlgebraic R ((algebraMap R A) ↑n)","state_after":"no goals","tactic":"exact isAlgebraic_algebraMap (Int.cast n)","premises":[{"full_name":"Int.cast","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[353,47],"def_end_pos":[353,55]},{"full_name":"isAlgebraic_algebraMap","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[111,8],"def_end_pos":[111,30]}]}]}
{"url":"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean","commit":"","full_name":"zmultiplesHom_ker_eq","start":[699,0],"end":[705,85],"file_path":"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean","tactics":[{"state_before":"α : Type u\na : α\nG : Type u_1\nH : Type u_2\ninst✝ : AddGroup G\ng : G\n⊢ ((zmultiplesHom G) g).ker = zmultiples ↑(addOrderOf g)","state_after":"case h\nα : Type u\na : α\nG : Type u_1\nH : Type u_2\ninst✝ : AddGroup G\ng : G\nx✝ : ℤ\n⊢ x✝ ∈ ((zmultiplesHom G) g).ker ↔ x✝ ∈ zmultiples ↑(addOrderOf g)","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u\na : α\nG : Type u_1\nH : Type u_2\ninst✝ : AddGroup G\ng : G\nx✝ : ℤ\n⊢ x✝ ∈ ((zmultiplesHom G) g).ker ↔ x✝ ∈ zmultiples ↑(addOrderOf g)","state_after":"no goals","tactic":"simp_rw [AddMonoidHom.mem_ker, mem_zmultiples_iff, zmultiplesHom_apply,\n    ← addOrderOf_dvd_iff_zsmul_eq_zero, zsmul_eq_mul', Int.cast_id, dvd_def, eq_comm]","premises":[{"full_name":"AddMonoidHom.mem_ker","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2097,2],"def_end_pos":[2097,13]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Int.cast_id","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[123,25],"def_end_pos":[123,32]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"addOrderOf_dvd_iff_zsmul_eq_zero","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[547,2],"def_end_pos":[547,13]},{"full_name":"dvd_def","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[54,8],"def_end_pos":[54,15]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"zmultiplesHom_apply","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[304,6],"def_end_pos":[304,25]},{"full_name":"zsmul_eq_mul'","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[107,6],"def_end_pos":[107,26]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"InnerProductSpace.Core.inner_neg_right","start":[256,0],"end":[257,86],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ⟪x, -y⟫_𝕜 = -⟪x, y⟫_𝕜","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ (starRingEnd 𝕜) (-⟪y, x⟫_𝕜) = -⟪x, y⟫_𝕜","tactic":"rw [← inner_conj_symm, inner_neg_left]","premises":[{"full_name":"InnerProductSpace.Core.inner_conj_symm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[196,8],"def_end_pos":[196,23]},{"full_name":"InnerProductSpace.Core.inner_neg_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[252,8],"def_end_pos":[252,22]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ (starRingEnd 𝕜) (-⟪y, x⟫_𝕜) = -⟪x, y⟫_𝕜","state_after":"no goals","tactic":"simp only [RingHom.map_neg, inner_conj_symm]","premises":[{"full_name":"InnerProductSpace.Core.inner_conj_symm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[196,8],"def_end_pos":[196,23]},{"full_name":"RingHom.map_neg","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[522,18],"def_end_pos":[522,25]}]}]}
{"url":"Mathlib/GroupTheory/Perm/Support.lean","commit":"","full_name":"Equiv.Perm.support_extend_domain","start":[469,0],"end":[492,29],"file_path":"Mathlib/GroupTheory/Perm/Support.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\n⊢ (g.extendDomain f).support = map f.asEmbedding g.support","state_after":"case a\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\n⊢ b ∈ (g.extendDomain f).support ↔ b ∈ map f.asEmbedding g.support","tactic":"ext b","premises":[]},{"state_before":"case a\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\n⊢ b ∈ (g.extendDomain f).support ↔ b ∈ map f.asEmbedding g.support","state_after":"case a\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\n⊢ ¬(g.extendDomain f) b = b ↔ ∃ a, ¬g a = a ∧ f.asEmbedding a = b","tactic":"simp only [exists_prop, Function.Embedding.coeFn_mk, toEmbedding_apply, mem_map, Ne,\n    Function.Embedding.trans_apply, mem_support]","premises":[{"full_name":"Equiv.Perm.mem_support","def_path":"Mathlib/GroupTheory/Perm/Support.lean","def_pos":[271,8],"def_end_pos":[271,19]},{"full_name":"Equiv.toEmbedding_apply","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[76,8],"def_end_pos":[76,31]},{"full_name":"Finset.mem_map","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[70,8],"def_end_pos":[70,15]},{"full_name":"Function.Embedding.coeFn_mk","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[117,8],"def_end_pos":[117,16]},{"full_name":"Function.Embedding.trans_apply","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[136,9],"def_end_pos":[136,14]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case a\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\n⊢ ¬(g.extendDomain f) b = b ↔ ∃ a, ¬g a = a ∧ f.asEmbedding a = b","state_after":"case pos\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\npb : p b\n⊢ ¬(g.extendDomain f) b = b ↔ ∃ a, ¬g a = a ∧ f.asEmbedding a = b\n\ncase neg\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\npb : ¬p b\n⊢ ¬(g.extendDomain f) b = b ↔ ∃ a, ¬g a = a ∧ f.asEmbedding a = b","tactic":"by_cases pb : p b","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","commit":"","full_name":"seminormFromBounded_nonzero","start":[328,0],"end":[337,10],"file_path":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\n⊢ seminormFromBounded' f ≠ 0","state_after":"case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f ≠ 0","tactic":"obtain ⟨x, hx⟩ := Function.ne_iff.mp f_ne_zero","premises":[{"full_name":"Function.ne_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[57,8],"def_end_pos":[57,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f ≠ 0","state_after":"case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ ∃ a, seminormFromBounded' f a ≠ 0 a","tactic":"rw [Function.ne_iff]","premises":[{"full_name":"Function.ne_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[57,8],"def_end_pos":[57,14]}]},{"state_before":"case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ ∃ a, seminormFromBounded' f a ≠ 0 a","state_after":"case h\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f x ≠ 0 x","tactic":"use x","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f x ≠ 0 x","state_after":"case h\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ ¬f x = 0","tactic":"rw [ne_eq, Pi.zero_apply, seminormFromBounded_eq_zero_iff f_nonneg f_mul x]","premises":[{"full_name":"Pi.zero_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[59,2],"def_end_pos":[59,13]},{"full_name":"ne_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[89,16],"def_end_pos":[89,21]},{"full_name":"seminormFromBounded_eq_zero_iff","def_path":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","def_pos":[150,8],"def_end_pos":[150,39]}]},{"state_before":"case h\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ ¬f x = 0","state_after":"no goals","tactic":"exact hx","premises":[]}]}
{"url":"Mathlib/Data/Real/EReal.lean","commit":"","full_name":"EReal.neg_lt_iff_neg_lt","start":[918,0],"end":[919,32],"file_path":"Mathlib/Data/Real/EReal.lean","tactics":[{"state_before":"a b : EReal\n⊢ -a < b ↔ -b < a","state_after":"no goals","tactic":"rw [← neg_lt_neg_iff, neg_neg]","premises":[{"full_name":"EReal.neg_lt_neg_iff","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[898,16],"def_end_pos":[898,30]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Restrict.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen","start":[182,0],"end":[184,53],"file_path":"Mathlib/AlgebraicGeometry/Restrict.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX : Scheme\nU : X.Opens\nr : ↑Γ(↑U, ⊤)\n⊢ U.ι ''ᵁ (↑U).basicOpen r = X.basicOpen r","state_after":"no goals","tactic":"rw [Scheme.map_basicOpen', Scheme.basicOpen_res_eq]","premises":[{"full_name":"AlgebraicGeometry.Scheme.basicOpen_res_eq","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[472,8],"def_end_pos":[472,24]},{"full_name":"AlgebraicGeometry.Scheme.map_basicOpen'","def_path":"Mathlib/AlgebraicGeometry/Restrict.lean","def_pos":[172,6],"def_end_pos":[172,27]}]}]}
{"url":"Mathlib/Algebra/LinearRecurrence.lean","commit":"","full_name":"LinearRecurrence.mkSol_eq_init","start":[85,0],"end":[89,51],"file_path":"Mathlib/Algebra/LinearRecurrence.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : CommSemiring α\nE : LinearRecurrence α\ninit : Fin E.order → α\n⊢ ∀ (n : Fin E.order), E.mkSol init ↑n = init n","state_after":"α : Type u_1\ninst✝ : CommSemiring α\nE : LinearRecurrence α\ninit : Fin E.order → α\nn : Fin E.order\n⊢ E.mkSol init ↑n = init n","tactic":"intro n","premises":[]},{"state_before":"α : Type u_1\ninst✝ : CommSemiring α\nE : LinearRecurrence α\ninit : Fin E.order → α\nn : Fin E.order\n⊢ E.mkSol init ↑n = init n","state_after":"α : Type u_1\ninst✝ : CommSemiring α\nE : LinearRecurrence α\ninit : Fin E.order → α\nn : Fin E.order\n⊢ (if h : ↑n < E.order then init ⟨↑n, h⟩\n    else\n      ∑ k : Fin E.order,\n        let_fun x := ⋯;\n        E.coeffs k * E.mkSol init (↑n - E.order + ↑k)) =\n    init n","tactic":"rw [mkSol]","premises":[{"full_name":"LinearRecurrence.mkSol","def_path":"Mathlib/Algebra/LinearRecurrence.lean","def_pos":[67,4],"def_end_pos":[67,9]}]},{"state_before":"α : Type u_1\ninst✝ : CommSemiring α\nE : LinearRecurrence α\ninit : Fin E.order → α\nn : Fin E.order\n⊢ (if h : ↑n < E.order then init ⟨↑n, h⟩\n    else\n      ∑ k : Fin E.order,\n        let_fun x := ⋯;\n        E.coeffs k * E.mkSol init (↑n - E.order + ↑k)) =\n    init n","state_after":"no goals","tactic":"simp only [n.is_lt, dif_pos, Fin.mk_val, Fin.eta]","premises":[{"full_name":"Fin.eta","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[38,26],"def_end_pos":[38,29]},{"full_name":"Fin.is_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[31,16],"def_end_pos":[31,21]},{"full_name":"Fin.mk_val","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[57,8],"def_end_pos":[57,14]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"PartialHomeomorph.map_extend_nhdsWithin","start":[882,0],"end":[886,15],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ny : M\nhy : y ∈ f.source\n⊢ map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y","state_after":"no goals","tactic":"rw [map_extend_nhdsWithin_eq_image f I hy, nhdsWithin_inter, ←\n    nhdsWithin_extend_target_eq _ _ hy, ← nhdsWithin_inter, (f.extend I).image_source_inter_eq',\n    inter_comm]","premises":[{"full_name":"PartialEquiv.image_source_inter_eq'","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[416,8],"def_end_pos":[416,30]},{"full_name":"PartialHomeomorph.extend","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[749,4],"def_end_pos":[749,10]},{"full_name":"PartialHomeomorph.map_extend_nhdsWithin_eq_image","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[865,8],"def_end_pos":[865,38]},{"full_name":"PartialHomeomorph.nhdsWithin_extend_target_eq","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[835,8],"def_end_pos":[835,35]},{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]},{"full_name":"nhdsWithin_inter","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[207,8],"def_end_pos":[207,24]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","commit":"","full_name":"CategoryTheory.InjectiveResolution.desc_commutes","start":[92,0],"end":[97,34],"file_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nY Z : C\nf : Z ⟶ Y\nI : InjectiveResolution Y\nJ : InjectiveResolution Z\n⊢ J.ι ≫ desc f I J = (CochainComplex.single₀ C).map f ≫ I.ι","state_after":"case hfg\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nY Z : C\nf : Z ⟶ Y\nI : InjectiveResolution Y\nJ : InjectiveResolution Z\n⊢ (J.ι ≫ desc f I J).f 0 = ((CochainComplex.single₀ C).map f ≫ I.ι).f 0","tactic":"ext","premises":[]},{"state_before":"case hfg\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nY Z : C\nf : Z ⟶ Y\nI : InjectiveResolution Y\nJ : InjectiveResolution Z\n⊢ (J.ι ≫ desc f I J).f 0 = ((CochainComplex.single₀ C).map f ≫ I.ι).f 0","state_after":"no goals","tactic":"simp [desc, descFOne, descFZero]","premises":[{"full_name":"CategoryTheory.InjectiveResolution.desc","def_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","def_pos":[87,4],"def_end_pos":[87,8]},{"full_name":"CategoryTheory.InjectiveResolution.descFOne","def_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","def_pos":[65,4],"def_end_pos":[65,12]},{"full_name":"CategoryTheory.InjectiveResolution.descFZero","def_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","def_pos":[50,4],"def_end_pos":[50,13]}]}]}
{"url":"Mathlib/Data/ZMod/Basic.lean","commit":"","full_name":"ZMod.ringEquivCongr_intCast","start":[500,0],"end":[503,17],"file_path":"Mathlib/Data/ZMod/Basic.lean","tactics":[{"state_before":"n : ℕ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R n\na b : ℕ\nh : a = b\nz : ℤ\n⊢ (ringEquivCongr h) ↑z = ↑z","state_after":"n : ℕ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R n\na : ℕ\nz : ℤ\n⊢ (ringEquivCongr ⋯) ↑z = ↑z","tactic":"subst h","premises":[]},{"state_before":"n : ℕ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R n\na : ℕ\nz : ℤ\n⊢ (ringEquivCongr ⋯) ↑z = ↑z","state_after":"no goals","tactic":"cases a <;> rfl","premises":[]}]}
{"url":"Mathlib/Topology/Homotopy/HSpaces.lean","commit":"","full_name":"Path.delayReflRight_zero","start":[218,0],"end":[225,29],"file_path":"Mathlib/Topology/Homotopy/HSpaces.lean","tactics":[{"state_before":"X : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\n⊢ delayReflRight 0 γ = γ.trans (refl y)","state_after":"case a.h\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\n⊢ (delayReflRight 0 γ) t = (γ.trans (refl y)) t","tactic":"ext t","premises":[]},{"state_before":"case a.h\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\n⊢ (delayReflRight 0 γ) t = (γ.trans (refl y)) t","state_after":"case a.h\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\n⊢ γ (qRight (t, 0)) = if h : ↑t ≤ 1 / 2 then γ ⟨2 * ↑t, ⋯⟩ else y","tactic":"simp only [delayReflRight, trans_apply, refl_extend, Path.coe_mk_mk, Function.comp_apply,\n    refl_apply]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Path.coe_mk_mk","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[104,8],"def_end_pos":[104,17]},{"full_name":"Path.delayReflRight","def_path":"Mathlib/Topology/Homotopy/HSpaces.lean","def_pos":[203,4],"def_end_pos":[203,18]},{"full_name":"Path.refl_apply","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[145,8],"def_end_pos":[145,13]},{"full_name":"Path.refl_extend","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[270,8],"def_end_pos":[270,19]},{"full_name":"Path.trans_apply","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[298,8],"def_end_pos":[298,19]}]},{"state_before":"case a.h\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\n⊢ γ (qRight (t, 0)) = if h : ↑t ≤ 1 / 2 then γ ⟨2 * ↑t, ⋯⟩ else y","state_after":"case pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩\n\ncase neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = y","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩\n\ncase neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = y","state_after":"case neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = y\n\ncase pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩","tactic":"swap","premises":[]},{"state_before":"case neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = y\n\ncase pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩","state_after":"case neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ 1\n\ncase pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩","tactic":"on_goal 1 => conv_rhs => rw [← γ.target]","premises":[{"full_name":"Path.target","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[120,18],"def_end_pos":[120,24]}]},{"state_before":"case neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ 1\n\ncase pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩","state_after":"case neg.a\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑1\n\ncase pos.a\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑⟨2 * ↑t, ⋯⟩","tactic":"all_goals apply congr_arg γ; ext1; rw [qRight_zero_right]","premises":[{"full_name":"unitInterval.qRight_zero_right","def_path":"Mathlib/Topology/Homotopy/HSpaces.lean","def_pos":[179,8],"def_end_pos":[179,25]}]},{"state_before":"case neg.a\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑1\n\ncase pos.a\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑⟨2 * ↑t, ⋯⟩","state_after":"no goals","tactic":"exacts [if_neg h, if_pos h]","premises":[{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]}]}]}
{"url":"Mathlib/Data/Seq/Seq.lean","commit":"","full_name":"Stream'.Seq1.ret_bind","start":[861,0],"end":[865,35],"file_path":"Mathlib/Data/Seq/Seq.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na : α\nf : α → Seq1 β\n⊢ (ret a).bind f = f a","state_after":"α : Type u\nβ : Type v\nγ : Type w\na : α\nf : α → Seq1 β\n⊢ join (f a, nil) = f a","tactic":"simp only [bind, map, ret.eq_1, map_nil]","premises":[{"full_name":"Stream'.Seq.map_nil","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[611,8],"def_end_pos":[611,15]},{"full_name":"Stream'.Seq1.bind","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[846,4],"def_end_pos":[846,8]},{"full_name":"Stream'.Seq1.map","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[810,4],"def_end_pos":[810,7]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na : α\nf : α → Seq1 β\n⊢ join (f a, nil) = f a","state_after":"case mk\nα : Type u\nβ : Type v\nγ : Type w\na✝ : α\nf : α → Seq1 β\na : β\ns : Seq β\n⊢ join ((a, s), nil) = (a, s)","tactic":"cases' f a with a s","premises":[]},{"state_before":"case mk\nα : Type u\nβ : Type v\nγ : Type w\na✝ : α\nf : α → Seq1 β\na : β\ns : Seq β\n⊢ join ((a, s), nil) = (a, s)","state_after":"no goals","tactic":"apply recOn s <;> intros <;> simp","premises":[{"full_name":"Stream'.Seq.recOn","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[234,4],"def_end_pos":[234,9]}]}]}
{"url":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.prod_lt_prod_of_nonempty'","start":[388,0],"end":[391,51],"file_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : s.Nonempty\nhlt : ∀ i ∈ s, f i < g i\n⊢ s.val ≠ ∅","state_after":"no goals","tactic":"aesop","premises":[]}]}
{"url":"Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean","commit":"","full_name":"LinearMap.antilipschitz_of_comap_nhds_le","start":[65,0],"end":[85,45],"file_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\n⊢ ∃ K, AntilipschitzWith K ⇑f","state_after":"case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ⇑f ⁻¹' ball 0 ε ⊆ ball 0 1\n⊢ ∃ K, AntilipschitzWith K ⇑f","tactic":"rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩","premises":[{"full_name":"Filter.HasBasis.comap","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[697,8],"def_end_pos":[697,22]},{"full_name":"Filter.HasBasis.le_basis_iff","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[414,8],"def_end_pos":[414,29]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Metric.nhds_basis_ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[805,8],"def_end_pos":[805,23]}]},{"state_before":"case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ⇑f ⁻¹' ball 0 ε ⊆ ball 0 1\n⊢ ∃ K, AntilipschitzWith K ⇑f","state_after":"case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ∀ (x : E), ‖f x‖ < ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ⇑f","tactic":"simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε","premises":[{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]},{"full_name":"Set.subset_def","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[283,8],"def_end_pos":[283,18]},{"full_name":"mem_ball_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[513,2],"def_end_pos":[513,13]}]},{"state_before":"case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ∀ (x : E), ‖f x‖ < ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ⇑f","state_after":"case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ⇑f","tactic":"lift ε to ℝ≥0 using ε0.le","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ⇑f","state_after":"case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ ∃ K, AntilipschitzWith K ⇑f","tactic":"rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩","premises":[{"full_name":"NormedField.exists_one_lt_norm","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[907,8],"def_end_pos":[907,26]}]},{"state_before":"case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ ∃ K, AntilipschitzWith K ⇑f","state_after":"case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","tactic":"refine ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => ?_⟩","premises":[{"full_name":"AddMonoidHomClass.antilipschitz_of_bound","def_path":"Mathlib/Analysis/Normed/Group/Uniform.lean","def_pos":[123,2],"def_end_pos":[123,13]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"NNNorm.nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]}]},{"state_before":"case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","state_after":"case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖\n\ncase neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","tactic":"by_cases hx : f x = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","state_after":"case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","tactic":"have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"norm_pos_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1216,29],"def_end_pos":[1216,41]}]},{"state_before":"case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","state_after":"case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","tactic":"rw [← h.1] at hc","premises":[{"full_name":"RingHomIsometric.is_iso","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[1075,2],"def_end_pos":[1075,8]}]},{"state_before":"case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","state_after":"case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖σ₁₂ c ^ n • f x‖ < ↑ε\nhle : ‖σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖σ₁₂ c‖ * ‖f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","tactic":"rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩","premises":[{"full_name":"rescale_to_shell_zpow","def_path":"Mathlib/Analysis/Seminorm.lean","def_pos":[1317,6],"def_end_pos":[1317,27]}]},{"state_before":"case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖σ₁₂ c ^ n • f x‖ < ↑ε\nhle : ‖σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖σ₁₂ c‖ * ‖f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","state_after":"case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","tactic":"simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle","premises":[{"full_name":"MulActionSemiHomClass.map_smulₛₗ","def_path":"Mathlib/GroupTheory/GroupAction/Hom.lean","def_pos":[92,2],"def_end_pos":[92,12]},{"full_name":"RingHomIsometric.is_iso","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[1075,2],"def_end_pos":[1075,8]},{"full_name":"map_zpow₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean","def_pos":[108,8],"def_end_pos":[108,17]}]},{"state_before":"case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖","state_after":"no goals","tactic":"calc\n    ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖ := by\n      rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]\n    _ ≤ ‖c ^ n‖⁻¹ * 1 := (mul_le_mul_of_nonneg_left (hε _ hlt).le (inv_nonneg.2 (norm_nonneg _)))\n    _ ≤ ε⁻¹ * ‖c‖ * ‖f x‖ := by rwa [mul_one]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"inv_nonneg","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[29,14],"def_end_pos":[29,24]},{"full_name":"inv_smul_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[35,8],"def_end_pos":[35,22]},{"full_name":"mul_le_mul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[190,8],"def_end_pos":[190,33]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"norm_inv","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[716,8],"def_end_pos":[716,16]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]},{"full_name":"norm_smul","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[79,8],"def_end_pos":[79,17]},{"full_name":"zpow_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[358,6],"def_end_pos":[358,18]}]}]}
{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.sdiff_eq_self","start":[2345,0],"end":[2346,57],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ns t : Finset α\ninst✝ : DecidableEq α\ns₁ s₂ : Finset α\n⊢ s₁ \\ s₂ = s₁ ↔ s₁ ∩ s₂ ⊆ ∅","state_after":"no goals","tactic":"simp [Subset.antisymm_iff, disjoint_iff_inter_eq_empty]","premises":[{"full_name":"Finset.Subset.antisymm_iff","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[343,8],"def_end_pos":[343,27]},{"full_name":"Finset.disjoint_iff_inter_eq_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1174,8],"def_end_pos":[1174,35]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","commit":"","full_name":"setIntegral_withDensity_eq_setIntegral_smul₀","start":[1400,0],"end":[1403,73],"file_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : X → ℝ≥0\ns : Set X\nhf : AEMeasurable f (μ.restrict s)\ng : X → E\nhs : MeasurableSet s\n⊢ (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ","state_after":"no goals","tactic":"rw [restrict_withDensity hs, integral_withDensity_eq_integral_smul₀ hf]","premises":[{"full_name":"MeasureTheory.restrict_withDensity","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[192,8],"def_end_pos":[192,28]},{"full_name":"integral_withDensity_eq_integral_smul₀","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[1377,8],"def_end_pos":[1377,46]}]}]}
{"url":"Mathlib/FieldTheory/KummerExtension.lean","commit":"","full_name":"irreducible_X_pow_sub_C_of_root_adjoin_eq_top","start":[582,0],"end":[594,61],"file_path":"Mathlib/FieldTheory/KummerExtension.lean","tactics":[{"state_before":"K : Type u\ninst✝⁵ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\nhn : 0 < n\na✝ : K\nH : Irreducible (X ^ n - C a✝)\nL : Type u_1\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : IsGalois K L\ninst✝¹ : FiniteDimensional K L\ninst✝ : IsCyclic (L ≃ₐ[K] L)\nhK : (primitiveRoots (finrank K L) K).Nonempty\na : K\nα : L\nha : α ^ finrank K L = (algebraMap K L) a\nhα : K⟮α⟯ = ⊤\nthis : X ^ finrank K L - C a = minpoly K α\n⊢ Irreducible (X ^ finrank K L - C a)","state_after":"no goals","tactic":"exact this ▸ minpoly.irreducible (IsIntegral.of_finite K α)","premises":[{"full_name":"IsIntegral.of_finite","def_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","def_pos":[43,8],"def_end_pos":[43,28]},{"full_name":"minpoly.irreducible","def_path":"Mathlib/FieldTheory/Minpoly/Basic.lean","def_pos":[254,8],"def_end_pos":[254,19]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Dickson.lean","commit":"","full_name":"Polynomial.dickson_two_zero","start":[97,0],"end":[105,73],"file_path":"Mathlib/RingTheory/Polynomial/Dickson.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\n⊢ dickson 2 0 0 = X ^ 0","state_after":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\n⊢ 3 - ↑2 = 1","tactic":"simp only [dickson_zero, pow_zero]","premises":[{"full_name":"Polynomial.dickson_zero","def_path":"Mathlib/RingTheory/Polynomial/Dickson.lean","def_pos":[67,8],"def_end_pos":[67,20]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\n⊢ 3 - ↑2 = 1","state_after":"no goals","tactic":"norm_num","premises":[]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\n⊢ dickson 2 0 1 = X ^ 1","state_after":"no goals","tactic":"simp only [dickson_one, pow_one]","premises":[{"full_name":"Polynomial.dickson_one","def_path":"Mathlib/RingTheory/Polynomial/Dickson.lean","def_pos":[71,8],"def_end_pos":[71,19]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nn : ℕ\n⊢ dickson 2 0 (n + 2) = X ^ (n + 2)","state_after":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nn : ℕ\n⊢ X * dickson 2 0 (n + 1) = X ^ (n + 2)","tactic":"simp only [dickson_add_two, C_0, zero_mul, sub_zero]","premises":[{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"Polynomial.C_0","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[440,8],"def_end_pos":[440,11]},{"full_name":"Polynomial.dickson_add_two","def_path":"Mathlib/RingTheory/Polynomial/Dickson.lean","def_pos":[78,8],"def_end_pos":[78,23]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nn : ℕ\n⊢ X * dickson 2 0 (n + 1) = X ^ (n + 2)","state_after":"no goals","tactic":"rw [dickson_two_zero (n + 1), pow_add X (n + 1) 1, mul_comm, pow_one]","premises":[{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]}]}
{"url":"Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","commit":"","full_name":"Asymptotics.superpolynomialDecay_iff_isLittleO","start":[311,0],"end":[321,89],"file_path":"Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\n⊢ SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), f =o[l] fun a => k a ^ z","state_after":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\n⊢ f =o[l] fun a => k a ^ z","tactic":"refine ⟨fun h z => ?_, fun h => (superpolynomialDecay_iff_isBigO f hk).2 fun z => (h z).isBigO⟩","premises":[{"full_name":"Asymptotics.IsLittleO.isBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[193,8],"def_end_pos":[193,24]},{"full_name":"Asymptotics.superpolynomialDecay_iff_isBigO","def_path":"Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","def_pos":[293,8],"def_end_pos":[293,39]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\n⊢ f =o[l] fun a => k a ^ z","state_after":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\n⊢ f =o[l] fun a => k a ^ z","tactic":"have hk0 : ∀ᶠ x in l, k x ≠ 0 := hk.eventually_ne_atTop 0","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Tendsto.eventually_ne_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[184,18],"def_end_pos":[184,45]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\n⊢ f =o[l] fun a => k a ^ z","state_after":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis : (fun x => 1) =o[l] k\n⊢ f =o[l] fun a => k a ^ z","tactic":"have : (fun _ : α => (1 : β)) =o[l] k :=\n    isLittleO_of_tendsto' (hk0.mono fun x hkx hkx' => absurd hkx' hkx)\n      (by simpa using hk.inv_tendsto_atTop)","premises":[{"full_name":"Asymptotics.IsLittleO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[156,16],"def_end_pos":[156,25]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.Tendsto.inv_tendsto_atTop","def_path":"Mathlib/Topology/Algebra/Order/Field.lean","def_pos":[136,8],"def_end_pos":[136,40]},{"full_name":"absurd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[246,20],"def_end_pos":[246,26]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis : (fun x => 1) =o[l] k\n⊢ f =o[l] fun a => k a ^ z","state_after":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis✝ : (fun x => 1) =o[l] k\nthis : f =o[l] fun x => k x * k x ^ (z - 1)\n⊢ f =o[l] fun a => k a ^ z","tactic":"have : f =o[l] fun x : α => k x * k x ^ (z - 1) := by\n    simpa using this.mul_isBigO ((superpolynomialDecay_iff_isBigO f hk).1 h <| z - 1)","premises":[{"full_name":"Asymptotics.IsLittleO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[156,16],"def_end_pos":[156,25]},{"full_name":"Asymptotics.IsLittleO.mul_isBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1348,8],"def_end_pos":[1348,28]},{"full_name":"Asymptotics.superpolynomialDecay_iff_isBigO","def_path":"Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","def_pos":[293,8],"def_end_pos":[293,39]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis✝ : (fun x => 1) =o[l] k\nthis : f =o[l] fun x => k x * k x ^ (z - 1)\n⊢ f =o[l] fun a => k a ^ z","state_after":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis✝ : (fun x => 1) =o[l] k\nthis : f =o[l] fun x => k x * k x ^ (z - 1)\nx : α\nhkx : k x ≠ 0\n⊢ ‖k x * k x ^ (z - 1)‖ = 1 * ‖k x ^ z‖","tactic":"refine this.trans_isBigO (IsBigO.of_bound 1 (hk0.mono fun x hkx => le_of_eq ?_))","premises":[{"full_name":"Asymptotics.IsBigO.of_bound","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[141,8],"def_end_pos":[141,23]},{"full_name":"Asymptotics.IsLittleO.trans_isBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[449,8],"def_end_pos":[449,30]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝¹ : NormedLinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis✝ : (fun x => 1) =o[l] k\nthis : f =o[l] fun x => k x * k x ^ (z - 1)\nx : α\nhkx : k x ≠ 0\n⊢ ‖k x * k x ^ (z - 1)‖ = 1 * ‖k x ^ z‖","state_after":"no goals","tactic":"rw [one_mul, zpow_sub_one₀ hkx, mul_comm (k x), mul_assoc, inv_mul_cancel hkx, mul_one]","premises":[{"full_name":"inv_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[50,8],"def_end_pos":[50,22]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"zpow_sub_one₀","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[396,6],"def_end_pos":[396,19]}]}]}
{"url":"Mathlib/RingTheory/Finiteness.lean","commit":"","full_name":"Ideal.fg_ker_comp","start":[462,0],"end":[471,98],"file_path":"Mathlib/RingTheory/Finiteness.lean","tactics":[{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis : Algebra R S := f.toAlgebra\n⊢ (RingHom.ker (g.comp f)).FG","tactic":"letI : Algebra R S := RingHom.toAlgebra f","premises":[{"full_name":"Algebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[100,6],"def_end_pos":[100,13]},{"full_name":"RingHom.toAlgebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[198,4],"def_end_pos":[198,21]}]},{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis : Algebra R S := f.toAlgebra\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝ : Algebra R S := f.toAlgebra\nthis : Algebra R A := (g.comp f).toAlgebra\n⊢ (RingHom.ker (g.comp f)).FG","tactic":"letI : Algebra R A := RingHom.toAlgebra (g.comp f)","premises":[{"full_name":"Algebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[100,6],"def_end_pos":[100,13]},{"full_name":"RingHom.comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[563,4],"def_end_pos":[563,8]},{"full_name":"RingHom.toAlgebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[198,4],"def_end_pos":[198,21]}]},{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝ : Algebra R S := f.toAlgebra\nthis : Algebra R A := (g.comp f).toAlgebra\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝¹ : Algebra R S := f.toAlgebra\nthis✝ : Algebra R A := (g.comp f).toAlgebra\nthis : Algebra S A := g.toAlgebra\n⊢ (RingHom.ker (g.comp f)).FG","tactic":"letI : Algebra S A := RingHom.toAlgebra g","premises":[{"full_name":"Algebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[100,6],"def_end_pos":[100,13]},{"full_name":"RingHom.toAlgebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[198,4],"def_end_pos":[198,21]}]},{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝¹ : Algebra R S := f.toAlgebra\nthis✝ : Algebra R A := (g.comp f).toAlgebra\nthis : Algebra S A := g.toAlgebra\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : Algebra R A := (g.comp f).toAlgebra\nthis✝ : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\n⊢ (RingHom.ker (g.comp f)).FG","tactic":"letI : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun _ => rfl","premises":[{"full_name":"IsScalarTower","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[214,6],"def_end_pos":[214,19]},{"full_name":"IsScalarTower.of_algebraMap_eq","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[105,8],"def_end_pos":[105,24]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : Algebra R A := (g.comp f).toAlgebra\nthis✝ : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : Algebra R A := (g.comp f).toAlgebra\nthis✝ : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf₁ : R →ₗ[R] S := Algebra.linearMap R S\n⊢ (RingHom.ker (g.comp f)).FG","tactic":"let f₁ := Algebra.linearMap R S","premises":[{"full_name":"Algebra.linearMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[337,14],"def_end_pos":[337,23]}]},{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : Algebra R A := (g.comp f).toAlgebra\nthis✝ : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf₁ : R →ₗ[R] S := Algebra.linearMap R S\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : Algebra R A := (g.comp f).toAlgebra\nthis✝ : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf₁ : R →ₗ[R] S := Algebra.linearMap R S\ng₁ : S →ₗ[R] A := (IsScalarTower.toAlgHom R S A).toLinearMap\n⊢ (RingHom.ker (g.comp f)).FG","tactic":"let g₁ := (IsScalarTower.toAlgHom R S A).toLinearMap","premises":[{"full_name":"AlgHom.toLinearMap","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[302,4],"def_end_pos":[302,15]},{"full_name":"IsScalarTower.toAlgHom","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[133,4],"def_end_pos":[133,12]}]},{"state_before":"R✝ : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing A\nf : R →+* S\ng : S →+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective ⇑f\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : Algebra R A := (g.comp f).toAlgebra\nthis✝ : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf₁ : R →ₗ[R] S := Algebra.linearMap R S\ng₁ : S →ₗ[R] A := (IsScalarTower.toAlgHom R S A).toLinearMap\n⊢ (RingHom.ker (g.comp f)).FG","state_after":"no goals","tactic":"exact Submodule.fg_ker_comp f₁ g₁ hf (Submodule.fg_restrictScalars (RingHom.ker g) hg hsur) hsur","premises":[{"full_name":"RingHom.ker","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[520,4],"def_end_pos":[520,7]},{"full_name":"Submodule.fg_ker_comp","def_path":"Mathlib/RingTheory/Finiteness.lean","def_pos":[321,8],"def_end_pos":[321,19]},{"full_name":"Submodule.fg_restrictScalars","def_path":"Mathlib/RingTheory/Finiteness.lean","def_pos":[331,8],"def_end_pos":[331,26]}]}]}
{"url":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","commit":"","full_name":"constFormalMultilinearSeries_zero","start":[331,0],"end":[340,94],"file_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","tactics":[{"state_before":"𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\n⊢ constFormalMultilinearSeries 𝕜 E 0 = 0","state_after":"case h.H\n𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : Fin n → E\n⊢ (constFormalMultilinearSeries 𝕜 E 0 n) x = (0 n) x","tactic":"ext n x","premises":[]},{"state_before":"case h.H\n𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : Fin n → E\n⊢ (constFormalMultilinearSeries 𝕜 E 0 n) x = (0 n) x","state_after":"case h.H\n𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : Fin n → E\n⊢ (match n with\n      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0\n      | x => 0)\n      x =\n    0","tactic":"simp only [FormalMultilinearSeries.zero_apply, ContinuousMultilinearMap.zero_apply,\n    constFormalMultilinearSeries]","premises":[{"full_name":"ContinuousMultilinearMap.zero_apply","def_path":"Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean","def_pos":[131,8],"def_end_pos":[131,18]},{"full_name":"FormalMultilinearSeries.zero_apply","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[73,8],"def_end_pos":[73,18]},{"full_name":"constFormalMultilinearSeries","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[318,4],"def_end_pos":[318,32]}]},{"state_before":"case h.H\n𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : Fin n → E\n⊢ (match n with\n      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0\n      | x => 0)\n      x =\n    0","state_after":"case h.H.zero\n𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nx : Fin 0 → E\n⊢ (match 0 with\n      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0\n      | x => 0)\n      x =\n    0\n\ncase h.H.succ\n𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nn✝ : ℕ\na✝ :\n  ∀ (x : Fin n✝ → E),\n    (match n✝ with\n        | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0\n        | x => 0)\n        x =\n      0\nx : Fin (n✝ + 1) → E\n⊢ (match n✝ + 1 with\n      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0\n      | x => 0)\n      x =\n    0","tactic":"induction n","premises":[]}]}
{"url":"Mathlib/RingTheory/Adjoin/Field.lean","commit":"","full_name":"Subalgebra.adjoin_rank_le","start":[108,0],"end":[117,52],"file_path":"Mathlib/RingTheory/Adjoin/Field.lean","tactics":[{"state_before":"R : Type u_1\nK✝ : Type u_2\nL✝ : Type u_3\nM : Type u_4\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : Field K✝\ninst✝¹⁶ : Field L✝\ninst✝¹⁵ : CommRing M\ninst✝¹⁴ : Algebra R K✝\ninst✝¹³ : Algebra R L✝\ninst✝¹² : Algebra R M\nx✝ : L✝\nint✝ : IsIntegral R x✝\nh : Splits (algebraMap R K✝) (minpoly R x✝)\ninst✝¹¹ : Algebra K✝ M\ninst✝¹⁰ : IsScalarTower R K✝ M\nx : M\nint : IsIntegral R x\nF : Type u_5\nE : Type u_6\nK : Type u_7\ninst✝⁹ : CommRing F\ninst✝⁸ : StrongRankCondition F\ninst✝⁷ : CommRing E\ninst✝⁶ : StrongRankCondition E\ninst✝⁵ : Ring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : Algebra F K\ninst✝¹ : IsScalarTower F E K\nL : Subalgebra F K\ninst✝ : Module.Free F ↥L\n⊢ Module.rank E ↥(Algebra.adjoin E ↑L) ≤ Module.rank F ↥L","state_after":"R : Type u_1\nK✝ : Type u_2\nL✝ : Type u_3\nM : Type u_4\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : Field K✝\ninst✝¹⁶ : Field L✝\ninst✝¹⁵ : CommRing M\ninst✝¹⁴ : Algebra R K✝\ninst✝¹³ : Algebra R L✝\ninst✝¹² : Algebra R M\nx✝ : L✝\nint✝ : IsIntegral R x✝\nh : Splits (algebraMap R K✝) (minpoly R x✝)\ninst✝¹¹ : Algebra K✝ M\ninst✝¹⁰ : IsScalarTower R K✝ M\nx : M\nint : IsIntegral R x\nF : Type u_5\nE : Type u_6\nK : Type u_7\ninst✝⁹ : CommRing F\ninst✝⁸ : StrongRankCondition F\ninst✝⁷ : CommRing E\ninst✝⁶ : StrongRankCondition E\ninst✝⁵ : Ring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : Algebra F K\ninst✝¹ : IsScalarTower F E K\nL : Subalgebra F K\ninst✝ : Module.Free F ↥L\n⊢ Module.rank E ↥(Submodule.span E (Set.range fun i => ↑((Module.Free.chooseBasis F ↥L) i))) ≤\n    Cardinal.mk (Module.Free.ChooseBasisIndex F ↥L)","tactic":"rw [← rank_toSubmodule, Module.Free.rank_eq_card_chooseBasisIndex F L,\n    L.adjoin_eq_span_basis E (Module.Free.chooseBasis F L)]","premises":[{"full_name":"Module.Free.chooseBasis","def_path":"Mathlib/LinearAlgebra/FreeModule/Basic.lean","def_pos":[78,18],"def_end_pos":[78,29]},{"full_name":"Module.Free.rank_eq_card_chooseBasisIndex","def_path":"Mathlib/LinearAlgebra/Dimension/Free.lean","def_pos":[77,8],"def_end_pos":[77,37]},{"full_name":"Subalgebra.adjoin_eq_span_basis","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[494,8],"def_end_pos":[494,39]},{"full_name":"Subalgebra.rank_toSubmodule","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[481,8],"def_end_pos":[481,35]}]},{"state_before":"R : Type u_1\nK✝ : Type u_2\nL✝ : Type u_3\nM : Type u_4\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : Field K✝\ninst✝¹⁶ : Field L✝\ninst✝¹⁵ : CommRing M\ninst✝¹⁴ : Algebra R K✝\ninst✝¹³ : Algebra R L✝\ninst✝¹² : Algebra R M\nx✝ : L✝\nint✝ : IsIntegral R x✝\nh : Splits (algebraMap R K✝) (minpoly R x✝)\ninst✝¹¹ : Algebra K✝ M\ninst✝¹⁰ : IsScalarTower R K✝ M\nx : M\nint : IsIntegral R x\nF : Type u_5\nE : Type u_6\nK : Type u_7\ninst✝⁹ : CommRing F\ninst✝⁸ : StrongRankCondition F\ninst✝⁷ : CommRing E\ninst✝⁶ : StrongRankCondition E\ninst✝⁵ : Ring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : Algebra F K\ninst✝¹ : IsScalarTower F E K\nL : Subalgebra F K\ninst✝ : Module.Free F ↥L\n⊢ Module.rank E ↥(Submodule.span E (Set.range fun i => ↑((Module.Free.chooseBasis F ↥L) i))) ≤\n    Cardinal.mk (Module.Free.ChooseBasisIndex F ↥L)","state_after":"no goals","tactic":"exact rank_span_le _ |>.trans Cardinal.mk_range_le","premises":[{"full_name":"Cardinal.mk_range_le","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1672,8],"def_end_pos":[1672,19]},{"full_name":"rank_span_le","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[404,8],"def_end_pos":[404,20]}]}]}
{"url":"Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean","commit":"","full_name":"AlgebraicTopology.DoldKan.compatibility_N₂_N₁_karoubi","start":[66,0],"end":[90,65],"file_path":"Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Preadditive C\n⊢ N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor =\n    karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙\n      N₁ ⋙\n        (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙\n          (KaroubiKaroubi.equivalence C).inverse.mapHomologicalComplex (ComplexShape.down ℕ)","state_after":"case refine_1\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\n⊢ (N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor).obj P =\n    (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙\n          N₁ ⋙\n            (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙\n              (KaroubiKaroubi.equivalence C).inverse.mapHomologicalComplex (ComplexShape.down ℕ)).obj\n      P\n\ncase refine_2\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P ⟶ Q\n⊢ (N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor).map f =\n    eqToHom ⋯ ≫\n      (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙\n              N₁ ⋙\n                (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙\n                  (KaroubiKaroubi.equivalence C).inverse.mapHomologicalComplex (ComplexShape.down ℕ)).map\n          f ≫\n        eqToHom ⋯","tactic":"refine CategoryTheory.Functor.ext (fun P => ?_) fun P Q f => ?_","premises":[{"full_name":"CategoryTheory.Functor.ext","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[179,8],"def_end_pos":[179,11]}]}]}
{"url":"Mathlib/Data/Nat/Prime/Basic.lean","commit":"","full_name":"Nat.dvd_of_forall_prime_mul_dvd","start":[73,0],"end":[78,58],"file_path":"Mathlib/Data/Nat/Prime/Basic.lean","tactics":[{"state_before":"n a b : ℕ\nhdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b\n⊢ a ∣ b","state_after":"case inl\nn b : ℕ\nhdvd : ∀ (p : ℕ), Prime p → p ∣ 1 → p * 1 ∣ b\n⊢ 1 ∣ b\n\ncase inr\nn a b : ℕ\nhdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b\nha : a ≠ 1\n⊢ a ∣ b","tactic":"obtain rfl | ha := eq_or_ne a 1","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nn a b : ℕ\nhdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b\nha : a ≠ 1\n⊢ a ∣ b","state_after":"case inr.intro\nn a b : ℕ\nhdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b\nha : a ≠ 1\np : ℕ\nhp : Prime p ∧ p ∣ a\n⊢ a ∣ b","tactic":"obtain ⟨p, hp⟩ := exists_prime_and_dvd ha","premises":[{"full_name":"Nat.exists_prime_and_dvd","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[395,8],"def_end_pos":[395,28]}]},{"state_before":"case inr.intro\nn a b : ℕ\nhdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b\nha : a ≠ 1\np : ℕ\nhp : Prime p ∧ p ∣ a\n⊢ a ∣ b","state_after":"no goals","tactic":"exact _root_.trans (dvd_mul_left a p) (hdvd p hp.1 hp.2)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"dvd_mul_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[176,8],"def_end_pos":[176,20]},{"full_name":"trans","def_path":"Mathlib/Init/Algebra/Classes.lean","def_pos":[261,8],"def_end_pos":[261,13]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Smeval.lean","commit":"","full_name":"Polynomial.smeval_X_pow","start":[90,0],"end":[93,97],"file_path":"Mathlib/Algebra/Polynomial/Smeval.lean","tactics":[{"state_before":"R : Type u_1\ninst✝³ : Semiring R\nr : R\np : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : MulActionWithZero R S\nx : S\nn : ℕ\n⊢ (X ^ n).smeval x = x ^ n","state_after":"no goals","tactic":"simp only [smeval_eq_sum, smul_pow, X_pow_eq_monomial, zero_smul, sum_monomial_index, one_smul]","premises":[{"full_name":"Polynomial.X_pow_eq_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[773,8],"def_end_pos":[773,25]},{"full_name":"Polynomial.smeval_eq_sum","def_path":"Mathlib/Algebra/Polynomial/Smeval.lean","def_pos":[52,8],"def_end_pos":[52,21]},{"full_name":"Polynomial.smul_pow","def_path":"Mathlib/Algebra/Polynomial/Smeval.lean","def_pos":[46,4],"def_end_pos":[46,12]},{"full_name":"Polynomial.sum_monomial_index","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[837,8],"def_end_pos":[837,26]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]}]}
{"url":"Mathlib/GroupTheory/Complement.lean","commit":"","full_name":"AddSubgroup.isComplement_univ_right","start":[127,0],"end":[137,37],"file_path":"Mathlib/GroupTheory/Complement.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ IsComplement S univ ↔ ∃ g, S = {g}","state_after":"case refine_1\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement S univ\n⊢ S.Nonempty\n\ncase refine_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement S univ\na : G\nha : a ∈ S\nb : G\nhb : b ∈ S\n⊢ a = b\n\ncase refine_3\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ (∃ g, S = {g}) → IsComplement S univ","tactic":"refine\n    ⟨fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr ⟨?_, fun a ha b hb => ?_⟩, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.exists_eq_singleton_iff_nonempty_subsingleton","def_path":"Mathlib/Data/Set/Subsingleton.lean","def_pos":[89,8],"def_end_pos":[89,53]}]}]}
{"url":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean","commit":"","full_name":"ProjectiveSpectrum.vanishingIdeal_univ","start":[183,0],"end":[185,32],"file_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ vanishingIdeal ∅ = ⊤","state_after":"no goals","tactic":"simpa using (gc_ideal _).u_top","premises":[{"full_name":"GaloisConnection.u_top","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[199,8],"def_end_pos":[199,13]},{"full_name":"ProjectiveSpectrum.gc_ideal","def_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean","def_pos":[119,8],"def_end_pos":[119,16]}]}]}
{"url":"Mathlib/Algebra/Order/ToIntervalMod.lean","commit":"","full_name":"toIocDiv_neg'","start":[334,0],"end":[335,54],"file_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp (-a) b = -(toIcoDiv hp a (-b) + 1)","state_after":"no goals","tactic":"simpa only [neg_neg] using toIocDiv_neg hp (-a) (-b)","premises":[{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"toIocDiv_neg","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[331,8],"def_end_pos":[331,20]}]}]}
{"url":"Mathlib/LinearAlgebra/LinearIndependent.lean","commit":"","full_name":"linearIndependent_iff_not_mem_span","start":[1207,0],"end":[1215,54],"file_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","tactics":[{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\n⊢ LinearIndependent K v ↔ ∀ (i : ι), v i ∉ span K (v '' (univ \\ {i}))","state_after":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\n⊢ (∀ (i : ι) (a : K), a • v i ∈ span K (v '' (univ \\ {i})) → a = 0) ↔ ∀ (i : ι), v i ∉ span K (v '' (univ \\ {i}))","tactic":"apply linearIndependent_iff_not_smul_mem_span.trans","premises":[{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"linearIndependent_iff_not_smul_mem_span","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[877,8],"def_end_pos":[877,47]}]},{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\n⊢ (∀ (i : ι) (a : K), a • v i ∈ span K (v '' (univ \\ {i})) → a = 0) ↔ ∀ (i : ι), v i ∉ span K (v '' (univ \\ {i}))","state_after":"case mp\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\n⊢ (∀ (i : ι) (a : K), a • v i ∈ span K (v '' (univ \\ {i})) → a = 0) → ∀ (i : ι), v i ∉ span K (v '' (univ \\ {i}))\n\ncase mpr\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\n⊢ (∀ (i : ι), v i ∉ span K (v '' (univ \\ {i}))) → ∀ (i : ι) (a : K), a • v i ∈ span K (v '' (univ \\ {i})) → a = 0","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Coeff.lean","commit":"","full_name":"Polynomial.coeff_X_pow_mul","start":[236,0],"end":[239,46],"file_path":"Mathlib/Algebra/Polynomial/Coeff.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ (X ^ n * p).coeff (d + n) = p.coeff d","state_after":"no goals","tactic":"rw [(commute_X_pow p n).eq, coeff_mul_X_pow]","premises":[{"full_name":"Commute.eq","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[52,18],"def_end_pos":[52,20]},{"full_name":"Polynomial.coeff_mul_X_pow","def_path":"Mathlib/Algebra/Polynomial/Coeff.lean","def_pos":[224,8],"def_end_pos":[224,23]},{"full_name":"Polynomial.commute_X_pow","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[531,8],"def_end_pos":[531,21]}]}]}
{"url":"Mathlib/GroupTheory/Submonoid/Inverses.lean","commit":"","full_name":"Submonoid.leftInvEquiv_mul","start":[163,0],"end":[165,76],"file_path":"Mathlib/GroupTheory/Submonoid/Inverses.lean","tactics":[{"state_before":"M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S.leftInv\n⊢ ↑((S.leftInvEquiv hS) x) * ↑x = 1","state_after":"no goals","tactic":"simpa only [leftInvEquiv_apply, fromCommLeftInv] using fromLeftInv_mul S x","premises":[{"full_name":"Submonoid.fromCommLeftInv","def_path":"Mathlib/GroupTheory/Submonoid/Inverses.lean","def_pos":[124,18],"def_end_pos":[124,33]},{"full_name":"Submonoid.fromLeftInv_mul","def_path":"Mathlib/GroupTheory/Submonoid/Inverses.lean","def_pos":[108,8],"def_end_pos":[108,23]},{"full_name":"Submonoid.leftInvEquiv_apply","def_path":"Mathlib/GroupTheory/Submonoid/Inverses.lean","def_pos":[135,29],"def_end_pos":[135,34]}]}]}
{"url":"Mathlib/Analysis/Convex/Strict.lean","commit":"","full_name":"StrictConvex.inter","start":[70,0],"end":[74,64],"file_path":"Mathlib/Analysis/Convex/Strict.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx y : E\na b : 𝕜\nt : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\n⊢ StrictConvex 𝕜 (s ∩ t)","state_after":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx✝ y✝ : E\na✝ b✝ : 𝕜\nt : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\nx : E\nhx : x ∈ s ∩ t\ny : E\nhy : y ∈ s ∩ t\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior (s ∩ t)","tactic":"intro x hx y hy hxy a b ha hb hab","premises":[]},{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx✝ y✝ : E\na✝ b✝ : 𝕜\nt : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\nx : E\nhx : x ∈ s ∩ t\ny : E\nhy : y ∈ s ∩ t\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior (s ∩ t)","state_after":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx✝ y✝ : E\na✝ b✝ : 𝕜\nt : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\nx : E\nhx : x ∈ s ∩ t\ny : E\nhy : y ∈ s ∩ t\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior s ∩ interior t","tactic":"rw [interior_inter]","premises":[{"full_name":"interior_inter","def_path":"Mathlib/Topology/Basic.lean","def_pos":[268,8],"def_end_pos":[268,22]}]},{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx✝ y✝ : E\na✝ b✝ : 𝕜\nt : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\nx : E\nhx : x ∈ s ∩ t\ny : E\nhy : y ∈ s ∩ t\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior s ∩ interior t","state_after":"no goals","tactic":"exact ⟨hs hx.1 hy.1 hxy ha hb hab, ht hx.2 hy.2 hxy ha hb hab⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","commit":"","full_name":"Complex.GammaSeq_eq_betaIntegral_of_re_pos","start":[234,0],"end":[236,74],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","tactics":[{"state_before":"s : ℂ\nhs : 0 < s.re\nn : ℕ\n⊢ s.GammaSeq n = ↑n ^ s * s.betaIntegral (↑n + 1)","state_after":"no goals","tactic":"rw [GammaSeq, betaIntegral_eval_nat_add_one_right hs n, ← mul_div_assoc]","premises":[{"full_name":"Complex.GammaSeq","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","def_pos":[231,18],"def_end_pos":[231,26]},{"full_name":"Complex.betaIntegral_eval_nat_add_one_right","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","def_pos":[202,8],"def_end_pos":[202,43]},{"full_name":"mul_div_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[330,8],"def_end_pos":[330,21]}]}]}
{"url":"Mathlib/Logic/Relation.lean","commit":"","full_name":"Relation.TransGen.head'_iff","start":[394,0],"end":[400,30],"file_path":"Mathlib/Logic/Relation.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nζ : Type u_6\nr : α → α → Prop\na b c d : α\n⊢ TransGen r a c ↔ ∃ b, r a b ∧ ReflTransGen r b c","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nζ : Type u_6\nr : α → α → Prop\na b c d : α\nh : TransGen r a c\n⊢ ∃ b, r a b ∧ ReflTransGen r b c","tactic":"refine ⟨fun h ↦ ?_, fun ⟨b, hab, hbc⟩ ↦ head' hab hbc⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Relation.TransGen.head'","def_path":"Mathlib/Logic/Relation.lean","def_pos":[349,8],"def_end_pos":[349,13]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nζ : Type u_6\nr : α → α → Prop\na b c d : α\nh : TransGen r a c\n⊢ ∃ b, r a b ∧ ReflTransGen r b c","state_after":"no goals","tactic":"induction h with\n  | single hac => exact ⟨_, hac, by rfl⟩\n  | tail _ hbc IH =>\n  rcases IH with ⟨d, had, hdb⟩\n  exact ⟨_, had, hdb.tail hbc⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Relation.ReflTransGen.tail","def_path":"Mathlib/Logic/Relation.lean","def_pos":[222,4],"def_end_pos":[222,8]},{"full_name":"Relation.TransGen.single","def_path":"Mathlib/Logic/Relation.lean","def_pos":[235,4],"def_end_pos":[235,10]},{"full_name":"Relation.TransGen.tail","def_path":"Mathlib/Logic/Relation.lean","def_pos":[236,4],"def_end_pos":[236,8]}]}]}
{"url":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","commit":"","full_name":"Ideal.quotientEquiv_symm_apply","start":[482,0],"end":[503,78],"file_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","tactics":[{"state_before":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ I ≤ comap (↑f) J","state_after":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ I ≤ comap (↑f) (map (↑f) I)","tactic":"rw [hIJ]","premises":[]},{"state_before":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ I ≤ comap (↑f) (map (↑f) I)","state_after":"no goals","tactic":"exact le_comap_map","premises":[{"full_name":"Ideal.le_comap_map","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[141,8],"def_end_pos":[141,20]}]},{"state_before":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ J ≤ comap (↑f.symm) I","state_after":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ map (↑f) I ≤ comap (↑f.symm) I","tactic":"rw [hIJ]","premises":[]},{"state_before":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ map (↑f) I ≤ comap (↑f.symm) I","state_after":"no goals","tactic":"exact le_of_eq (map_comap_of_equiv I f)","premises":[{"full_name":"Ideal.map_comap_of_equiv","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[325,8],"def_end_pos":[325,26]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]},{"state_before":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ LeftInverse (⇑(quotientMap I ↑f.symm ⋯)) (↑↑__src✝).toFun","state_after":"case mk\nR : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\nx✝ : R ⧸ I\nr : R\n⊢ (quotientMap I ↑f.symm ⋯) ((↑↑__src✝).toFun (Quot.mk Setoid.r r)) = Quot.mk Setoid.r r","tactic":"rintro ⟨r⟩","premises":[]},{"state_before":"case mk\nR : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\nx✝ : R ⧸ I\nr : R\n⊢ (quotientMap I ↑f.symm ⋯) ((↑↑__src✝).toFun (Quot.mk Setoid.r r)) = Quot.mk Setoid.r r","state_after":"no goals","tactic":"simp only [Submodule.Quotient.quot_mk_eq_mk, Quotient.mk_eq_mk, RingHom.toFun_eq_coe,\n        quotientMap_mk, RingEquiv.coe_toRingHom, RingEquiv.symm_apply_apply]","premises":[{"full_name":"Ideal.Quotient.mk_eq_mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[110,8],"def_end_pos":[110,16]},{"full_name":"Ideal.quotientMap_mk","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[469,8],"def_end_pos":[469,22]},{"full_name":"RingEquiv.coe_toRingHom","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[604,8],"def_end_pos":[604,21]},{"full_name":"RingEquiv.symm_apply_apply","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[312,8],"def_end_pos":[312,24]},{"full_name":"RingHom.toFun_eq_coe","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[384,8],"def_end_pos":[384,20]},{"full_name":"Submodule.Quotient.quot_mk_eq_mk","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[66,8],"def_end_pos":[66,21]}]},{"state_before":"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\n⊢ Function.RightInverse (⇑(quotientMap I ↑f.symm ⋯)) (↑↑__src✝).toFun","state_after":"case mk\nR : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\nx✝ : S ⧸ J\ns : S\n⊢ (↑↑__src✝).toFun ((quotientMap I ↑f.symm ⋯) (Quot.mk Setoid.r s)) = Quot.mk Setoid.r s","tactic":"rintro ⟨s⟩","premises":[]},{"state_before":"case mk\nR : Type u\nS✝ : Type v\nF : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : CommRing A\ninst✝² : Algebra R₁ A\ninst✝¹ : Algebra R₂ A\nS : Type v\ninst✝ : CommRing S\nI : Ideal R\nJ : Ideal S\nf : R ≃+* S\nhIJ : J = map (↑f) I\nx✝ : S ⧸ J\ns : S\n⊢ (↑↑__src✝).toFun ((quotientMap I ↑f.symm ⋯) (Quot.mk Setoid.r s)) = Quot.mk Setoid.r s","state_after":"no goals","tactic":"simp only [Submodule.Quotient.quot_mk_eq_mk, Quotient.mk_eq_mk, RingHom.toFun_eq_coe,\n        quotientMap_mk, RingEquiv.coe_toRingHom, RingEquiv.apply_symm_apply]","premises":[{"full_name":"Ideal.Quotient.mk_eq_mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[110,8],"def_end_pos":[110,16]},{"full_name":"Ideal.quotientMap_mk","def_path":"Mathlib/RingTheory/Ideal/QuotientOperations.lean","def_pos":[469,8],"def_end_pos":[469,22]},{"full_name":"RingEquiv.apply_symm_apply","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[308,8],"def_end_pos":[308,24]},{"full_name":"RingEquiv.coe_toRingHom","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[604,8],"def_end_pos":[604,21]},{"full_name":"RingHom.toFun_eq_coe","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[384,8],"def_end_pos":[384,20]},{"full_name":"Submodule.Quotient.quot_mk_eq_mk","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[66,8],"def_end_pos":[66,21]}]}]}
{"url":"Mathlib/Algebra/Lie/Basic.lean","commit":"","full_name":"LieModuleHom.ext_iff","start":[710,0],"end":[713,13],"file_path":"Mathlib/Algebra/Lie/Basic.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\nP : Type w₂\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieRingModule L N\ninst✝³ : LieRingModule L P\ninst✝² : LieModule R L M\ninst✝¹ : LieModule R L N\ninst✝ : LieModule R L P\nf g : M →ₗ⁅R,L⁆ N\n⊢ f = g → ∀ (m : M), f m = g m","state_after":"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\nP : Type w₂\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieRingModule L N\ninst✝³ : LieRingModule L P\ninst✝² : LieModule R L M\ninst✝¹ : LieModule R L N\ninst✝ : LieModule R L P\nf : M →ₗ⁅R,L⁆ N\nm : M\n⊢ f m = f m","tactic":"rintro rfl m","premises":[]},{"state_before":"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\nP : Type w₂\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieRingModule L N\ninst✝³ : LieRingModule L P\ninst✝² : LieModule R L M\ninst✝¹ : LieModule R L N\ninst✝ : LieModule R L P\nf : M →ₗ⁅R,L⁆ N\nm : M\n⊢ f m = f m","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Derivation.lean","commit":"","full_name":"Polynomial.mkDerivation_one_eq_derivative","start":[73,0],"end":[75,5],"file_path":"Mathlib/Algebra/Polynomial/Derivation.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Module R[X] A\ninst✝ : IsScalarTower R R[X] A\nf : R[X]\n⊢ ((mkDerivation R) 1) f = derivative f","state_after":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Module R[X] A\ninst✝ : IsScalarTower R R[X] A\nf : R[X]\n⊢ derivative' f = derivative f","tactic":"rw [mkDerivation_one_eq_derivative']","premises":[{"full_name":"Polynomial.mkDerivation_one_eq_derivative'","def_path":"Mathlib/Algebra/Polynomial/Derivation.lean","def_pos":[69,6],"def_end_pos":[69,37]}]},{"state_before":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Module R[X] A\ninst✝ : IsScalarTower R R[X] A\nf : R[X]\n⊢ derivative' f = derivative f","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/Separation.lean","commit":"","full_name":"R1Space.sInf","start":[1174,0],"end":[1182,82],"file_path":"Mathlib/Topology/Separation.lean","tactics":[{"state_before":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx y : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\n⊢ R1Space X","state_after":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx y : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\n⊢ R1Space X","tactic":"let _ := sInf T","premises":[{"full_name":"InfSet.sInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[48,2],"def_end_pos":[48,6]}]},{"state_before":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx y : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\n⊢ R1Space X","state_after":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx✝¹ y✝ : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\n⊢ x ⤳ y ∨ Disjoint (𝓝 x) (𝓝 y)","tactic":"refine ⟨fun x y ↦ ?_⟩","premises":[]},{"state_before":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx✝¹ y✝ : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\n⊢ x ⤳ y ∨ Disjoint (𝓝 x) (𝓝 y)","state_after":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx✝¹ y✝ : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\n⊢ ⨅ t ∈ T, 𝓝 x ≤ ⨅ t ∈ T, 𝓝 y ∨ Disjoint (⨅ t ∈ T, 𝓝 x) (⨅ t ∈ T, 𝓝 y)","tactic":"simp only [Specializes, nhds_sInf]","premises":[{"full_name":"Specializes","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[180,4],"def_end_pos":[180,15]},{"full_name":"nhds_sInf","def_path":"Mathlib/Topology/Order.lean","def_pos":[585,8],"def_end_pos":[585,17]}]},{"state_before":"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx✝¹ y✝ : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\n⊢ ⨅ t ∈ T, 𝓝 x ≤ ⨅ t ∈ T, 𝓝 y ∨ Disjoint (⨅ t ∈ T, 𝓝 x) (⨅ t ∈ T, 𝓝 y)","state_after":"case inl.intro.intro\nX✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx✝¹ y✝ : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\nt : TopologicalSpace X\nhtT : t ∈ T\nhtd : Disjoint (𝓝 x) (𝓝 y)\n⊢ ⨅ t ∈ T, 𝓝 x ≤ ⨅ t ∈ T, 𝓝 y ∨ Disjoint (⨅ t ∈ T, 𝓝 x) (⨅ t ∈ T, 𝓝 y)\n\ncase inr\nX✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : FirstCountableTopology X✝\ninst✝ : R1Space X✝\nx✝¹ y✝ : X✝\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\nhTd : ¬∃ t ∈ T, Disjoint (𝓝 x) (𝓝 y)\n⊢ ⨅ t ∈ T, 𝓝 x ≤ ⨅ t ∈ T, 𝓝 y ∨ Disjoint (⨅ t ∈ T, 𝓝 x) (⨅ t ∈ T, 𝓝 y)","tactic":"rcases em (∃ t ∈ T, Disjoint (@nhds X t x) (@nhds X t y)) with ⟨t, htT, htd⟩ | hTd","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Disjoint","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[40,4],"def_end_pos":[40,12]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]}]}
{"url":"Mathlib/Algebra/Ring/Equiv.lean","commit":"","full_name":"RingEquiv.symm_toRingHom_comp_toRingHom","start":[673,0],"end":[676,6],"file_path":"Mathlib/Algebra/Ring/Equiv.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nS : Type u_5\nS' : Type u_6\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring S'\ne : R ≃+* S\n⊢ e.symm.toRingHom.comp e.toRingHom = RingHom.id R","state_after":"case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nS : Type u_5\nS' : Type u_6\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring S'\ne : R ≃+* S\nx✝ : R\n⊢ (e.symm.toRingHom.comp e.toRingHom) x✝ = (RingHom.id R) x✝","tactic":"ext","premises":[]},{"state_before":"case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nS : Type u_5\nS' : Type u_6\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring S'\ne : R ≃+* S\nx✝ : R\n⊢ (e.symm.toRingHom.comp e.toRingHom) x✝ = (RingHom.id R) x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Asymptotics/Theta.lean","commit":"","full_name":"Asymptotics.isTheta_zero_left","start":[235,0],"end":[237,71],"file_path":"Mathlib/Analysis/Asymptotics/Theta.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nR : Type u_12\nR' : Type u_13\n𝕜 : Type u_14\n𝕜' : Type u_15\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nl l' : Filter α\n⊢ (fun x => 0) =Θ[l] g'' ↔ g'' =ᶠ[l] 0","state_after":"no goals","tactic":"simp only [IsTheta, isBigO_zero, isBigO_zero_right_iff, true_and_iff]","premises":[{"full_name":"Asymptotics.IsTheta","def_path":"Mathlib/Analysis/Asymptotics/Theta.lean","def_pos":[41,4],"def_end_pos":[41,11]},{"full_name":"Asymptotics.isBigO_zero","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1011,8],"def_end_pos":[1011,19]},{"full_name":"Asymptotics.isBigO_zero_right_iff","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1028,8],"def_end_pos":[1028,29]},{"full_name":"true_and_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[94,8],"def_end_pos":[94,20]}]}]}
{"url":"Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean","commit":"","full_name":"cardinal_mk_eq_cardinal_mk_field_pow_rank","start":[227,0],"end":[230,74],"file_path":"Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean","tactics":[{"state_before":"K✝ : Type u\nV✝ : Type v\ninst✝⁹ : Ring K✝\ninst✝⁸ : StrongRankCondition K✝\ninst✝⁷ : AddCommGroup V✝\ninst✝⁶ : Module K✝ V✝\nK V : Type u\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Module.Free K V\ninst✝ : Module.Finite K V\n⊢ #V = #K ^ Module.rank K V","state_after":"no goals","tactic":"simpa using lift_cardinal_mk_eq_lift_cardinal_mk_field_pow_lift_rank K V","premises":[{"full_name":"lift_cardinal_mk_eq_lift_cardinal_mk_field_pow_lift_rank","def_path":"Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean","def_pos":[215,8],"def_end_pos":[215,64]}]}]}
{"url":"Mathlib/Analysis/Analytic/Basic.lean","commit":"","full_name":"FormalMultilinearSeries.changeOriginIndexEquiv_apply_snd","start":[1099,0],"end":[1130,78],"file_path":"Mathlib/Analysis/Analytic/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\n⊢ Function.LeftInverse (fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩)\n    fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩","state_after":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ (fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩)\n      ((fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩) ⟨k, ⟨l, ⟨s, hs⟩⟩⟩) =\n    ⟨k, ⟨l, ⟨s, hs⟩⟩⟩","tactic":"rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]}]},{"state_before":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ (fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩)\n      ((fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩) ⟨k, ⟨l, ⟨s, hs⟩⟩⟩) =\n    ⟨k, ⟨l, ⟨s, hs⟩⟩⟩","state_after":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ ⟨k + l - s.card, ⟨s.card, ⟨Finset.map (finCongr ⋯).toEmbedding s, ⋯⟩⟩⟩ = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩","tactic":"dsimp only [Subtype.coe_mk]","premises":[{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]}]},{"state_before":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ ⟨k + l - s.card, ⟨s.card, ⟨Finset.map (finCongr ⋯).toEmbedding s, ⋯⟩⟩⟩ = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩","state_after":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ ∀ (k' l' : ℕ),\n    k' = k →\n      l' = l →\n        ∀ (hkl : k + l = k' + l') (hs' : (Finset.map (finCongr hkl).toEmbedding s).card = l'),\n          ⟨k', ⟨l', ⟨Finset.map (finCongr hkl).toEmbedding s, hs'⟩⟩⟩ = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩","tactic":"suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),\n        (⟨k', l', ⟨s.map (finCongr hkl).toEmbedding, hs'⟩⟩ :\n          Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by\n      apply this <;> simp only [hs, add_tsub_cancel_right]","premises":[{"full_name":"Equiv.toEmbedding","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[69,14],"def_end_pos":[69,31]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Finset.map","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[56,4],"def_end_pos":[56,7]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[173,2],"def_end_pos":[173,4]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"add_tsub_cancel_right","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[305,8],"def_end_pos":[305,29]},{"full_name":"finCongr","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[598,4],"def_end_pos":[598,19]}]},{"state_before":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ ∀ (k' l' : ℕ),\n    k' = k →\n      l' = l →\n        ∀ (hkl : k + l = k' + l') (hs' : (Finset.map (finCongr hkl).toEmbedding s).card = l'),\n          ⟨k', ⟨l', ⟨Finset.map (finCongr hkl).toEmbedding s, hs'⟩⟩⟩ = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩","state_after":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk'✝ l'✝ : ℕ\ns : Finset (Fin (k'✝ + l'✝))\nhs : s.card = l'✝\nhkl : k'✝ + l'✝ = k'✝ + l'✝\nhs' : (Finset.map (finCongr hkl).toEmbedding s).card = l'✝\n⊢ ⟨k'✝, ⟨l'✝, ⟨Finset.map (finCongr hkl).toEmbedding s, hs'⟩⟩⟩ = ⟨k'✝, ⟨l'✝, ⟨s, hs⟩⟩⟩","tactic":"rintro _ _ rfl rfl hkl hs'","premises":[]},{"state_before":"case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk'✝ l'✝ : ℕ\ns : Finset (Fin (k'✝ + l'✝))\nhs : s.card = l'✝\nhkl : k'✝ + l'✝ = k'✝ + l'✝\nhs' : (Finset.map (finCongr hkl).toEmbedding s).card = l'✝\n⊢ ⟨k'✝, ⟨l'✝, ⟨Finset.map (finCongr hkl).toEmbedding s, hs'⟩⟩⟩ = ⟨k'✝, ⟨l'✝, ⟨s, hs⟩⟩⟩","state_after":"no goals","tactic":"simp only [Equiv.refl_toEmbedding, finCongr_refl, Finset.map_refl, eq_self_iff_true,\n      OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]","premises":[{"full_name":"Equiv.refl_toEmbedding","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[392,8],"def_end_pos":[392,24]},{"full_name":"Finset.map_refl","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[118,8],"def_end_pos":[118,16]},{"full_name":"OrderIso.refl_toEquiv","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[750,8],"def_end_pos":[750,20]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]},{"full_name":"finCongr_refl","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[608,6],"def_end_pos":[608,26]},{"full_name":"heq_iff_eq","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[37,8],"def_end_pos":[37,18]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\n⊢ Function.RightInverse (fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩)\n    fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩","state_after":"case mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nn : ℕ\ns : Finset (Fin n)\n⊢ (fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩)\n      ((fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩) ⟨n, s⟩) =\n    ⟨n, s⟩","tactic":"rintro ⟨n, s⟩","premises":[]},{"state_before":"case mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nn : ℕ\ns : Finset (Fin n)\n⊢ (fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩)\n      ((fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩) ⟨n, s⟩) =\n    ⟨n, s⟩","state_after":"no goals","tactic":"simp [tsub_add_cancel_of_le (card_finset_fin_le s), finCongr_eq_equivCast]","premises":[{"full_name":"card_finset_fin_le","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[304,8],"def_end_pos":[304,26]},{"full_name":"finCongr_eq_equivCast","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[618,6],"def_end_pos":[618,34]},{"full_name":"tsub_add_cancel_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[28,8],"def_end_pos":[28,29]}]}]}
{"url":"Mathlib/Probability/Distributions/Geometric.lean","commit":"","full_name":"ProbabilityTheory.geometricPMFRealSum","start":[38,0],"end":[45,12],"file_path":"Mathlib/Probability/Distributions/Geometric.lean","tactics":[{"state_before":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\n⊢ HasSum (fun n => geometricPMFReal p n) 1","state_after":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","tactic":"unfold geometricPMFReal","premises":[{"full_name":"ProbabilityTheory.geometricPMFReal","def_path":"Mathlib/Probability/Distributions/Geometric.lean","def_pos":[36,4],"def_end_pos":[36,20]}]},{"state_before":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","state_after":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun n => (1 - p) ^ n) (1 - (1 - p))⁻¹\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","tactic":"have := hasSum_geometric_of_lt_one (sub_nonneg.mpr hp_le_one) (sub_lt_self 1 hp_pos)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"hasSum_geometric_of_lt_one","def_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","def_pos":[251,8],"def_end_pos":[251,34]},{"full_name":"sub_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[480,29],"def_end_pos":[480,39]}]},{"state_before":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun n => (1 - p) ^ n) (1 - (1 - p))⁻¹\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","state_after":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) ((1 - (1 - p))⁻¹ * p)\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","tactic":"apply (hasSum_mul_right_iff (hp_pos.ne')).mpr at this","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"hasSum_mul_right_iff","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Ring.lean","def_pos":[80,8],"def_end_pos":[80,28]}]},{"state_before":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) ((1 - (1 - p))⁻¹ * p)\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","state_after":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) (p⁻¹ * p)\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","tactic":"simp only [sub_sub_cancel] at this","premises":[{"full_name":"sub_sub_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[910,2],"def_end_pos":[910,13]}]},{"state_before":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) (p⁻¹ * p)\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","state_after":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) 1\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","tactic":"rw [inv_mul_eq_div, div_self hp_pos.ne'] at this","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"div_self","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[251,14],"def_end_pos":[251,22]},{"full_name":"inv_mul_eq_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[525,8],"def_end_pos":[525,22]}]},{"state_before":"p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) 1\n⊢ HasSum (fun n => (1 - p) ^ n * p) 1","state_after":"no goals","tactic":"exact this","premises":[]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Fintype.prod_bijective","start":[1911,0],"end":[1920,56],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι✝ : Type u_1\nκ✝ : Type u_2\nα✝ : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α✝\na : α✝\nf✝ g✝ : α✝ → β\nι : Type u_6\nκ : Type u_7\nα : Type u_8\ninst✝² : Fintype ι\ninst✝¹ : Fintype κ\ninst✝ : CommMonoid α\ne : ι → κ\nhe : Bijective e\nf : ι → α\ng : κ → α\nh : ∀ (x : ι), f x = g (e x)\n⊢ ∀ (i : ι), i ∈ univ ↔ (Equiv.ofBijective e he) i ∈ univ","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"ι✝ : Type u_1\nκ✝ : Type u_2\nα✝ : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α✝\na : α✝\nf✝ g✝ : α✝ → β\nι : Type u_6\nκ : Type u_7\nα : Type u_8\ninst✝² : Fintype ι\ninst✝¹ : Fintype κ\ninst✝ : CommMonoid α\ne : ι → κ\nhe : Bijective e\nf : ι → α\ng : κ → α\nh : ∀ (x : ι), f x = g (e x)\n⊢ ∀ i ∈ univ, f i = g ((Equiv.ofBijective e he) i)","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","commit":"","full_name":"CategoryTheory.Limits.Multicofork.ofπ_pt","start":[512,0],"end":[527,17],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nI✝ : MultispanIndex C\nK : Multicofork I✝\nI : MultispanIndex C\nP : C\nπ : (b : I.R) → I.right b ⟶ P\nw : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)\n⊢ ∀ ⦃X Y : WalkingMultispan I.fstFrom I.sndFrom⦄ (f : X ⟶ Y),\n    I.multispan.map f ≫\n        (fun x =>\n            match x with\n            | WalkingMultispan.left a => I.fst a ≫ π (I.fstFrom a)\n            | WalkingMultispan.right b => π b)\n          Y =\n      (fun x =>\n            match x with\n            | WalkingMultispan.left a => I.fst a ≫ π (I.fstFrom a)\n            | WalkingMultispan.right b => π b)\n          X ≫\n        ((Functor.const (WalkingMultispan I.fstFrom I.sndFrom)).obj P).map f","state_after":"case left.right.snd\nC : Type u\ninst✝ : Category.{v, u} C\nI✝ : MultispanIndex C\nK : Multicofork I✝\nI : MultispanIndex C\nP : C\nπ : (b : I.R) → I.right b ⟶ P\nw : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)\na✝ : I.L\n⊢ I.snd a✝ ≫ π (I.sndFrom a✝) = I.fst a✝ ≫ π (I.fstFrom a✝)","tactic":"rintro (_ | _) (_ | _) (_ | _ | _) <;> dsimp <;>\n          simp only [Functor.map_id, MultispanIndex.multispan_obj_left,\n            Category.id_comp, Category.comp_id, MultispanIndex.multispan_obj_right]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Limits.MultispanIndex.multispan_obj_left","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[232,8],"def_end_pos":[232,26]},{"full_name":"CategoryTheory.Limits.MultispanIndex.multispan_obj_right","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","def_pos":[236,8],"def_end_pos":[236,27]}]},{"state_before":"case left.right.snd\nC : Type u\ninst✝ : Category.{v, u} C\nI✝ : MultispanIndex C\nK : Multicofork I✝\nI : MultispanIndex C\nP : C\nπ : (b : I.R) → I.right b ⟶ P\nw : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)\na✝ : I.L\n⊢ I.snd a✝ ≫ π (I.sndFrom a✝) = I.fst a✝ ≫ π (I.fstFrom a✝)","state_after":"case left.right.snd\nC : Type u\ninst✝ : Category.{v, u} C\nI✝ : MultispanIndex C\nK : Multicofork I✝\nI : MultispanIndex C\nP : C\nπ : (b : I.R) → I.right b ⟶ P\nw : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)\na✝ : I.L\n⊢ I.fst a✝ ≫ π (I.fstFrom a✝) = I.snd a✝ ≫ π (I.sndFrom a✝)","tactic":"symm","premises":[]},{"state_before":"case left.right.snd\nC : Type u\ninst✝ : Category.{v, u} C\nI✝ : MultispanIndex C\nK : Multicofork I✝\nI : MultispanIndex C\nP : C\nπ : (b : I.R) → I.right b ⟶ P\nw : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)\na✝ : I.L\n⊢ I.fst a✝ ≫ π (I.fstFrom a✝) = I.snd a✝ ≫ π (I.sndFrom a✝)","state_after":"no goals","tactic":"apply w","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/Gluing.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.OpenCover.gluedCover_V","start":[311,0],"end":[326,27],"file_path":"Mathlib/AlgebraicGeometry/Gluing.lean","tactics":[{"state_before":"X : Scheme\n𝒰 : X.OpenCover\nx : 𝒰.J\n⊢ (fun x y => (pullbackSymmetry (𝒰.map x) (𝒰.map y)).hom) x x =\n    𝟙\n      ((fun x =>\n          match x with\n          | (x, y) => pullback (𝒰.map x) (𝒰.map y))\n        (x, x))","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.J\n⊢ (fun x y z => 𝒰.gluedCoverT' x y z) x y z ≫\n      pullback.snd ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) y z)\n        ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) y x) =\n    pullback.fst ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) x y)\n        ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) x z) ≫\n      (fun x y => (pullbackSymmetry (𝒰.map x) (𝒰.map y)).hom) x y","state_after":"no goals","tactic":"apply pullback.hom_ext <;> simp","premises":[{"full_name":"CategoryTheory.Limits.pullback.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[212,8],"def_end_pos":[212,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean","commit":"","full_name":"CategoryTheory.Limits.diagramIsoParallelFamily_inv_app","start":[132,0],"end":[138,34],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean","tactics":[{"state_before":"J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\nF : WalkingParallelFamily J ⥤ C\nj : WalkingParallelFamily J\n⊢ F.obj j = (parallelFamily fun j => F.map (line j)).obj j","state_after":"no goals","tactic":"cases j <;> aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\nF : WalkingParallelFamily J ⥤ C\n⊢ ∀ {X Y : WalkingParallelFamily J} (f : X ⟶ Y),\n    F.map f ≫ ((fun j => eqToIso ⋯) Y).hom =\n      ((fun j => eqToIso ⋯) X).hom ≫ (parallelFamily fun j => F.map (line j)).map f","state_after":"no goals","tactic":"rintro _ _ (_|_) <;> aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.dropLast_take","start":[510,0],"end":[514,72],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\nxs : Stream' α\n⊢ (take n xs).dropLast = take (n - 1) xs","state_after":"no goals","tactic":"cases n with\n  | zero => simp\n  | succ n => rw [take_succ', List.dropLast_concat, Nat.add_one_sub_one]","premises":[{"full_name":"List.dropLast_concat","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1825,21],"def_end_pos":[1825,36]},{"full_name":"Nat.add_one_sub_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[118,18],"def_end_pos":[118,33]},{"full_name":"Stream'.take_succ'","def_path":"Mathlib/Data/Stream/Init.lean","def_pos":[485,8],"def_end_pos":[485,18]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Gamma.lean","commit":"","full_name":"integral_rpow_mul_exp_neg_rpow","start":[21,0],"end":[37,20],"file_path":"Mathlib/MeasureTheory/Integral/Gamma.lean","tactics":[{"state_before":"p q : ℝ\nhp : 0 < p\nhq : -1 < q\n⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * Gamma ((q + 1) / p)","state_after":"no goals","tactic":"calc\n    _ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by\n      rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),\n        abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]\n      refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)\n      rw [← rpow_mul (le_of_lt hx) _ p, one_div_mul_cancel (ne_of_gt hp), rpow_one]\n    _ = ∫ (x : ℝ) in Ioi 0, 1 / p * exp (-x) * x ^ (1 / p - 1 + q / p) := by\n      simp_rw [smul_eq_mul, mul_assoc]\n      refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)\n      rw [← rpow_mul (le_of_lt hx), div_mul_eq_mul_div, one_mul, rpow_add hx]\n      ring_nf\n    _ = (1 / p) * Gamma ((q + 1) / p) := by\n      rw [Gamma_eq_integral (div_pos (neg_lt_iff_pos_add.mp hq) hp)]\n      simp_rw [show 1 / p - 1 + q / p = (q + 1) / p - 1 by field_simp; ring, ← integral_mul_left,\n        ← mul_assoc]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"MeasureTheory.integral","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[714,16],"def_end_pos":[714,24]},{"full_name":"MeasureTheory.integral_comp_rpow_Ioi","def_path":"Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean","def_pos":[1055,8],"def_end_pos":[1055,30]},{"full_name":"MeasureTheory.integral_mul_left","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[831,8],"def_end_pos":[831,25]},{"full_name":"MeasureTheory.setIntegral_congr","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[90,8],"def_end_pos":[90,25]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.Gamma","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[473,4],"def_end_pos":[473,9]},{"full_name":"Real.Gamma_eq_integral","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[476,8],"def_end_pos":[476,25]},{"full_name":"Real.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[102,11],"def_end_pos":[102,14]},{"full_name":"Real.rpow_add","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[172,8],"def_end_pos":[172,16]},{"full_name":"Real.rpow_mul","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[366,8],"def_end_pos":[366,16]},{"full_name":"Real.rpow_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[134,8],"def_end_pos":[134,16]},{"full_name":"Set.Ioi","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[70,4],"def_end_pos":[70,7]},{"full_name":"abs_eq_self","def_path":"Mathlib/Algebra/Order/Group/Abs.lean","def_pos":[196,8],"def_end_pos":[196,19]},{"full_name":"div_mul_eq_mul_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[558,8],"def_end_pos":[558,26]},{"full_name":"div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[45,6],"def_end_pos":[45,13]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"measurableSet_Ioi","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[170,8],"def_end_pos":[170,25]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"ne_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[85,8],"def_end_pos":[85,16]},{"full_name":"neg_lt_iff_pos_add","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[180,2],"def_end_pos":[180,13]},{"full_name":"one_div_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[293,6],"def_end_pos":[293,24]},{"full_name":"one_div_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[349,8],"def_end_pos":[349,23]},{"full_name":"one_div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[37,6],"def_end_pos":[37,17]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
{"url":"Mathlib/RingTheory/PrimeSpectrum.lean","commit":"","full_name":"PrimeSpectrum.zeroLocus_singleton_mul","start":[341,0],"end":[343,60],"file_path":"Mathlib/RingTheory/PrimeSpectrum.lean","tactics":[{"state_before":"R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf g : R\nx : PrimeSpectrum R\n⊢ x ∈ zeroLocus {f * g} ↔ x ∈ zeroLocus {f} ∪ zeroLocus {g}","state_after":"no goals","tactic":"simpa using x.2.mul_mem_iff_mem_or_mem","premises":[{"full_name":"Ideal.IsPrime.mul_mem_iff_mem_or_mem","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[521,8],"def_end_pos":[521,38]},{"full_name":"PrimeSpectrum.isPrime","def_path":"Mathlib/RingTheory/PrimeSpectrum.lean","def_pos":[62,2],"def_end_pos":[62,9]}]}]}
{"url":"Mathlib/Algebra/Order/Rearrangement.lean","commit":"","full_name":"MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff","start":[173,0],"end":[180,42],"file_path":"Mathlib/Algebra/Order/Rearrangement.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ ⇑σ) g ↑s","state_after":"no goals","tactic":"simp [← hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ, lt_iff_le_and_ne,\n    hfg.sum_comp_perm_smul_le_sum_smul hσ]","premises":[{"full_name":"MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[156,8],"def_end_pos":[156,53]},{"full_name":"MonovaryOn.sum_comp_perm_smul_le_sum_smul","def_path":"Mathlib/Algebra/Order/Rearrangement.lean","def_pos":[147,8],"def_end_pos":[147,49]},{"full_name":"lt_iff_le_and_ne","def_path":"Mathlib/Order/Basic.lean","def_pos":[309,8],"def_end_pos":[309,24]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","commit":"","full_name":"CategoryTheory.ShortComplex.unop_g","start":[240,0],"end":[243,64],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.60429, u_1} C\ninst✝² : Category.{?u.60433, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS✝ S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nS : ShortComplex Cᵒᵖ\n⊢ S.g.unop ≫ S.f.unop = 0","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.60429, u_1} C\ninst✝² : Category.{?u.60433, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS✝ S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nS : ShortComplex Cᵒᵖ\n⊢ Quiver.Hom.unop 0 = 0","tactic":"simp only [← unop_comp, S.zero]","premises":[{"full_name":"CategoryTheory.ShortComplex.zero","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[41,2],"def_end_pos":[41,6]},{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.60429, u_1} C\ninst✝² : Category.{?u.60433, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS✝ S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nS : ShortComplex Cᵒᵖ\n⊢ Quiver.Hom.unop 0 = 0","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/SetToL1.lean","commit":"","full_name":"MeasureTheory.L1.setToL1_congr_left'","start":[963,0],"end":[972,49],"file_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf : ↥(Lp E 1 μ)\n⊢ (setToL1 hT) f = (setToL1 hT') f","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf : ↥(Lp E 1 μ)\n⊢ setToL1 hT = setToL1 hT'","tactic":"suffices setToL1 hT = setToL1 hT' by rw [this]","premises":[{"full_name":"MeasureTheory.L1.setToL1","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[923,4],"def_end_pos":[923,11]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf : ↥(Lp E 1 μ)\n⊢ setToL1 hT = setToL1 hT'","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf : ↥(Lp E 1 μ)\n⊢ (setToL1 hT').comp (coeToLp α E ℝ) = setToL1SCLM α E μ hT","tactic":"refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_","premises":[{"full_name":"ContinuousLinearMap.extend_unique","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean","def_pos":[216,8],"def_end_pos":[216,21]},{"full_name":"MeasureTheory.L1.SimpleFunc.setToL1SCLM","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[800,4],"def_end_pos":[800,15]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf : ↥(Lp E 1 μ)\n⊢ (setToL1 hT').comp (coeToLp α E ℝ) = setToL1SCLM α E μ hT","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf✝ : ↥(Lp E 1 μ)\nf : ↥(simpleFunc E 1 μ)\n⊢ ((setToL1 hT').comp (coeToLp α E ℝ)) f = (setToL1SCLM α E μ hT) f","tactic":"ext1 f","premises":[]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf✝ : ↥(Lp E 1 μ)\nf : ↥(simpleFunc E 1 μ)\n⊢ ((setToL1 hT').comp (coeToLp α E ℝ)) f = (setToL1SCLM α E μ hT) f","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf✝ : ↥(Lp E 1 μ)\nf : ↥(simpleFunc E 1 μ)\n⊢ (setToL1 hT') ↑f = (setToL1SCLM α E μ hT) f","tactic":"suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; rfl","premises":[{"full_name":"MeasureTheory.L1.SimpleFunc.setToL1SCLM","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[800,4],"def_end_pos":[800,15]},{"full_name":"MeasureTheory.L1.setToL1","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[923,4],"def_end_pos":[923,11]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf✝ : ↥(Lp E 1 μ)\nf : ↥(simpleFunc E 1 μ)\n⊢ (setToL1 hT') ↑f = (setToL1SCLM α E μ hT) f","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf✝ : ↥(Lp E 1 μ)\nf : ↥(simpleFunc E 1 μ)\n⊢ (setToL1SCLM α E μ hT') f = (setToL1SCLM α E μ hT) f","tactic":"rw [setToL1_eq_setToL1SCLM]","premises":[{"full_name":"MeasureTheory.L1.setToL1_eq_setToL1SCLM","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[927,8],"def_end_pos":[927,30]}]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT✝ T'✝ T'' : Set α → E →L[ℝ] F\nC✝ C'✝ C'' : ℝ\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s\nf✝ : ↥(Lp E 1 μ)\nf : ↥(simpleFunc E 1 μ)\n⊢ (setToL1SCLM α E μ hT') f = (setToL1SCLM α E μ hT) f","state_after":"no goals","tactic":"exact (setToL1SCLM_congr_left' hT hT' h f).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left'","def_path":"Mathlib/MeasureTheory/Integral/SetToL1.lean","def_pos":[825,8],"def_end_pos":[825,31]}]}]}
{"url":"Mathlib/Analysis/Convex/Star.lean","commit":"","full_name":"StarConvex.mem_smul","start":[384,0],"end":[387,54],"file_path":"Mathlib/Analysis/Convex/Star.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ x ∈ t • s","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ t⁻¹ • x ∈ s","tactic":"rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Set.mem_smul_set_iff_inv_smul_mem₀","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[824,8],"def_end_pos":[824,38]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ t⁻¹ • x ∈ s","state_after":"no goals","tactic":"exact hs.smul_mem hx (by positivity) (inv_le_one ht)","premises":[{"full_name":"StarConvex.smul_mem","def_path":"Mathlib/Analysis/Convex/Star.lean","def_pos":[304,8],"def_end_pos":[304,27]},{"full_name":"inv_le_one","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[191,8],"def_end_pos":[191,18]}]}]}
{"url":"Mathlib/Data/ZMod/Basic.lean","commit":"","full_name":"ZMod.natCast_toNat","start":[670,0],"end":[673,36],"file_path":"Mathlib/Data/ZMod/Basic.lean","tactics":[{"state_before":"m n✝ p n : ℕ\n_h : 0 ≤ ↑n\n⊢ ↑(↑n).toNat = ↑↑n","state_after":"no goals","tactic":"simp only [Int.cast_natCast, Int.toNat_natCast]","premises":[{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.toNat_natCast","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[544,14],"def_end_pos":[544,27]}]},{"state_before":"m n✝ p n : ℕ\nh : 0 ≤ Int.negSucc n\n⊢ ↑(Int.negSucc n).toNat = ↑(Int.negSucc n)","state_after":"no goals","tactic":"simp at h","premises":[]}]}
{"url":"Mathlib/Algebra/Lie/Matrix.lean","commit":"","full_name":"Matrix.lieConj_symm_apply","start":[68,0],"end":[72,31],"file_path":"Mathlib/Algebra/Lie/Matrix.lean","tactics":[{"state_before":"R : Type u\ninst✝² : CommRing R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nP A : Matrix n n R\nh : Invertible P\n⊢ (P.lieConj h).symm A = P⁻¹ * A * P","state_after":"no goals","tactic":"simp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,\n    LinearMap.toMatrix'_toLin']","premises":[{"full_name":"LinearEquiv.symm_conj_apply","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[559,8],"def_end_pos":[559,23]},{"full_name":"LinearMap.toMatrix'_comp","def_path":"Mathlib/LinearAlgebra/Matrix/ToLin.lean","def_pos":[381,8],"def_end_pos":[381,32]},{"full_name":"LinearMap.toMatrix'_toLin'","def_path":"Mathlib/LinearAlgebra/Matrix/ToLin.lean","def_pos":[320,8],"def_end_pos":[320,34]},{"full_name":"Matrix.lieConj","def_path":"Mathlib/Algebra/Lie/Matrix.lean","def_pos":[59,4],"def_end_pos":[59,18]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","commit":"","full_name":"SimpleGraph.right_nonuniformWitnesses_card","start":[140,0],"end":[143,63],"file_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","tactics":[{"state_before":"α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬G.IsUniform ε s t\n⊢ ↑t.card * ε ≤ ↑(G.nonuniformWitnesses ε s t).2.card","state_after":"α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬G.IsUniform ε s t\n⊢ ↑t.card * ε ≤ ↑(⋯.choose, ⋯.choose).2.card","tactic":"rw [nonuniformWitnesses, dif_pos h]","premises":[{"full_name":"SimpleGraph.nonuniformWitnesses","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","def_pos":[120,18],"def_end_pos":[120,37]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]},{"state_before":"α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬G.IsUniform ε s t\n⊢ ↑t.card * ε ≤ ↑(⋯.choose, ⋯.choose).2.card","state_after":"no goals","tactic":"exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.2.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Exists.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[177,8],"def_end_pos":[177,26]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"SimpleGraph.not_isUniform_iff","def_path":"Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean","def_pos":[107,8],"def_end_pos":[107,25]}]}]}
{"url":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","commit":"","full_name":"FractionalIdeal.spanSingleton_inv_mul","start":[157,0],"end":[159,43],"file_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Field K\nR₁ : Type u_4\ninst✝⁶ : CommRing R₁\ninst✝⁵ : IsDomain R₁\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nI J : FractionalIdeal R₁⁰ K\nK' : Type u_5\ninst✝² : Field K'\ninst✝¹ : Algebra R₁ K'\ninst✝ : IsFractionRing R₁ K'\nx : K\nhx : x ≠ 0\n⊢ (spanSingleton R₁⁰ x)⁻¹ * spanSingleton R₁⁰ x = 1","state_after":"no goals","tactic":"rw [mul_comm, spanSingleton_mul_inv K hx]","premises":[{"full_name":"FractionalIdeal.spanSingleton_mul_inv","def_path":"Mathlib/RingTheory/DedekindDomain/Ideal.lean","def_pos":[146,8],"def_end_pos":[146,29]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.sub_self","start":[506,0],"end":[507,95],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal.{u_4}\n⊢ a - a = 0","state_after":"no goals","tactic":"simpa only [add_zero] using add_sub_cancel a 0","premises":[{"full_name":"Ordinal.add_sub_cancel","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[473,8],"def_end_pos":[473,22]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Pointwise.lean","commit":"","full_name":"ball_zero_add_singleton","start":[118,0],"end":[119,90],"file_path":"Mathlib/Analysis/Normed/Group/Pointwise.lean","tactics":[{"state_before":"E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ ball 1 δ * {x} = ball x δ","state_after":"no goals","tactic":"simp [ball_mul_singleton]","premises":[{"full_name":"ball_mul_singleton","def_path":"Mathlib/Analysis/Normed/Group/Pointwise.lean","def_pos":[104,8],"def_end_pos":[104,26]}]}]}
{"url":"Mathlib/GroupTheory/ClassEquation.lean","commit":"","full_name":"Group.card_center_add_sum_card_noncenter_eq_card","start":[72,0],"end":[81,8],"file_path":"Mathlib/GroupTheory/ClassEquation.lean","tactics":[{"state_before":"G✝ : Type u_1\ninst✝⁵ : Group G✝\nG : Type u_2\ninst✝⁴ : Group G\ninst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier\ninst✝² : Fintype G\ninst✝¹ : Fintype ↥(Subgroup.center G)\ninst✝ : Fintype ↑(noncenter G)\n⊢ Fintype.card ↥(Subgroup.center G) + ∑ x ∈ (noncenter G).toFinset, x.carrier.toFinset.card = Fintype.card G","state_after":"case h.e'_2.h.e'_5\nG✝ : Type u_1\ninst✝⁵ : Group G✝\nG : Type u_2\ninst✝⁴ : Group G\ninst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier\ninst✝² : Fintype G\ninst✝¹ : Fintype ↥(Subgroup.center G)\ninst✝ : Fintype ↑(noncenter G)\n⊢ Fintype.card ↥(Subgroup.center G) = Nat.card ↥(Subgroup.center G)\n\ncase h.e'_2.h.e'_6\nG✝ : Type u_1\ninst✝⁵ : Group G✝\nG : Type u_2\ninst✝⁴ : Group G\ninst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier\ninst✝² : Fintype G\ninst✝¹ : Fintype ↥(Subgroup.center G)\ninst✝ : Fintype ↑(noncenter G)\n⊢ ∑ x ∈ (noncenter G).toFinset, x.carrier.toFinset.card =\n    ∑ᶠ (x : ConjClasses G) (_ : x ∈ noncenter G), Nat.card ↑x.carrier\n\ncase h.e'_3\nG✝ : Type u_1\ninst✝⁵ : Group G✝\nG : Type u_2\ninst✝⁴ : Group G\ninst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier\ninst✝² : Fintype G\ninst✝¹ : Fintype ↥(Subgroup.center G)\ninst✝ : Fintype ↑(noncenter G)\n⊢ Fintype.card G = Nat.card G","tactic":"convert Group.nat_card_center_add_sum_card_noncenter_eq_card G using 2","premises":[{"full_name":"Group.nat_card_center_add_sum_card_noncenter_eq_card","def_path":"Mathlib/GroupTheory/ClassEquation.lean","def_pos":[47,8],"def_end_pos":[47,60]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.filter_attach'","start":[2513,0],"end":[2521,76],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\ns : Multiset α\np : { a // a ∈ s } → Prop\ninst✝¹ : DecidableEq α\ninst✝ : DecidablePred p\n⊢ filter p s.attach = map (Subtype.map id ⋯) (filter (fun x => ∃ (h : x ∈ s), p ⟨x, h⟩) s).attach","state_after":"no goals","tactic":"classical\n  refine Multiset.map_injective Subtype.val_injective ?_\n  rw [map_filter' _ Subtype.val_injective]\n  simp only [Function.comp, Subtype.exists, coe_mk, Subtype.map,\n    exists_and_right, exists_eq_right, attach_map_val, map_map, map_coe, id]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Multiset.attach_map_val","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1354,8],"def_end_pos":[1354,22]},{"full_name":"Multiset.map_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1031,25],"def_end_pos":[1031,32]},{"full_name":"Multiset.map_filter'","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1829,6],"def_end_pos":[1829,17]},{"full_name":"Multiset.map_injective","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[2510,8],"def_end_pos":[2510,21]},{"full_name":"Multiset.map_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1129,8],"def_end_pos":[1129,15]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"Subtype.exists","def_path":"Mathlib/Data/Subtype.lean","def_pos":[50,18],"def_end_pos":[50,26]},{"full_name":"Subtype.map","def_path":"Mathlib/Data/Subtype.lean","def_pos":[171,4],"def_end_pos":[171,7]},{"full_name":"Subtype.val_injective","def_path":"Mathlib/Data/Subtype.lean","def_pos":[104,8],"def_end_pos":[104,21]},{"full_name":"exists_and_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[291,16],"def_end_pos":[291,32]},{"full_name":"exists_eq_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[285,16],"def_end_pos":[285,31]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Real.sqrtTwoAddSeries_two","start":[623,0],"end":[623,74],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"x : ℝ\n⊢ sqrtTwoAddSeries 0 2 = √(2 + √2)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","commit":"","full_name":"CategoryTheory.InjectiveResolution.iso_inv_naturality","start":[245,0],"end":[252,75],"file_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasInjectiveResolutions C\nX Y : C\nf : X ⟶ Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\nφ : I.cocomplex ⟶ J.cocomplex\ncomm : I.ι.f 0 ≫ φ.f 0 = f ≫ J.ι.f 0\n⊢ I.iso.inv ≫ (injectiveResolutions C).map f = (HomotopyCategory.quotient C (ComplexShape.up ℕ)).map φ ≫ J.iso.inv","state_after":"no goals","tactic":"rw [← cancel_mono (J.iso).hom, Category.assoc, iso_hom_naturality f I J φ comm,\n    Iso.inv_hom_id_assoc, Category.assoc, Iso.inv_hom_id, Category.comp_id]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.InjectiveResolution.iso","def_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","def_pos":[230,4],"def_end_pos":[230,27]},{"full_name":"CategoryTheory.InjectiveResolution.iso_hom_naturality","def_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","def_pos":[236,6],"def_end_pos":[236,44]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]},{"full_name":"CategoryTheory.cancel_mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[263,8],"def_end_pos":[263,19]}]}]}
{"url":"Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean","commit":"","full_name":"groupCohomology.mem_twoCocycles_def","start":[277,0],"end":[282,67],"file_path":"Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean","tactics":[{"state_before":"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : G × G → CoeSort.coe A\n⊢ (dTwo A) f = 0 ↔ ∀ (g h j : G), (A.ρ g) (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0","state_after":"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : G × G → CoeSort.coe A\n⊢ (∀ (a : G × G × G), (dTwo A) f a = 0 a) ↔\n    ∀ (g h j : G), (A.ρ g) (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0","tactic":"rw [Function.funext_iff]","premises":[{"full_name":"Function.funext_iff","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[63,8],"def_end_pos":[63,27]}]},{"state_before":"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : G × G → CoeSort.coe A\n⊢ (∀ (a : G × G × G), (dTwo A) f a = 0 a) ↔\n    ∀ (g h j : G), (A.ρ g) (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0","state_after":"no goals","tactic":"simp only [dTwo_apply, Prod.mk.eta, Pi.zero_apply, Prod.forall]","premises":[{"full_name":"Pi.zero_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[59,2],"def_end_pos":[59,13]},{"full_name":"Prod.forall","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[28,8],"def_end_pos":[28,16]},{"full_name":"Prod.mk.eta","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[24,8],"def_end_pos":[24,14]},{"full_name":"groupCohomology.dTwo_apply","def_path":"Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean","def_pos":[121,2],"def_end_pos":[121,7]}]}]}
{"url":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","commit":"","full_name":"ContMDiffOn.sub","start":[157,0],"end":[160,47],"file_path":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁸ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nF : Type u_4\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\nJ : ModelWithCorners 𝕜 F F\nG : Type u_5\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : ChartedSpace H G\ninst✝¹¹ : Group G\ninst✝¹⁰ : LieGroup I G\nE' : Type u_6\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H' M\nE'' : Type u_9\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_10\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM' : Type u_11\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H'' M'\nn : ℕ∞\nf g : M → G\ns : Set M\nhf : ContMDiffOn I' I n f s\nhg : ContMDiffOn I' I n g s\n⊢ ContMDiffOn I' I n (fun x => f x / g x) s","state_after":"𝕜 : Type u_1\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁸ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nF : Type u_4\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\nJ : ModelWithCorners 𝕜 F F\nG : Type u_5\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : ChartedSpace H G\ninst✝¹¹ : Group G\ninst✝¹⁰ : LieGroup I G\nE' : Type u_6\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H' M\nE'' : Type u_9\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_10\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM' : Type u_11\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H'' M'\nn : ℕ∞\nf g : M → G\ns : Set M\nhf : ContMDiffOn I' I n f s\nhg : ContMDiffOn I' I n g s\n⊢ ContMDiffOn I' I n (fun x => f x * (g x)⁻¹) s","tactic":"simp_rw [div_eq_mul_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁸ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nF : Type u_4\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\nJ : ModelWithCorners 𝕜 F F\nG : Type u_5\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : ChartedSpace H G\ninst✝¹¹ : Group G\ninst✝¹⁰ : LieGroup I G\nE' : Type u_6\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H' M\nE'' : Type u_9\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_10\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM' : Type u_11\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H'' M'\nn : ℕ∞\nf g : M → G\ns : Set M\nhf : ContMDiffOn I' I n f s\nhg : ContMDiffOn I' I n g s\n⊢ ContMDiffOn I' I n (fun x => f x * (g x)⁻¹) s","state_after":"no goals","tactic":"exact hf.mul hg.inv","premises":[{"full_name":"ContMDiffOn.inv","def_path":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","def_pos":[120,8],"def_end_pos":[120,23]},{"full_name":"ContMDiffOn.mul","def_path":"Mathlib/Geometry/Manifold/Algebra/Monoid.lean","def_pos":[100,8],"def_end_pos":[100,23]}]}]}
{"url":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","commit":"","full_name":"IsPrimitiveRoot.norm_pow_sub_one_of_prime_pow_ne_two","start":[379,0],"end":[443,62],"file_path":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","tactics":[{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\n⊢ (Algebra.norm K) (ζ ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑p ^ (k - s + 1)) K)\n⊢ (Algebra.norm K) (ζ ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s","tactic":"have hirr₁ : Irreducible (cyclotomic ((p : ℕ) ^ (k - s + 1)) K) :=\n    cyclotomic_irreducible_pow_of_irreducible_pow hpri.1 (by simp) hirr","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Irreducible","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[174,10],"def_end_pos":[174,21]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Polynomial.cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[231,4],"def_end_pos":[231,14]},{"full_name":"Polynomial.cyclotomic_irreducible_pow_of_irreducible_pow","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean","def_pos":[96,8],"def_end_pos":[96,53]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑p ^ (k - s + 1)) K)\n⊢ (Algebra.norm K) (ζ ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\n⊢ (Algebra.norm K) (ζ ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s","tactic":"rw [← PNat.pow_coe] at hirr₁","premises":[{"full_name":"PNat.pow_coe","def_path":"Mathlib/Data/PNat/Basic.lean","def_pos":[226,8],"def_end_pos":[226,15]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\n⊢ (Algebra.norm K) (ζ ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","tactic":"set η := ζ ^ (p : ℕ) ^ s - 1","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","tactic":"let η₁ : K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η","premises":[{"full_name":"IntermediateField.AdjoinSimple.gen","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[533,4],"def_end_pos":[533,20]},{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nhη : IsPrimitiveRoot (η + 1) (↑p ^ (k + 1 - s))\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","tactic":"have hη : IsPrimitiveRoot (η + 1) ((p : ℕ) ^ (k + 1 - s)) := by\n    rw [sub_add_cancel]\n    refine IsPrimitiveRoot.pow (p ^ (k + 1)).pos hζ ?_\n    rw [PNat.pow_coe, ← pow_add, add_comm s, Nat.sub_add_cancel (le_trans hs (Nat.le_succ k))]","premises":[{"full_name":"IsPrimitiveRoot","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[259,10],"def_end_pos":[259,25]},{"full_name":"IsPrimitiveRoot.pow","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[443,8],"def_end_pos":[443,11]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1684,8],"def_end_pos":[1684,19]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"PNat.pos","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[129,8],"def_end_pos":[129,11]},{"full_name":"PNat.pow_coe","def_path":"Mathlib/Data/PNat/Basic.lean","def_pos":[226,8],"def_end_pos":[226,15]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]},{"full_name":"sub_add_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[691,2],"def_end_pos":[691,13]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nhη : IsPrimitiveRoot (η + 1) (↑p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","tactic":"replace hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1)) := by\n    apply coe_submonoidClass_iff.1\n    convert hη using 1\n    rw [Nat.sub_add_comm hs, pow_coe]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsPrimitiveRoot","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[259,10],"def_end_pos":[259,25]},{"full_name":"IsPrimitiveRoot.coe_submonoidClass_iff","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[352,8],"def_end_pos":[352,30]},{"full_name":"Nat.sub_add_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[125,18],"def_end_pos":[125,30]},{"full_name":"PNat.pow_coe","def_path":"Mathlib/Data/PNat/Basic.lean","def_pos":[226,8],"def_end_pos":[226,15]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis : FiniteDimensional K L\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","tactic":"haveI := IsCyclotomicExtension.finiteDimensional {p ^ (k + 1)} K L","premises":[{"full_name":"IsCyclotomicExtension.finiteDimensional","def_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","def_pos":[333,8],"def_end_pos":[333,25]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis : FiniteDimensional K L\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","tactic":"haveI := IsCyclotomicExtension.isGalois (p ^ (k + 1)) K L","premises":[{"full_name":"IsCyclotomicExtension.isGalois","def_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","def_pos":[454,8],"def_end_pos":[454,16]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ (Algebra.norm K) η = ↑↑p ^ ↑p ^ s","state_after":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","tactic":"rw [norm_eq_norm_adjoin K]","premises":[{"full_name":"Algebra.norm_eq_norm_adjoin","def_path":"Mathlib/RingTheory/Norm/Basic.lean","def_pos":[134,8],"def_end_pos":[134,27]}]},{"state_before":"p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","state_after":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) (η₁ + 1 - 1) = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s\n\ncase refine_1\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ IsPrimePow ↑(p ^ (k - s + 1))","tactic":"have H := hη.sub_one_norm_isPrimePow ?_ hirr₁ htwo","premises":[{"full_name":"IsPrimitiveRoot.sub_one_norm_isPrimePow","def_path":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","def_pos":[349,8],"def_end_pos":[349,31]}]},{"state_before":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) (η₁ + 1 - 1) = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s\n\ncase refine_1\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ IsPrimePow ↑(p ^ (k - s + 1))","state_after":"case refine_1\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ IsPrimePow ↑(p ^ (k - s + 1))\n\ncase refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) (η₁ + 1 - 1) = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","tactic":"swap","premises":[]},{"state_before":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) (η₁ + 1 - 1) = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","state_after":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","tactic":"rw [add_sub_cancel_right] at H","premises":[{"full_name":"add_sub_cancel_right","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[698,2],"def_end_pos":[698,13]}]},{"state_before":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","state_after":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ ↑(↑(p ^ (k - s + 1))).minFac ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","tactic":"rw [H]","premises":[]},{"state_before":"case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ ↑(↑(p ^ (k - s + 1))).minFac ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s","state_after":"case refine_2.e_a.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (↑(p ^ (k - s + 1))).minFac = ↑p\n\ncase refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","tactic":"congr","premises":[]},{"state_before":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","state_after":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : FiniteDimensional.finrank K ↥K⟮η⟯ * FiniteDimensional.finrank (↥K⟮η⟯) L = FiniteDimensional.finrank K L\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","tactic":"have := FiniteDimensional.finrank_mul_finrank K K⟮η⟯ L","premises":[{"full_name":"FiniteDimensional.finrank_mul_finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Free.lean","def_pos":[59,8],"def_end_pos":[59,45]},{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : FiniteDimensional.finrank K ↥K⟮η⟯ * FiniteDimensional.finrank (↥K⟮η⟯) L = FiniteDimensional.finrank K L\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","state_after":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : (↑p - 1) * (↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L) = (↑p - 1) * ↑p ^ (k.succ - 1)\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","tactic":"rw [IsCyclotomicExtension.finrank L hirr, IsCyclotomicExtension.finrank K⟮η⟯ hirr₁,\n    PNat.pow_coe, PNat.pow_coe, Nat.totient_prime_pow hpri.out (k - s).succ_pos,\n    Nat.totient_prime_pow hpri.out k.succ_pos, mul_comm _ ((p : ℕ) - 1), mul_assoc,\n    mul_comm ((p : ℕ) ^ (k.succ - 1))] at this","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"IsCyclotomicExtension.finrank","def_path":"Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean","def_pos":[173,8],"def_end_pos":[173,15]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Nat.succ_pos","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1693,8],"def_end_pos":[1693,20]},{"full_name":"Nat.totient_prime_pow","def_path":"Mathlib/Data/Nat/Totient.lean","def_pos":[199,8],"def_end_pos":[199,25]},{"full_name":"PNat.pow_coe","def_path":"Mathlib/Data/PNat/Basic.lean","def_pos":[226,8],"def_end_pos":[226,15]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : (↑p - 1) * (↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L) = (↑p - 1) * ↑p ^ (k.succ - 1)\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","state_after":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","tactic":"replace this := mul_left_cancel₀ (tsub_pos_iff_lt.2 hpri.out.one_lt).ne' this","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Nat.Prime.one_lt","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"mul_left_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[48,8],"def_end_pos":[48,24]},{"full_name":"tsub_pos_iff_lt","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[356,8],"def_end_pos":[356,23]}]},{"state_before":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","state_after":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)\nHex : k.succ - 1 = (k - s).succ - 1 + s\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","tactic":"have Hex : k.succ - 1 = (k - s).succ - 1 + s := by\n    simp only [Nat.succ_sub_succ_eq_sub, tsub_zero]\n    exact (Nat.sub_add_cancel hs).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Nat.succ_sub_succ_eq_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[276,8],"def_end_pos":[276,28]},{"full_name":"tsub_zero","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[384,8],"def_end_pos":[384,17]}]},{"state_before":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)\nHex : k.succ - 1 = (k - s).succ - 1 + s\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","state_after":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ ((k - s).succ - 1) * ↑p ^ s\nHex : k.succ - 1 = (k - s).succ - 1 + s\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","tactic":"rw [Hex, pow_add] at this","premises":[{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]}]},{"state_before":"case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ ((k - s).succ - 1) * ↑p ^ s\nHex : k.succ - 1 = (k - s).succ - 1 + s\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s","state_after":"no goals","tactic":"exact mul_left_cancel₀ (pow_ne_zero _ hpri.out.ne_zero) this","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Nat.Prime.ne_zero","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[48,8],"def_end_pos":[48,21]},{"full_name":"mul_left_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[48,8],"def_end_pos":[48,24]},{"full_name":"pow_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[172,6],"def_end_pos":[172,17]}]}]}
{"url":"Mathlib/Algebra/Order/Group/Int.lean","commit":"","full_name":"Int.emod_lt","start":[99,0],"end":[100,55],"file_path":"Mathlib/Algebra/Order/Group/Int.lean","tactics":[{"state_before":"a b : ℤ\nH : b ≠ 0\n⊢ a % b < |b|","state_after":"a b : ℤ\nH : b ≠ 0\n⊢ a % |b| < |b|","tactic":"rw [← emod_abs]","premises":[{"full_name":"Int.emod_abs","def_path":"Mathlib/Algebra/Order/Group/Int.lean","def_pos":[96,8],"def_end_pos":[96,16]}]},{"state_before":"a b : ℤ\nH : b ≠ 0\n⊢ a % |b| < |b|","state_after":"no goals","tactic":"exact emod_lt_of_pos _ (abs_pos.2 H)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.emod_lt_of_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[432,8],"def_end_pos":[432,22]},{"full_name":"abs_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[208,29],"def_end_pos":[208,36]}]}]}
{"url":"Mathlib/Order/Category/DistLat.lean","commit":"","full_name":"DistLat.Iso.mk_inv_toSupHom_toFun","start":[58,0],"end":[69,30],"file_path":"Mathlib/Order/Category/DistLat.lean","tactics":[{"state_before":"α β : DistLat\ne : ↑α ≃o ↑β\n⊢ { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } = 𝟙 α","state_after":"case w\nα β : DistLat\ne : ↑α ≃o ↑β\nx✝ : (forget DistLat).obj α\n⊢ ({ toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }) x✝ = (𝟙 α) x✝","tactic":"ext","premises":[]},{"state_before":"case w\nα β : DistLat\ne : ↑α ≃o ↑β\nx✝ : (forget DistLat).obj α\n⊢ ({ toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }) x✝ = (𝟙 α) x✝","state_after":"no goals","tactic":"exact e.symm_apply_apply _","premises":[{"full_name":"OrderIso.symm_apply_apply","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[761,8],"def_end_pos":[761,24]}]},{"state_before":"α β : DistLat\ne : ↑α ≃o ↑β\n⊢ { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } = 𝟙 β","state_after":"case w\nα β : DistLat\ne : ↑α ≃o ↑β\nx✝ : (forget DistLat).obj β\n⊢ ({ toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }) x✝ = (𝟙 β) x✝","tactic":"ext","premises":[]},{"state_before":"case w\nα β : DistLat\ne : ↑α ≃o ↑β\nx✝ : (forget DistLat).obj β\n⊢ ({ toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }) x✝ = (𝟙 β) x✝","state_after":"no goals","tactic":"exact e.apply_symm_apply _","premises":[{"full_name":"OrderIso.apply_symm_apply","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[757,8],"def_end_pos":[757,24]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.leftUnshift_add","start":[264,0],"end":[268,22],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ γ₁✝ γ₂✝ : Cochain K L n✝\nn' a : ℤ\nγ₁ γ₂ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\n⊢ (γ₁ + γ₂).leftUnshift n hn = γ₁.leftUnshift n hn + γ₂.leftUnshift n hn","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ γ₁✝ γ₂✝ : Cochain K L n✝\nn' a : ℤ\nγ₁ γ₂ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\n⊢ (leftShiftAddEquiv K L n a n' hn).symm (γ₁ + γ₂) = γ₁.leftUnshift n hn + γ₂.leftUnshift n hn","tactic":"change (leftShiftAddEquiv K L n a n' hn).symm (γ₁ + γ₂) = _","premises":[{"full_name":"AddEquiv.symm","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[251,2],"def_end_pos":[251,13]},{"full_name":"CochainComplex.HomComplex.Cochain.leftShiftAddEquiv","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[181,4],"def_end_pos":[181,21]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ γ₁✝ γ₂✝ : Cochain K L n✝\nn' a : ℤ\nγ₁ γ₂ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn : n + a = n'\n⊢ (leftShiftAddEquiv K L n a n' hn).symm (γ₁ + γ₂) = γ₁.leftUnshift n hn + γ₂.leftUnshift n hn","state_after":"no goals","tactic":"apply _root_.map_add","premises":[{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean","commit":"","full_name":"CategoryTheory.Limits.colimitLimitToLimitColimit_injective","start":[67,0],"end":[140,31],"file_path":"Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean","tactics":[{"state_before":"J : Type u₁\nK : Type u₂\ninst✝⁴ : Category.{v₁, u₁} J\ninst✝³ : Category.{v₂, u₂} K\ninst✝² : Small.{v, u₂} K\nF : J × K ⥤ Type v\ninst✝¹ : IsFiltered K\ninst✝ : Finite J\n⊢ Function.Injective (colimitLimitToLimitColimit F)","state_after":"no goals","tactic":"classical\n    cases nonempty_fintype J\n    -- Suppose we have two terms `x y` in the colimit (over `K`) of the limits (over `J`),\n    -- and that these have the same image under `colimitLimitToLimitColimit F`.\n    intro x y h\n    -- These elements of the colimit have representatives somewhere:\n    obtain ⟨kx, x, rfl⟩ := jointly_surjective' x\n    obtain ⟨ky, y, rfl⟩ := jointly_surjective' y\n    dsimp at x y\n    -- Since the images of `x` and `y` are equal in a limit, they are equal componentwise\n    -- (indexed by `j : J`),\n    replace h := fun j => congr_arg (limit.π (curry.obj F ⋙ colim) j) h\n    -- and they are equations in a filtered colimit,\n    -- so for each `j` we have some place `k j` to the right of both `kx` and `ky`\n    simp? [colimit_eq_iff] at h says\n      simp only [Functor.comp_obj, colim_obj, ι_colimitLimitToLimitColimit_π_apply,\n        colimit_eq_iff, curry_obj_obj_obj, curry_obj_obj_map] at h\n    let k j := (h j).choose\n    let f : ∀ j, kx ⟶ k j := fun j => (h j).choose_spec.choose\n    let g : ∀ j, ky ⟶ k j := fun j => (h j).choose_spec.choose_spec.choose\n    -- where the images of the components of the representatives become equal:\n    have w :\n      ∀ j, F.map ((𝟙 j, f j) :\n        (j, kx) ⟶ (j, k j)) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =\n          F.map ((𝟙 j, g j) : (j, ky) ⟶ (j, k j))\n            (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y) :=\n      fun j => (h j).choose_spec.choose_spec.choose_spec\n    -- We now use that `K` is filtered, picking some point to the right of all these\n    -- morphisms `f j` and `g j`.\n    let O : Finset K := Finset.univ.image k ∪ {kx, ky}\n    have kxO : kx ∈ O := Finset.mem_union.mpr (Or.inr (by simp))\n    have kyO : ky ∈ O := Finset.mem_union.mpr (Or.inr (by simp))\n    have kjO : ∀ j, k j ∈ O := fun j => Finset.mem_union.mpr (Or.inl (by simp))\n    let H : Finset (Σ' (X Y : K) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=\n      (Finset.univ.image fun j : J =>\n          ⟨kx, k j, kxO, Finset.mem_union.mpr (Or.inl (by simp)), f j⟩) ∪\n        Finset.univ.image fun j : J => ⟨ky, k j, kyO, Finset.mem_union.mpr (Or.inl (by simp)), g j⟩\n    obtain ⟨S, T, W⟩ := IsFiltered.sup_exists O H\n    have fH : ∀ j, (⟨kx, k j, kxO, kjO j, f j⟩ : Σ' (X Y : K) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) ∈ H :=\n      fun j =>\n      Finset.mem_union.mpr\n        (Or.inl\n          (by\n            simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n              Finset.mem_image, heq_iff_eq]\n            refine ⟨j, ?_⟩\n            simp only [heq_iff_eq] ))\n    have gH :\n      ∀ j, (⟨ky, k j, kyO, kjO j, g j⟩ : Σ' (X Y : K) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) ∈ H :=\n      fun j =>\n      Finset.mem_union.mpr\n        (Or.inr\n          (by\n            simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n              Finset.mem_image, heq_iff_eq]\n            refine ⟨j, ?_⟩\n            simp only [heq_iff_eq]))\n    -- Our goal is now an equation between equivalence classes of representatives of a colimit,\n    -- and so it suffices to show those representative become equal somewhere, in particular at `S`.\n    apply colimit_sound' (T kxO) (T kyO)\n    -- We can check if two elements of a limit (in `Type`)\n    -- are equal by comparing them componentwise.\n    ext j\n    -- Now it's just a calculation using `W` and `w`.\n    simp only [Functor.comp_map, Limit.map_π_apply, curry_obj_map_app, swap_map]\n    rw [← W _ _ (fH j), ← W _ _ (gH j)]\n    -- Porting note(#10745): had to add `Limit.map_π_apply`\n    -- (which was un-tagged simp since \"simp can prove it\")\n    simp [Limit.map_π_apply, w]","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Functor.comp_map","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[109,8],"def_end_pos":[109,16]},{"full_name":"CategoryTheory.Functor.comp_obj","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[99,8],"def_end_pos":[99,11]},{"full_name":"CategoryTheory.IsFiltered.sup_exists","def_path":"Mathlib/CategoryTheory/Filtered/Basic.lean","def_pos":[240,8],"def_end_pos":[240,18]},{"full_name":"CategoryTheory.Limits.Types.FilteredColimit.colimit_eq_iff","def_path":"Mathlib/CategoryTheory/Limits/TypesFiltered.lean","def_pos":[121,8],"def_end_pos":[121,22]},{"full_name":"CategoryTheory.Limits.Types.Limit.map_π_apply","def_path":"Mathlib/CategoryTheory/Limits/Types.lean","def_pos":[284,8],"def_end_pos":[284,25]},{"full_name":"CategoryTheory.Limits.Types.colimit_sound'","def_path":"Mathlib/CategoryTheory/Limits/Types.lean","def_pos":[545,8],"def_end_pos":[545,22]},{"full_name":"CategoryTheory.Limits.Types.jointly_surjective'","def_path":"Mathlib/CategoryTheory/Limits/Types.lean","def_pos":[573,8],"def_end_pos":[573,27]},{"full_name":"CategoryTheory.Limits.colim","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[977,4],"def_end_pos":[977,9]},{"full_name":"CategoryTheory.Limits.colim_obj","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[976,2],"def_end_pos":[976,7]},{"full_name":"CategoryTheory.Limits.limit.π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[139,4],"def_end_pos":[139,11]},{"full_name":"CategoryTheory.Limits.ι_colimitLimitToLimitColimit_π_apply","def_path":"Mathlib/CategoryTheory/Limits/ColimitLimit.lean","def_pos":[91,8],"def_end_pos":[91,44]},{"full_name":"CategoryTheory.Prod.swap","def_path":"Mathlib/CategoryTheory/Products/Basic.lean","def_pos":[136,4],"def_end_pos":[136,8]},{"full_name":"CategoryTheory.Prod.swap_map","def_path":"Mathlib/CategoryTheory/Products/Basic.lean","def_pos":[135,2],"def_end_pos":[135,7]},{"full_name":"CategoryTheory.curry","def_path":"Mathlib/CategoryTheory/Functor/Currying.lean","def_pos":[63,4],"def_end_pos":[63,9]},{"full_name":"CategoryTheory.curry_obj_map_app","def_path":"Mathlib/CategoryTheory/Functor/Currying.lean","def_pos":[62,33],"def_end_pos":[62,44]},{"full_name":"CategoryTheory.curry_obj_obj_map","def_path":"Mathlib/CategoryTheory/Functor/Currying.lean","def_pos":[62,21],"def_end_pos":[62,32]},{"full_name":"CategoryTheory.curry_obj_obj_obj","def_path":"Mathlib/CategoryTheory/Functor/Currying.lean","def_pos":[62,9],"def_end_pos":[62,20]},{"full_name":"Exists.choose","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[174,31],"def_end_pos":[174,44]},{"full_name":"Exists.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[177,8],"def_end_pos":[177,26]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[289,4],"def_end_pos":[289,9]},{"full_name":"Finset.mem_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[303,8],"def_end_pos":[303,17]},{"full_name":"Finset.mem_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1191,8],"def_end_pos":[1191,17]},{"full_name":"Finset.mem_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"PSigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[195,10],"def_end_pos":[195,16]},{"full_name":"PSigma.mk","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[199,2],"def_end_pos":[199,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]},{"full_name":"exists_prop_of_true","def_path":"Mathlib/Logic/Basic.lean","def_pos":[641,8],"def_end_pos":[641,27]},{"full_name":"heq_iff_eq","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[37,8],"def_end_pos":[37,18]},{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]},{"full_name":"true_and_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[94,8],"def_end_pos":[94,20]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean","commit":"","full_name":"CategoryTheory.Limits.Cotrident.app_one","start":[188,0],"end":[190,46],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean","tactics":[{"state_before":"J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ns : Cotrident f\nj : J\n⊢ f j ≫ s.ι.app one = s.ι.app zero","state_after":"no goals","tactic":"rw [← s.w (line j), parallelFamily_map_left]","premises":[{"full_name":"CategoryTheory.Limits.Cocone.w","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[166,8],"def_end_pos":[166,16]},{"full_name":"CategoryTheory.Limits.WalkingParallelFamily.Hom.line","def_path":"Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean","def_pos":[75,4],"def_end_pos":[75,8]},{"full_name":"CategoryTheory.Limits.parallelFamily_map_left","def_path":"Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean","def_pos":[129,8],"def_end_pos":[129,31]}]}]}
{"url":"Mathlib/Data/Matrix/Reflection.lean","commit":"","full_name":"Matrix.mulᵣ_eq","start":[136,0],"end":[151,5],"file_path":"Mathlib/Data/Matrix/Reflection.lean","tactics":[{"state_before":"l m n : ℕ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\nA : Matrix (Fin l) (Fin m) α\nB : Matrix (Fin m) (Fin n) α\n⊢ A.mulᵣ B = A * B","state_after":"l m n : ℕ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\nA : Matrix (Fin l) (Fin m) α\nB : Matrix (Fin m) (Fin n) α\n⊢ (of fun x x_1 => A x ⬝ᵥ of (fun x y => B y x) x_1) = A * B","tactic":"simp [mulᵣ, Function.comp, Matrix.transpose]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Matrix.mulᵣ","def_path":"Mathlib/Data/Matrix/Reflection.lean","def_pos":[132,4],"def_end_pos":[132,8]},{"full_name":"Matrix.transpose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[140,4],"def_end_pos":[140,13]}]},{"state_before":"l m n : ℕ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\nA : Matrix (Fin l) (Fin m) α\nB : Matrix (Fin m) (Fin n) α\n⊢ (of fun x x_1 => A x ⬝ᵥ of (fun x y => B y x) x_1) = A * B","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/ModelTheory/FinitelyGenerated.lean","commit":"","full_name":"FirstOrder.Language.Structure.FG.range","start":[187,0],"end":[189,26],"file_path":"Mathlib/ModelTheory/FinitelyGenerated.lean","tactics":[{"state_before":"L : Language\nM : Type u_1\ninst✝¹ : L.Structure M\nN : Type u_2\ninst✝ : L.Structure N\nh : FG L M\nf : M →[L] N\n⊢ f.range.FG","state_after":"L : Language\nM : Type u_1\ninst✝¹ : L.Structure M\nN : Type u_2\ninst✝ : L.Structure N\nh : FG L M\nf : M →[L] N\n⊢ (map f ⊤).FG","tactic":"rw [Hom.range_eq_map]","premises":[{"full_name":"FirstOrder.Language.Hom.range_eq_map","def_path":"Mathlib/ModelTheory/Substructures.lean","def_pos":[715,8],"def_end_pos":[715,20]}]},{"state_before":"L : Language\nM : Type u_1\ninst✝¹ : L.Structure M\nN : Type u_2\ninst✝ : L.Structure N\nh : FG L M\nf : M →[L] N\n⊢ (map f ⊤).FG","state_after":"no goals","tactic":"exact (fg_def.1 h).map f","premises":[{"full_name":"FirstOrder.Language.Structure.fg_def","def_path":"Mathlib/ModelTheory/FinitelyGenerated.lean","def_pos":[180,8],"def_end_pos":[180,14]},{"full_name":"FirstOrder.Language.Substructure.FG.map","def_path":"Mathlib/ModelTheory/FinitelyGenerated.lean","def_pos":[72,8],"def_end_pos":[72,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]}]}
{"url":"Mathlib/Order/LatticeIntervals.lean","commit":"","full_name":"Set.Iic.eq_top_iff","start":[101,0],"end":[103,24],"file_path":"Mathlib/Order/LatticeIntervals.lean","tactics":[{"state_before":"α : Type u_1\na : α\ninst✝ : Preorder α\nx : ↑(Iic a)\n⊢ x = ⊤ ↔ ↑x = a","state_after":"no goals","tactic":"simp [Subtype.ext_iff]","premises":[{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]}]}
{"url":"Mathlib/Data/List/Indexes.lean","commit":"","full_name":"List.foldrIdxM_eq_foldrM_enum","start":[344,0],"end":[347,36],"file_path":"Mathlib/Data/List/Indexes.lean","tactics":[{"state_before":"α : Type u\nβ✝ : Type v\nm : Type u → Type v\ninst✝¹ : Monad m\nβ : Type u\nf : ℕ → α → β → m β\nb : β\nas : List α\ninst✝ : LawfulMonad m\n⊢ foldrIdxM f b as = foldrM (uncurry f) b as.enum","state_after":"no goals","tactic":"simp (config := { unfoldPartialApp := true }) only [foldrIdxM, foldrM_eq_foldr,\n    foldrIdx_eq_foldr_enum, uncurry]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Function.uncurry","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[189,4],"def_end_pos":[189,11]},{"full_name":"List.foldrIdxM","def_path":"Mathlib/Data/List/Defs.lean","def_pos":[94,4],"def_end_pos":[94,13]},{"full_name":"List.foldrIdx_eq_foldr_enum","def_path":"Mathlib/Data/List/Indexes.lean","def_pos":[232,8],"def_end_pos":[232,30]},{"full_name":"List.foldrM_eq_foldr","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[1628,8],"def_end_pos":[1628,23]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/ZPowers.lean","commit":"","full_name":"AddSubgroup.intCast_mem_zmultiples_one","start":[127,0],"end":[129,36],"file_path":"Mathlib/Algebra/Group/Subgroup/ZPowers.lean","tactics":[{"state_before":"G : Type u_1\ninst✝³ : Group G\nA : Type u_2\ninst✝² : AddGroup A\nN : Type u_3\ninst✝¹ : Group N\nR : Type u_4\ninst✝ : Ring R\nr : R\nk : ℤ\n⊢ (fun x => x • 1) k = ↑k","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/RingTheory/LocalRing/Basic.lean","commit":"","full_name":"LocalRing.of_surjective'","start":[91,0],"end":[98,23],"file_path":"Mathlib/RingTheory/LocalRing/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\n⊢ ∀ (a : S), IsUnit a ∨ IsUnit (1 - a)","state_after":"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\nb : S\n⊢ IsUnit b ∨ IsUnit (1 - b)","tactic":"intro b","premises":[]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\nb : S\n⊢ IsUnit b ∨ IsUnit (1 - b)","state_after":"case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (f a) ∨ IsUnit (1 - f a)","tactic":"obtain ⟨a, rfl⟩ := hf b","premises":[]},{"state_before":"case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (f a) ∨ IsUnit (1 - f a)","state_after":"case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (1 - a) → IsUnit (1 - f a)","tactic":"apply (isUnit_or_isUnit_one_sub_self a).imp <| RingHom.isUnit_map _","premises":[{"full_name":"LocalRing.isUnit_or_isUnit_one_sub_self","def_path":"Mathlib/RingTheory/LocalRing/Basic.lean","def_pos":[82,8],"def_end_pos":[82,37]},{"full_name":"Or.imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[172,8],"def_end_pos":[172,14]},{"full_name":"RingHom.isUnit_map","def_path":"Mathlib/Algebra/Ring/Hom/Basic.lean","def_pos":[45,8],"def_end_pos":[45,18]}]},{"state_before":"case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (1 - a) → IsUnit (1 - f a)","state_after":"case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (1 - a) → IsUnit (f (1 - a))","tactic":"rw [← f.map_one, ← f.map_sub]","premises":[{"full_name":"RingHom.map_one","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[476,18],"def_end_pos":[476,25]},{"full_name":"RingHom.map_sub","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[526,18],"def_end_pos":[526,25]}]},{"state_before":"case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (1 - a) → IsUnit (f (1 - a))","state_after":"no goals","tactic":"apply f.isUnit_map","premises":[{"full_name":"RingHom.isUnit_map","def_path":"Mathlib/Algebra/Ring/Hom/Basic.lean","def_pos":[45,8],"def_end_pos":[45,18]}]}]}
{"url":"Mathlib/Algebra/GroupWithZero/Center.lean","commit":"","full_name":"Set.div_mem_centralizer₀","start":[54,0],"end":[56,84],"file_path":"Mathlib/Algebra/GroupWithZero/Center.lean","tactics":[{"state_before":"M₀ : Type u_1\nG₀ : Type u_2\ninst✝ : GroupWithZero G₀\ns : Set G₀\na b : G₀\nha : a ∈ s.centralizer\nhb : b ∈ s.centralizer\n⊢ a / b ∈ s.centralizer","state_after":"no goals","tactic":"simpa only [div_eq_mul_inv] using mul_mem_centralizer ha (inv_mem_centralizer₀ hb)","premises":[{"full_name":"Set.inv_mem_centralizer₀","def_path":"Mathlib/Algebra/GroupWithZero/Center.lean","def_pos":[47,14],"def_end_pos":[47,34]},{"full_name":"Set.mul_mem_centralizer","def_path":"Mathlib/Algebra/Group/Center.lean","def_pos":[176,6],"def_end_pos":[176,25]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/ENat.lean","commit":"","full_name":"Cardinal.ofENat_toENat_eq_self","start":[234,0],"end":[237,68],"file_path":"Mathlib/SetTheory/Cardinal/ENat.lean","tactics":[{"state_before":"a : Cardinal.{u_1}\n⊢ ↑(toENat a) = a ↔ a ≤ ℵ₀","state_after":"a : Cardinal.{u_1}\n⊢ (∃ a_1, a = ↑a_1) ↔ a ≤ ℵ₀","tactic":"rw [eq_comm, ← enat_gc.exists_eq_l]","premises":[{"full_name":"Cardinal.enat_gc","def_path":"Mathlib/SetTheory/Cardinal/ENat.lean","def_pos":[216,6],"def_end_pos":[216,13]},{"full_name":"GaloisConnection.exists_eq_l","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[186,8],"def_end_pos":[186,19]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]},{"state_before":"a : Cardinal.{u_1}\n⊢ (∃ a_1, a = ↑a_1) ↔ a ≤ ℵ₀","state_after":"no goals","tactic":"simpa only [mem_range, eq_comm] using Set.ext_iff.1 range_ofENat a","premises":[{"full_name":"Cardinal.range_ofENat","def_path":"Mathlib/SetTheory/Cardinal/ENat.lean","def_pos":[143,6],"def_end_pos":[143,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.ext_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[190,18],"def_end_pos":[190,25]},{"full_name":"Set.mem_range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[146,16],"def_end_pos":[146,25]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/Data/Seq/Parallel.lean","commit":"","full_name":"Computation.parallel_congr_left","start":[348,0],"end":[369,28],"file_path":"Mathlib/Data/Seq/Parallel.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel S\n⊢ a' ∈ parallel T","state_after":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel S\naa : a = a'\n⊢ a' ∈ parallel T","tactic":"have aa := parallel_promises h1 h","premises":[{"full_name":"Computation.parallel_promises","def_path":"Mathlib/Data/Seq/Parallel.lean","def_pos":[327,8],"def_end_pos":[327,25]}]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel S\naa : a = a'\n⊢ a' ∈ parallel T","state_after":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel S\naa : a = a'\n⊢ a ∈ parallel T","tactic":"rw [← aa]","premises":[]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel S\naa : a = a'\n⊢ a ∈ parallel T","state_after":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a ∈ parallel S\naa : a = a'\n⊢ a ∈ parallel T","tactic":"rw [← aa] at h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a ∈ parallel S\naa : a = a'\n⊢ a ∈ parallel T","state_after":"no goals","tactic":"exact\n      let ⟨s, sS, as⟩ := exists_of_mem_parallel h\n      let ⟨t, tT, st⟩ := WSeq.exists_of_liftRel_left H sS\n      let aT := (st _).1 as\n      mem_parallel h2 tT aT","premises":[{"full_name":"Computation.exists_of_mem_parallel","def_path":"Mathlib/Data/Seq/Parallel.lean","def_pos":[181,8],"def_end_pos":[181,30]},{"full_name":"Computation.mem_parallel","def_path":"Mathlib/Data/Seq/Parallel.lean","def_pos":[332,8],"def_end_pos":[332,20]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Stream'.WSeq.exists_of_liftRel_left","def_path":"Mathlib/Data/Seq/WSeq.lean","def_pos":[916,8],"def_end_pos":[916,30]}]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel T\n⊢ a' ∈ parallel S","state_after":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel T\naa : a = a'\n⊢ a' ∈ parallel S","tactic":"have aa := parallel_promises h2 h","premises":[{"full_name":"Computation.parallel_promises","def_path":"Mathlib/Data/Seq/Parallel.lean","def_pos":[327,8],"def_end_pos":[327,25]}]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel T\naa : a = a'\n⊢ a' ∈ parallel S","state_after":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel T\naa : a = a'\n⊢ a ∈ parallel S","tactic":"rw [← aa]","premises":[]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel T\naa : a = a'\n⊢ a ∈ parallel S","state_after":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a ∈ parallel T\naa : a = a'\n⊢ a ∈ parallel S","tactic":"rw [← aa] at h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a ∈ parallel T\naa : a = a'\n⊢ a ∈ parallel S","state_after":"no goals","tactic":"exact\n      let ⟨s, sS, as⟩ := exists_of_mem_parallel h\n      let ⟨t, tT, st⟩ := WSeq.exists_of_liftRel_right H sS\n      let aT := (st _).2 as\n      mem_parallel h1 tT aT","premises":[{"full_name":"Computation.exists_of_mem_parallel","def_path":"Mathlib/Data/Seq/Parallel.lean","def_pos":[181,8],"def_end_pos":[181,30]},{"full_name":"Computation.mem_parallel","def_path":"Mathlib/Data/Seq/Parallel.lean","def_pos":[332,8],"def_end_pos":[332,20]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Stream'.WSeq.exists_of_liftRel_right","def_path":"Mathlib/Data/Seq/WSeq.lean","def_pos":[926,8],"def_end_pos":[926,31]}]}]}
{"url":"Mathlib/Data/Multiset/Bind.lean","commit":"","full_name":"Multiset.coe_product","start":[239,0],"end":[244,6],"file_path":"Mathlib/Data/Multiset/Bind.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\nb : β\ns : Multiset α\nt : Multiset β\nl₁ : List α\nl₂ : List β\n⊢ ↑l₁ ×ˢ ↑l₂ = ↑(l₁ ×ˢ l₂)","state_after":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\nb : β\ns : Multiset α\nt : Multiset β\nl₁ : List α\nl₂ : List β\n⊢ (↑l₁).product ↑l₂ = ↑(l₁.product l₂)","tactic":"dsimp only [SProd.sprod]","premises":[{"full_name":"SProd.sprod","def_path":"Mathlib/Data/SProd.lean","def_pos":[29,2],"def_end_pos":[29,7]}]},{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\nb : β\ns : Multiset α\nt : Multiset β\nl₁ : List α\nl₂ : List β\n⊢ (↑l₁).product ↑l₂ = ↑(l₁.product l₂)","state_after":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\nb : β\ns : Multiset α\nt : Multiset β\nl₁ : List α\nl₂ : List β\n⊢ ((↑l₁).bind fun a => map (Prod.mk a) ↑l₂) = (↑l₁).bind fun a => ↑(List.map (Prod.mk a) l₂)","tactic":"rw [product, List.product, ← coe_bind]","premises":[{"full_name":"List.product","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[712,4],"def_end_pos":[712,11]},{"full_name":"Multiset.coe_bind","def_path":"Mathlib/Data/Multiset/Bind.lean","def_pos":[103,8],"def_end_pos":[103,16]},{"full_name":"Multiset.product","def_path":"Mathlib/Data/Multiset/Bind.lean","def_pos":[233,4],"def_end_pos":[233,11]}]},{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\nb : β\ns : Multiset α\nt : Multiset β\nl₁ : List α\nl₂ : List β\n⊢ ((↑l₁).bind fun a => map (Prod.mk a) ↑l₂) = (↑l₁).bind fun a => ↑(List.map (Prod.mk a) l₂)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/RingTheory/Multiplicity.lean","commit":"","full_name":"multiplicity.unique'","start":[120,0],"end":[122,68],"file_path":"Mathlib/RingTheory/Multiplicity.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\n⊢ k = (multiplicity a b).get ⋯","state_after":"no goals","tactic":"rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]","premises":[{"full_name":"PartENat.natCast_get","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[166,8],"def_end_pos":[166,19]},{"full_name":"PartENat.natCast_inj","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[110,8],"def_end_pos":[110,19]},{"full_name":"multiplicity.unique","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[113,8],"def_end_pos":[113,14]}]}]}
{"url":"Mathlib/Data/Set/Basic.lean","commit":"","full_name":"Set.disjoint_singleton","start":[1274,0],"end":[1275,6],"file_path":"Mathlib/Data/Set/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\n⊢ Disjoint {a} {b} ↔ a ≠ b","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Rat/Lemmas.lean","commit":"","full_name":"Rat.inv_intCast_den_of_pos","start":[235,0],"end":[239,30],"file_path":"Mathlib/Data/Rat/Lemmas.lean","tactics":[{"state_before":"a : ℤ\nha0 : 0 < a\n⊢ ↑(↑a)⁻¹.den = a","state_after":"a : ℤ\nha0 : 0 < a\n⊢ ↑(↑1 / ↑a).den = a","tactic":"rw [← ofInt_eq_cast, ofInt, mk_eq_divInt, Rat.inv_divInt', divInt_eq_div, Nat.cast_one]","premises":[{"full_name":"Nat.cast_one","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"Rat.divInt_eq_div","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[434,8],"def_end_pos":[434,21]},{"full_name":"Rat.inv_divInt'","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[238,14],"def_end_pos":[238,25]},{"full_name":"Rat.mk_eq_divInt","def_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","def_pos":[135,8],"def_end_pos":[135,20]},{"full_name":"Rat.ofInt","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[89,4],"def_end_pos":[89,9]},{"full_name":"Rat.ofInt_eq_cast","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[50,8],"def_end_pos":[50,21]}]},{"state_before":"a : ℤ\nha0 : 0 < a\n⊢ ↑(↑1 / ↑a).den = a","state_after":"a : ℤ\nha0 : 0 < a\n⊢ (Int.natAbs 1).Coprime a.natAbs","tactic":"apply den_div_eq_of_coprime ha0","premises":[{"full_name":"Rat.den_div_eq_of_coprime","def_path":"Mathlib/Data/Rat/Lemmas.lean","def_pos":[182,8],"def_end_pos":[182,29]}]},{"state_before":"a : ℤ\nha0 : 0 < a\n⊢ (Int.natAbs 1).Coprime a.natAbs","state_after":"a : ℤ\nha0 : 0 < a\n⊢ Nat.Coprime 1 a.natAbs","tactic":"rw [Int.natAbs_one]","premises":[{"full_name":"Int.natAbs_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[416,16],"def_end_pos":[416,26]}]},{"state_before":"a : ℤ\nha0 : 0 < a\n⊢ Nat.Coprime 1 a.natAbs","state_after":"no goals","tactic":"exact Nat.coprime_one_left _","premises":[{"full_name":"Nat.coprime_one_left","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[147,8],"def_end_pos":[147,24]}]}]}
{"url":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","commit":"","full_name":"Subsemigroup.closure_induction'","start":[301,0],"end":[310,28],"file_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\ns : Set M\np : (x : M) → x ∈ closure s → Prop\nmem : ∀ (x : M) (h : x ∈ s), p x ⋯\nmul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯\nx : M\nhx : x ∈ closure s\n⊢ p x hx","state_after":"M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\ns : Set M\np : (x : M) → x ∈ closure s → Prop\nmem : ∀ (x : M) (h : x ∈ s), p x ⋯\nmul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯\nx : M\nhx : x ∈ closure s\n⊢ ∃ (x_1 : x ∈ closure s), p x x_1","tactic":"refine Exists.elim ?_ fun (hx : x ∈ closure s) (hc : p x hx) => hc","premises":[{"full_name":"Exists.elim","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[827,8],"def_end_pos":[827,19]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Subsemigroup.closure","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[253,4],"def_end_pos":[253,11]}]},{"state_before":"M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\ns : Set M\np : (x : M) → x ∈ closure s → Prop\nmem : ∀ (x : M) (h : x ∈ s), p x ⋯\nmul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯\nx : M\nhx : x ∈ closure s\n⊢ ∃ (x_1 : x ∈ closure s), p x x_1","state_after":"no goals","tactic":"exact\n    closure_induction hx (fun x hx => ⟨_, mem x hx⟩) fun x y ⟨hx', hx⟩ ⟨hy', hy⟩ =>\n      ⟨_, mul _ _ _ _ hx hy⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Subsemigroup.closure_induction","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[297,8],"def_end_pos":[297,25]}]}]}
{"url":"Mathlib/CategoryTheory/Square.lean","commit":"","full_name":"CategoryTheory.Square.opFunctor_map_τ₃","start":[284,0],"end":[296,47],"file_path":"Mathlib/CategoryTheory/Square.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : (Square C)ᵒᵖ\nφ : X✝ ⟶ Y✝\n⊢ (((fun sq => (Opposite.unop sq).op) X✝).f₁₂ ≫ φ.unop.τ₂.op).unop =\n    (φ.unop.τ₄.op ≫ ((fun sq => (Opposite.unop sq).op) Y✝).f₁₂).unop","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : (Square C)ᵒᵖ\nφ : X✝ ⟶ Y✝\n⊢ (((fun sq => (Opposite.unop sq).op) X✝).f₁₃ ≫ φ.unop.τ₃.op).unop =\n    (φ.unop.τ₄.op ≫ ((fun sq => (Opposite.unop sq).op) Y✝).f₁₃).unop","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : (Square C)ᵒᵖ\nφ : X✝ ⟶ Y✝\n⊢ (((fun sq => (Opposite.unop sq).op) X✝).f₂₄ ≫ φ.unop.τ₁.op).unop =\n    (φ.unop.τ₂.op ≫ ((fun sq => (Opposite.unop sq).op) Y✝).f₂₄).unop","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ : (Square C)ᵒᵖ\nφ : X✝ ⟶ Y✝\n⊢ (((fun sq => (Opposite.unop sq).op) X✝).f₃₄ ≫ φ.unop.τ₁.op).unop =\n    (φ.unop.τ₃.op ≫ ((fun sq => (Opposite.unop sq).op) Y✝).f₃₄).unop","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Combinatorics/Young/YoungDiagram.lean","commit":"","full_name":"YoungDiagram.colLen_eq_card","start":[341,0],"end":[342,20],"file_path":"Mathlib/Combinatorics/Young/YoungDiagram.lean","tactics":[{"state_before":"μ : YoungDiagram\nj : ℕ\n⊢ μ.colLen j = (μ.col j).card","state_after":"no goals","tactic":"simp [col_eq_prod]","premises":[{"full_name":"YoungDiagram.col_eq_prod","def_path":"Mathlib/Combinatorics/Young/YoungDiagram.lean","def_pos":[334,8],"def_end_pos":[334,19]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","commit":"","full_name":"HasFDerivAt.le_of_lipschitzOn","start":[326,0],"end":[332,78],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖f'‖ ≤ ↑C","state_after":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ ↑C * ‖x - x₀‖","tactic":"refine hf.le_of_lip' C.coe_nonneg ?_","premises":[{"full_name":"HasFDerivAt.le_of_lip'","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[314,8],"def_end_pos":[314,30]},{"full_name":"NNReal.coe_nonneg","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[122,8],"def_end_pos":[122,18]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ ↑C * ‖x - x₀‖","state_after":"no goals","tactic":"filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs)","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"mem_of_mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[737,8],"def_end_pos":[737,23]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.pair_mem_product","start":[937,0],"end":[942,46],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nxs : List α\nys : List β\nx : α\ny : β\n⊢ (x, y) ∈ xs.product ys ↔ x ∈ xs ∧ y ∈ ys","state_after":"no goals","tactic":"simp only [product, and_imp, mem_map, Prod.mk.injEq,\n    exists_eq_right_right, mem_bind, iff_self]","premises":[{"full_name":"List.mem_bind","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1253,16],"def_end_pos":[1253,24]},{"full_name":"List.mem_map","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[750,16],"def_end_pos":[750,23]},{"full_name":"List.product","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[712,4],"def_end_pos":[712,11]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"exists_eq_right_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[301,16],"def_end_pos":[301,37]},{"full_name":"iff_self","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[126,16],"def_end_pos":[126,24]}]}]}
{"url":"Mathlib/Data/Nat/Factorization/Basic.lean","commit":"","full_name":"Nat.factorization_le_factorization_mul_left","start":[154,0],"end":[159,23],"file_path":"Mathlib/Data/Nat/Factorization/Basic.lean","tactics":[{"state_before":"a✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\n⊢ a.factorization ≤ (a * b).factorization","state_after":"case inl\na b✝ m n p b : ℕ\nhb : b ≠ 0\n⊢ factorization 0 ≤ (0 * b).factorization\n\ncase inr\na✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a.factorization ≤ (a * b).factorization","tactic":"rcases eq_or_ne a 0 with (rfl | ha)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\na✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a.factorization ≤ (a * b).factorization","state_after":"case inr\na✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a ∣ a * b","tactic":"rw [factorization_le_iff_dvd ha <| mul_ne_zero ha hb]","premises":[{"full_name":"Nat.factorization_le_iff_dvd","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[150,8],"def_end_pos":[150,32]},{"full_name":"mul_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[80,8],"def_end_pos":[80,19]}]},{"state_before":"case inr\na✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a ∣ a * b","state_after":"no goals","tactic":"exact Dvd.intro b rfl","premises":[{"full_name":"Dvd.intro","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[46,8],"def_end_pos":[46,17]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Data/Int/Cast/Lemmas.lean","commit":"","full_name":"MonoidHom.apply_mint","start":[315,0],"end":[317,74],"file_path":"Mathlib/Data/Int/Cast/Lemmas.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Group α\ninst✝ : AddGroup β\nf : Multiplicative ℤ →* α\nn : Multiplicative ℤ\n⊢ f n = f (ofAdd 1) ^ toAdd n","state_after":"no goals","tactic":"rw [← zpowersHom_symm_apply, ← zpowersHom_apply, Equiv.apply_symm_apply]","premises":[{"full_name":"Equiv.apply_symm_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[239,16],"def_end_pos":[239,32]},{"full_name":"zpowersHom_apply","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[309,6],"def_end_pos":[309,22]},{"full_name":"zpowersHom_symm_apply","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[312,6],"def_end_pos":[312,27]}]}]}
{"url":"Mathlib/Data/QPF/Univariate/Basic.lean","commit":"","full_name":"QPF.comp_map","start":[74,0],"end":[79,5],"file_path":"Mathlib/Data/QPF/Univariate/Basic.lean","tactics":[{"state_before":"F : Type u → Type u\nq : QPF F\nα β γ : Type u\nf : α → β\ng : β → γ\nx : F α\n⊢ (g ∘ f) <$> x = g <$> f <$> x","state_after":"F : Type u → Type u\nq : QPF F\nα β γ : Type u\nf : α → β\ng : β → γ\nx : F α\n⊢ (g ∘ f) <$> abs (repr x) = g <$> f <$> abs (repr x)","tactic":"rw [← abs_repr x]","premises":[{"full_name":"QPF.abs_repr","def_path":"Mathlib/Data/QPF/Univariate/Basic.lean","def_pos":[53,2],"def_end_pos":[53,10]}]},{"state_before":"F : Type u → Type u\nq : QPF F\nα β γ : Type u\nf : α → β\ng : β → γ\nx : F α\n⊢ (g ∘ f) <$> abs (repr x) = g <$> f <$> abs (repr x)","state_after":"case mk\nF : Type u → Type u\nq : QPF F\nα β γ : Type u\nf✝ : α → β\ng : β → γ\nx : F α\na : (P F).A\nf : (P F).B a → α\n⊢ (g ∘ f✝) <$> abs ⟨a, f⟩ = g <$> f✝ <$> abs ⟨a, f⟩","tactic":"cases' repr x with a f","premises":[{"full_name":"QPF.repr","def_path":"Mathlib/Data/QPF/Univariate/Basic.lean","def_pos":[52,2],"def_end_pos":[52,6]}]},{"state_before":"case mk\nF : Type u → Type u\nq : QPF F\nα β γ : Type u\nf✝ : α → β\ng : β → γ\nx : F α\na : (P F).A\nf : (P F).B a → α\n⊢ (g ∘ f✝) <$> abs ⟨a, f⟩ = g <$> f✝ <$> abs ⟨a, f⟩","state_after":"case mk\nF : Type u → Type u\nq : QPF F\nα β γ : Type u\nf✝ : α → β\ng : β → γ\nx : F α\na : (P F).A\nf : (P F).B a → α\n⊢ abs ((P F).map (g ∘ f✝) ⟨a, f⟩) = abs ((P F).map g ((P F).map f✝ ⟨a, f⟩))","tactic":"rw [← abs_map, ← abs_map, ← abs_map]","premises":[{"full_name":"QPF.abs_map","def_path":"Mathlib/Data/QPF/Univariate/Basic.lean","def_pos":[54,2],"def_end_pos":[54,9]}]},{"state_before":"case mk\nF : Type u → Type u\nq : QPF F\nα β γ : Type u\nf✝ : α → β\ng : β → γ\nx : F α\na : (P F).A\nf : (P F).B a → α\n⊢ abs ((P F).map (g ∘ f✝) ⟨a, f⟩) = abs ((P F).map g ((P F).map f✝ ⟨a, f⟩))","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","commit":"","full_name":"Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius","start":[411,0],"end":[427,43],"file_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","tactics":[{"state_before":"n : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1","state_after":"case inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic 0 R) =\n    ∏ i ∈ Nat.divisorsAntidiagonal 0, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1\n\ncase inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1","tactic":"rcases n.eq_zero_or_pos with (rfl | hpos)","premises":[{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]}]},{"state_before":"case inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1","state_after":"case inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\nh :\n  ∀ (n : ℕ),\n    0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1","tactic":"have h : ∀ n : ℕ, 0 < n → (∏ i ∈ Nat.divisors n, algebraMap _ (RatFunc R) (cyclotomic i R)) =\n      algebraMap _ _ (X ^ n - 1 : R[X]) := by\n    intro n hn\n    rw [← prod_cyclotomic_eq_X_pow_sub_one hn R, map_prod]","premises":[{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.divisors","def_path":"Mathlib/NumberTheory/Divisors.lean","def_pos":[41,4],"def_end_pos":[41,12]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.cyclotomic","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[231,4],"def_end_pos":[231,14]},{"full_name":"Polynomial.prod_cyclotomic_eq_X_pow_sub_one","def_path":"Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","def_pos":[326,8],"def_end_pos":[326,40]},{"full_name":"RatFunc","def_path":"Mathlib/FieldTheory/RatFunc/Defs.lean","def_pos":[65,10],"def_end_pos":[65,17]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"map_prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"case inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\nh :\n  ∀ (n : ℕ),\n    0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1","state_after":"n : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\nh :\n  ∀ (n : ℕ),\n    0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)\n⊢ ∀ (n : ℕ), 0 < n → ¬cyclotomic n R = 0\n\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\nh :\n  ∀ (n : ℕ),\n    0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)\n⊢ ∀ (n : ℕ), 0 < n → ¬X ^ n - 1 = 0","tactic":"rw [(prod_eq_iff_prod_pow_moebius_eq_of_nonzero (fun n hn => _) fun n hn => _).1 h n hpos] <;>\n    simp_rw [Ne, IsFractionRing.to_map_eq_zero_iff]","premises":[{"full_name":"ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_of_nonzero","def_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","def_pos":[1180,8],"def_end_pos":[1180,50]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsFractionRing.to_map_eq_zero_iff","def_path":"Mathlib/RingTheory/Localization/FractionRing.lean","def_pos":[72,8],"def_end_pos":[72,26]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","commit":"","full_name":"MeasureTheory.Measure.haar.add_prehaar_le_addIndex","start":[261,0],"end":[266,25],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : (interior U).Nonempty\n⊢ prehaar (↑K₀) U K ≤ ↑(index ↑K ↑K₀)","state_after":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : (interior U).Nonempty\n⊢ ↑(index (↑K) U) / ↑(index (↑K₀) U) ≤ ↑(index ↑K ↑K₀)","tactic":"unfold prehaar","premises":[{"full_name":"MeasureTheory.Measure.haar.prehaar","def_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","def_pos":[106,18],"def_end_pos":[106,25]}]},{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : (interior U).Nonempty\n⊢ ↑(index (↑K) U) / ↑(index (↑K₀) U) ≤ ↑(index ↑K ↑K₀)","state_after":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : (interior U).Nonempty\n⊢ index (↑K) U ≤ index ↑K ↑K₀ * index (↑K₀) U\n\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : (interior U).Nonempty\n⊢ 0 < index (↑K₀) U","tactic":"rw [div_le_iff] <;> norm_cast","premises":[{"full_name":"div_le_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[51,8],"def_end_pos":[51,18]}]}]}
{"url":"Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean","commit":"","full_name":"Behrend.exists_large_sphere","start":[244,0],"end":[261,24],"file_path":"Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nn✝ d✝ k N : ℕ\nx : Fin n✝ → ℕ\nn d : ℕ\n⊢ ∃ k, ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\n⊢ ∃ k, ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","tactic":"obtain ⟨k, -, hk⟩ := exists_large_sphere_aux n d","premises":[{"full_name":"Behrend.exists_large_sphere_aux","def_path":"Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean","def_pos":[235,8],"def_end_pos":[235,31]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\n⊢ ∃ k, ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","tactic":"refine ⟨k, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","state_after":"case intro.intro.inl\nα : Type u_1\nβ : Type u_2\nn d✝ k✝ N : ℕ\nx : Fin n → ℕ\nd k : ℕ\nhk : ↑(d ^ 0) / (↑(0 * (d - 1) ^ 2) + 1) ≤ ↑(sphere 0 d k).card\n⊢ ↑(d ^ 0) / ↑(0 * d ^ 2) ≤ ↑(sphere 0 d k).card\n\ncase intro.intro.inr\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","tactic":"obtain rfl | hn := n.eq_zero_or_pos","premises":[{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]}]},{"state_before":"case intro.intro.inr\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","state_after":"case intro.intro.inr.inl\nα : Type u_1\nβ : Type u_2\nn✝ d k✝ N : ℕ\nx : Fin n✝ → ℕ\nn k : ℕ\nhn : n > 0\nhk : ↑(0 ^ n) / (↑(n * (0 - 1) ^ 2) + 1) ≤ ↑(sphere n 0 k).card\n⊢ ↑(0 ^ n) / ↑(n * 0 ^ 2) ≤ ↑(sphere n 0 k).card\n\ncase intro.intro.inr.inr\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","tactic":"obtain rfl | hd := d.eq_zero_or_pos","premises":[{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]}]},{"state_before":"case intro.intro.inr.inr\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card","state_after":"case intro.intro.inr.inr.refine_1\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 0 ≤ ↑(d ^ n)\n\ncase intro.intro.inr.inr.refine_2\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 0 < ↑(n * (d - 1) ^ 2) + 1\n\ncase intro.intro.inr.inr.refine_3\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ ↑(n * (d - 1) ^ 2) + 1 ≤ ↑(n * d ^ 2)","tactic":"refine (div_le_div_of_nonneg_left ?_ ?_ ?_).trans hk","premises":[{"full_name":"div_le_div_of_nonneg_left","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[235,6],"def_end_pos":[235,31]}]},{"state_before":"case intro.intro.inr.inr.refine_3\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ ↑(n * (d - 1) ^ 2) + 1 ≤ ↑(n * d ^ 2)","state_after":"case intro.intro.inr.inr.refine_3\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ ↑n * (2 * ↑d - 1)","tactic":"simp only [← le_sub_iff_add_le', cast_mul, ← mul_sub, cast_pow, cast_sub hd, sub_sq, one_pow,\n    cast_one, mul_one, sub_add, sub_sub_self]","premises":[{"full_name":"Nat.cast_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[56,25],"def_end_pos":[56,33]},{"full_name":"Nat.cast_one","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"Nat.cast_pow","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[82,6],"def_end_pos":[82,14]},{"full_name":"Nat.cast_sub","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[30,8],"def_end_pos":[30,16]},{"full_name":"le_sub_iff_add_le'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[545,2],"def_end_pos":[545,13]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]},{"full_name":"sub_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[545,2],"def_end_pos":[545,13]},{"full_name":"sub_sq","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[192,6],"def_end_pos":[192,12]},{"full_name":"sub_sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[904,14],"def_end_pos":[904,26]}]},{"state_before":"case intro.intro.inr.inr.refine_3\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ ↑n * (2 * ↑d - 1)","state_after":"case intro.intro.inr.inr.refine_3.ha\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ ↑n\n\ncase intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ 2 * ↑d - 1","tactic":"apply one_le_mul_of_one_le_of_one_le","premises":[{"full_name":"one_le_mul_of_one_le_of_one_le","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[173,8],"def_end_pos":[173,38]}]},{"state_before":"case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ 2 * ↑d - 1","state_after":"case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 + 1 ≤ 2 * ↑d","tactic":"rw [_root_.le_sub_iff_add_le]","premises":[{"full_name":"le_sub_iff_add_le","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[492,2],"def_end_pos":[492,13]}]},{"state_before":"case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 + 1 ≤ 2 * ↑d","state_after":"case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ d","tactic":"norm_num","premises":[]},{"state_before":"case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ d","state_after":"no goals","tactic":"exact one_le_cast.2 hd","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.one_le_cast","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[89,8],"def_end_pos":[89,19]}]}]}
{"url":"Mathlib/Topology/Connected/Basic.lean","commit":"","full_name":"isPreconnected_of_forall_pair","start":[100,0],"end":[106,77],"file_path":"Mathlib/Topology/Connected/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nH : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\n⊢ IsPreconnected s","state_after":"case inl\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns t u v : Set α\nH : ∀ x ∈ ∅, ∀ y ∈ ∅, ∃ t ⊆ ∅, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\n⊢ IsPreconnected ∅\n\ncase inr.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nH : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\nx : α\nhx : x ∈ s\n⊢ IsPreconnected s","tactic":"rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩)","premises":[{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]}]},{"state_before":"case inl\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns t u v : Set α\nH : ∀ x ∈ ∅, ∀ y ∈ ∅, ∃ t ⊆ ∅, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\n⊢ IsPreconnected ∅\n\ncase inr.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nH : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\nx : α\nhx : x ∈ s\n⊢ IsPreconnected s","state_after":"no goals","tactic":"exacts [isPreconnected_empty, isPreconnected_of_forall x fun y => H x hx y]","premises":[{"full_name":"isPreconnected_empty","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[69,8],"def_end_pos":[69,28]},{"full_name":"isPreconnected_of_forall","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[83,8],"def_end_pos":[83,32]}]}]}
{"url":"Mathlib/Tactic/Ring/Basic.lean","commit":"","full_name":"Mathlib.Tactic.Ring.pow_bit0","start":[694,0],"end":[696,42],"file_path":"Mathlib/Tactic/Ring/Basic.lean","tactics":[{"state_before":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR : Type u_1\ninst✝ : CommSemiring R\na a' a₁ a₂ a₃ b b' b₁ b₂ b₃ c c₁ c₂ : R\nk : ℕ\nx✝¹ : a ^ k = b\nx✝ : b * b = c\n⊢ a ^ Nat.mul 2 k = c","state_after":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR : Type u_1\ninst✝ : CommSemiring R\na a' a₁ a₂ a₃ b' b₁ b₂ b₃ c₁ c₂ : R\nk : ℕ\n⊢ a ^ Nat.mul 2 k = a ^ k * a ^ k","tactic":"subst_vars","premises":[]},{"state_before":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR : Type u_1\ninst✝ : CommSemiring R\na a' a₁ a₂ a₃ b' b₁ b₂ b₃ c₁ c₂ : R\nk : ℕ\n⊢ a ^ Nat.mul 2 k = a ^ k * a ^ k","state_after":"no goals","tactic":"simp [Nat.succ_mul, pow_add]","premises":[{"full_name":"Nat.succ_mul","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[210,8],"def_end_pos":[210,16]},{"full_name":"pow_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[598,6],"def_end_pos":[598,13]}]}]}
{"url":"Mathlib/Data/Sym/Card.lean","commit":"","full_name":"Sym.card_sym_eq_choose","start":[110,0],"end":[114,50],"file_path":"Mathlib/Data/Sym/Card.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nn : ℕ\nα : Type u_3\ninst✝¹ : Fintype α\nk : ℕ\ninst✝ : Fintype (Sym α k)\n⊢ Fintype.card (Sym α k) = (Fintype.card α + k - 1).choose k","state_after":"no goals","tactic":"rw [card_sym_eq_multichoose, Nat.multichoose_eq]","premises":[{"full_name":"Nat.multichoose_eq","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[359,8],"def_end_pos":[359,22]},{"full_name":"Sym.card_sym_eq_multichoose","def_path":"Mathlib/Data/Sym/Card.lean","def_pos":[103,8],"def_end_pos":[103,31]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","commit":"","full_name":"Complex.arg_eq_zero_iff","start":[207,0],"end":[213,32],"file_path":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","tactics":[{"state_before":"a x z✝ z : ℂ\n⊢ z.arg = 0 ↔ 0 ≤ z.re ∧ z.im = 0","state_after":"case refine_1\na x z✝ z : ℂ\nh : z.arg = 0\n⊢ 0 ≤ z.re ∧ z.im = 0\n\ncase refine_2\na x z✝ z : ℂ\n⊢ 0 ≤ z.re ∧ z.im = 0 → z.arg = 0","tactic":"refine ⟨fun h => ?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.tendsto_comp_val_Ici_atTop","start":[1507,0],"end":[1510,65],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeSup α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f atTop l","state_after":"no goals","tactic":"rw [← map_val_Ici_atTop a, tendsto_map'_iff, Function.comp_def]","premises":[{"full_name":"Filter.map_val_Ici_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1441,8],"def_end_pos":[1441,25]},{"full_name":"Filter.tendsto_map'_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2676,8],"def_end_pos":[2676,24]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Probability/Kernel/MeasurableIntegral.lean","commit":"","full_name":"ProbabilityTheory.Kernel.measurable_lintegral_indicator_const","start":[129,0],"end":[139,66],"file_path":"Mathlib/Probability/Kernel/MeasurableIntegral.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\na : α\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun a => ∫⁻ (b : β), t.indicator (Function.const (α × β) c) (a, b) ∂κ a","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\na : α\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun x => c * (κ x) (Prod.mk x ⁻¹' t)","tactic":"conv =>\n    congr\n    ext\n    erw [lintegral_indicator_const_comp measurable_prod_mk_left ht _]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MeasureTheory.lintegral_indicator_const_comp","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1396,8],"def_end_pos":[1396,38]},{"full_name":"measurable_prod_mk_left","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[653,8],"def_end_pos":[653,31]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\na : α\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun x => c * (κ x) (Prod.mk x ⁻¹' t)","state_after":"no goals","tactic":"exact Measurable.const_mul (measurable_kernel_prod_mk_left ht) c","premises":[{"full_name":"Measurable.const_mul","def_path":"Mathlib/MeasureTheory/Group/Arithmetic.lean","def_pos":[94,8],"def_end_pos":[94,28]},{"full_name":"ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left","def_path":"Mathlib/Probability/Kernel/MeasurableIntegral.lean","def_pos":[99,8],"def_end_pos":[99,38]}]}]}
{"url":"Mathlib/Analysis/Convex/Combination.lean","commit":"","full_name":"Finset.centerMass_mem_convexHull_of_nonpos","start":[221,0],"end":[225,95],"file_path":"Mathlib/Analysis/Convex/Combination.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt✝ : Finset ι\nw : ι → R\nz : ι → E\nt : Finset ι\nhw₀ : ∀ i ∈ t, w i ≤ 0\nhws : ∑ i ∈ t, w i < 0\nhz : ∀ i ∈ t, z i ∈ s\n⊢ t.centerMass w z ∈ (convexHull R) s","state_after":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt✝ : Finset ι\nw : ι → R\nz : ι → E\nt : Finset ι\nhw₀ : ∀ i ∈ t, w i ≤ 0\nhws : ∑ i ∈ t, w i < 0\nhz : ∀ i ∈ t, z i ∈ s\n⊢ t.centerMass (-w) z ∈ (convexHull R) s","tactic":"rw [← centerMass_neg_left]","premises":[{"full_name":"Finset.centerMass_neg_left","def_path":"Mathlib/Analysis/Convex/Combination.lean","def_pos":[68,14],"def_end_pos":[68,40]}]},{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt✝ : Finset ι\nw : ι → R\nz : ι → E\nt : Finset ι\nhw₀ : ∀ i ∈ t, w i ≤ 0\nhws : ∑ i ∈ t, w i < 0\nhz : ∀ i ∈ t, z i ∈ s\n⊢ t.centerMass (-w) z ∈ (convexHull R) s","state_after":"no goals","tactic":"exact Finset.centerMass_mem_convexHull _ (fun _i hi ↦ neg_nonneg.2 <| hw₀ _ hi) (by simpa) hz","premises":[{"full_name":"Finset.centerMass_mem_convexHull","def_path":"Mathlib/Analysis/Convex/Combination.lean","def_pos":[216,8],"def_end_pos":[216,40]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
{"url":"Mathlib/CategoryTheory/FiberedCategory/HomLift.lean","commit":"","full_name":"CategoryTheory.IsHomLift.eqToHom_comp_lift_iff","start":[185,0],"end":[189,25],"file_path":"Mathlib/CategoryTheory/FiberedCategory/HomLift.lean","tactics":[{"state_before":"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₂} 𝒳\ninst✝ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b b' : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\nh : b = b'\nhφ' : p.IsHomLift f (φ ≫ eqToHom h)\n⊢ p.IsHomLift f φ","state_after":"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₂} 𝒳\ninst✝ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\nhφ' : p.IsHomLift f (φ ≫ eqToHom ⋯)\n⊢ p.IsHomLift f φ","tactic":"subst h","premises":[]},{"state_before":"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₂} 𝒳\ninst✝ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\nhφ' : p.IsHomLift f (φ ≫ eqToHom ⋯)\n⊢ p.IsHomLift f φ","state_after":"no goals","tactic":"simpa using hφ'","premises":[]}]}
{"url":"Mathlib/NumberTheory/PythagoreanTriples.lean","commit":"","full_name":"PythagoreanTriple.mul_isClassified","start":[107,0],"end":[116,24],"file_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","tactics":[{"state_before":"x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\nhc : h.IsClassified\n⊢ ⋯.IsClassified","state_after":"case intro.intro.intro.intro.inl.intro\nz k l m n : ℤ\nco : m.gcd n = 1\nh : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z\n⊢ ⋯.IsClassified\n\ncase intro.intro.intro.intro.inr.intro\nz k l m n : ℤ\nco : m.gcd n = 1\nh : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z\n⊢ ⋯.IsClassified","tactic":"obtain ⟨l, m, n, ⟨⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, co⟩⟩ := hc","premises":[]}]}
{"url":"Mathlib/Order/Filter/CountableInter.lean","commit":"","full_name":"Filter.le_countableGenerate_iff_of_countableInterFilter","start":[260,0],"end":[269,41],"file_path":"Mathlib/Order/Filter/CountableInter.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\n⊢ f ≤ countableGenerate g ↔ g ⊆ f.sets","state_after":"case mp\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : f ≤ countableGenerate g\n⊢ g ⊆ f.sets\n\ncase mpr\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\n⊢ f ≤ countableGenerate g","tactic":"constructor <;> intro h","premises":[]},{"state_before":"case mpr\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\n⊢ f ≤ countableGenerate g","state_after":"case mpr\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\nhs : s ∈ countableGenerate g\n⊢ s ∈ f","tactic":"intro s hs","premises":[]},{"state_before":"case mpr\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\nhs : s ∈ countableGenerate g\n⊢ s ∈ f","state_after":"case mpr.basic\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns✝ s : Set α\nhs : s ∈ g\n⊢ s ∈ f\n\ncase mpr.univ\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\n⊢ univ ∈ f\n\ncase mpr.superset\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns✝ s t : Set α\na✝ : CountableGenerateSets g s\nst : s ⊆ t\nih : s ∈ f\n⊢ t ∈ f\n\ncase mpr.sInter\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\nS : Set (Set α)\nSct : S.Countable\na✝ : ∀ s ∈ S, CountableGenerateSets g s\nih : ∀ s ∈ S, s ∈ f\n⊢ ⋂₀ S ∈ f","tactic":"induction' hs with s hs s t _ st ih S Sct _ ih","premises":[]},{"state_before":"case mpr.sInter\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\nS : Set (Set α)\nSct : S.Countable\na✝ : ∀ s ∈ S, CountableGenerateSets g s\nih : ∀ s ∈ S, s ∈ f\n⊢ ⋂₀ S ∈ f","state_after":"no goals","tactic":"exact (countable_sInter_mem Sct).mpr ih","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"countable_sInter_mem","def_path":"Mathlib/Order/Filter/CountableInter.lean","def_pos":[46,8],"def_end_pos":[46,28]}]}]}
{"url":"Mathlib/Data/Finmap.lean","commit":"","full_name":"Finmap.keys_erase_toFinset","start":[373,0],"end":[375,53],"file_path":"Mathlib/Data/Finmap.lean","tactics":[{"state_before":"α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : AList β\n⊢ ⟦AList.erase a s⟧.keys = ⟦s⟧.keys.erase a","state_after":"no goals","tactic":"simp [Finset.erase, keys, AList.erase, keys_kerase]","premises":[{"full_name":"AList.erase","def_path":"Mathlib/Data/List/AList.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Finmap.keys","def_path":"Mathlib/Data/Finmap.lean","def_pos":[169,4],"def_end_pos":[169,8]},{"full_name":"Finset.erase","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1574,4],"def_end_pos":[1574,9]},{"full_name":"List.keys_kerase","def_path":"Mathlib/Data/List/Sigma.lean","def_pos":[390,8],"def_end_pos":[390,19]}]}]}
{"url":"Mathlib/Algebra/Homology/TotalComplex.lean","commit":"","full_name":"HomologicalComplex₂.ιTotalOrZero_map","start":[434,0],"end":[438,32],"file_path":"Mathlib/Algebra/Homology/TotalComplex.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_5, u_1} C\ninst✝⁴ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝³ : DecidableEq I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : K.HasTotal c₁₂\ninst✝ : L.HasTotal c₁₂\ni₁ : I₁\ni₂ : I₂\ni₁₂ : I₁₂\n⊢ K.ιTotalOrZero c₁₂ i₁ i₂ i₁₂ ≫ (total.map φ c₁₂).f i₁₂ = (φ.f i₁).f i₂ ≫ L.ιTotalOrZero c₁₂ i₁ i₂ i₁₂","state_after":"no goals","tactic":"simp [total.map, ιTotalOrZero]","premises":[{"full_name":"HomologicalComplex₂.total.map","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[389,18],"def_end_pos":[389,21]},{"full_name":"HomologicalComplex₂.ιTotalOrZero","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[277,18],"def_end_pos":[277,30]}]}]}
{"url":"Mathlib/CategoryTheory/Monad/Adjunction.lean","commit":"","full_name":"CategoryTheory.Adjunction.toMonad_η","start":[40,0],"end":[56,8],"file_path":"Mathlib/CategoryTheory/Monad/Adjunction.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ (L ⋙ R).map ((whiskerRight (whiskerLeft L h.counit) R).app X) ≫ (whiskerRight (whiskerLeft L h.counit) R).app X =\n    (whiskerRight (whiskerLeft L h.counit) R).app ((L ⋙ R).obj X) ≫ (whiskerRight (whiskerLeft L h.counit) R).app X","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (R.map (h.counit.app (L.obj X)))) ≫ R.map (h.counit.app (L.obj X)) =\n    R.map (h.counit.app (L.obj (R.obj (L.obj X)))) ≫ R.map (h.counit.app (L.obj X))","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (R.map (h.counit.app (L.obj X)))) ≫ R.map (h.counit.app (L.obj X)) =\n    R.map (h.counit.app (L.obj (R.obj (L.obj X)))) ≫ R.map (h.counit.app (L.obj X))","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (R.map (h.counit.app (L.obj X))) ≫ h.counit.app (L.obj X)) =\n    R.map (h.counit.app (L.obj (R.obj (L.obj X)))) ≫ R.map (h.counit.app (L.obj X))","tactic":"rw [← R.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (R.map (h.counit.app (L.obj X))) ≫ h.counit.app (L.obj X)) =\n    R.map (h.counit.app (L.obj (R.obj (L.obj X)))) ≫ R.map (h.counit.app (L.obj X))","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ (L ⋙ R).map (h.unit.app X) ≫ (whiskerRight (whiskerLeft L h.counit) R).app X = 𝟙 ((L ⋙ R).obj ((𝟭 C).obj X))","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (h.unit.app X)) ≫ R.map (h.counit.app (L.obj X)) = 𝟙 (R.obj (L.obj X))","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (h.unit.app X)) ≫ R.map (h.counit.app (L.obj X)) = 𝟙 (R.obj (L.obj X))","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (h.unit.app X) ≫ h.counit.app (L.obj X)) = 𝟙 (R.obj (L.obj X))","tactic":"rw [← R.map_comp]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (h.unit.app X) ≫ h.counit.app (L.obj X)) = 𝟙 (R.obj (L.obj X))","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Nat/Choose/Factorization.lean","commit":"","full_name":"Nat.prod_pow_factorization_centralBinom","start":[130,0],"end":[135,7],"file_path":"Mathlib/Data/Nat/Choose/Factorization.lean","tactics":[{"state_before":"p n✝ k n : ℕ\n⊢ ∏ p ∈ Finset.range (2 * n + 1), p ^ n.centralBinom.factorization p = n.centralBinom","state_after":"case hkn\np n✝ k n : ℕ\n⊢ n ≤ 2 * n","tactic":"apply prod_pow_factorization_choose","premises":[{"full_name":"Nat.prod_pow_factorization_choose","def_path":"Mathlib/Data/Nat/Choose/Factorization.lean","def_pos":[116,8],"def_end_pos":[116,37]}]},{"state_before":"case hkn\np n✝ k n : ℕ\n⊢ n ≤ 2 * n","state_after":"no goals","tactic":"omega","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Combination.lean","commit":"","full_name":"Finset.centerMass_subset","start":[117,0],"end":[122,41],"file_path":"Mathlib/Analysis/Convex/Combination.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\n⊢ t.centerMass w z = t'.centerMass w z","state_after":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\n⊢ ∑ x ∈ t, (∑ x ∈ t', w x)⁻¹ • w x • z x = ∑ x ∈ t', (∑ i ∈ t', w i)⁻¹ • w x • z x","tactic":"rw [centerMass, sum_subset ht h, smul_sum, centerMass, smul_sum]","premises":[{"full_name":"Finset.centerMass","def_path":"Mathlib/Analysis/Convex/Combination.lean","def_pos":[41,4],"def_end_pos":[41,21]},{"full_name":"Finset.smul_sum","def_path":"Mathlib/GroupTheory/GroupAction/BigOperators.lean","def_pos":[45,8],"def_end_pos":[45,23]},{"full_name":"Finset.sum_subset","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[520,2],"def_end_pos":[520,13]}]},{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\n⊢ ∑ x ∈ t, (∑ x ∈ t', w x)⁻¹ • w x • z x = ∑ x ∈ t', (∑ i ∈ t', w i)⁻¹ • w x • z x","state_after":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\n⊢ ∀ x ∈ t', x ∉ t → (∑ x ∈ t', w x)⁻¹ • w x • z x = 0","tactic":"apply sum_subset ht","premises":[{"full_name":"Finset.sum_subset","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[520,2],"def_end_pos":[520,13]}]},{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\n⊢ ∀ x ∈ t', x ∉ t → (∑ x ∈ t', w x)⁻¹ • w x • z x = 0","state_after":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\ni : ι\nhit' : i ∈ t'\nhit : i ∉ t\n⊢ (∑ x ∈ t', w x)⁻¹ • w i • z i = 0","tactic":"intro i hit' hit","premises":[]},{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\ni : ι\nhit' : i ∈ t'\nhit : i ∉ t\n⊢ (∑ x ∈ t', w x)⁻¹ • w i • z i = 0","state_after":"no goals","tactic":"rw [h i hit' hit, zero_smul, smul_zero]","premises":[{"full_name":"smul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[63,8],"def_end_pos":[63,17]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]}]}
{"url":"Mathlib/NumberTheory/Rayleigh.lean","commit":"","full_name":"Irrational.beattySeq'_pos_eq","start":[161,0],"end":[168,49],"file_path":"Mathlib/NumberTheory/Rayleigh.lean","tactics":[{"state_before":"r : ℝ\nhr : Irrational r\n⊢ {x | ∃ k > 0, beattySeq' r k = x} = {x | ∃ k > 0, beattySeq r k = x}","state_after":"r : ℝ\nhr : Irrational r\n⊢ {x | ∃ k > 0, ⌈↑k * r⌉ - 1 = x} = {x | ∃ k > 0, ⌊↑k * r⌋ = x}","tactic":"dsimp only [beattySeq, beattySeq']","premises":[{"full_name":"beattySeq","def_path":"Mathlib/NumberTheory/Rayleigh.lean","def_pos":[49,18],"def_end_pos":[49,27]},{"full_name":"beattySeq'","def_path":"Mathlib/NumberTheory/Rayleigh.lean","def_pos":[53,18],"def_end_pos":[53,28]}]},{"state_before":"r : ℝ\nhr : Irrational r\n⊢ {x | ∃ k > 0, ⌈↑k * r⌉ - 1 = x} = {x | ∃ k > 0, ⌊↑k * r⌋ = x}","state_after":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝¹ x✝ : ℤ\n⊢ x✝ > 0 ∧ ⌈↑x✝ * r⌉ - 1 = x✝¹ ↔ x✝ > 0 ∧ ⌊↑x✝ * r⌋ = x✝¹","tactic":"congr! 4","premises":[]},{"state_before":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝¹ x✝ : ℤ\n⊢ x✝ > 0 ∧ ⌈↑x✝ * r⌉ - 1 = x✝¹ ↔ x✝ > 0 ∧ ⌊↑x✝ * r⌋ = x✝¹","state_after":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\n⊢ k > 0 ∧ ⌈↑k * r⌉ - 1 = x✝ ↔ k > 0 ∧ ⌊↑k * r⌋ = x✝","tactic":"rename_i k","premises":[]},{"state_before":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\n⊢ k > 0 ∧ ⌈↑k * r⌉ - 1 = x✝ ↔ k > 0 ∧ ⌊↑k * r⌋ = x✝","state_after":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\n⊢ k > 0 → (⌈↑k * r⌉ - 1 = x✝ ↔ ⌊↑k * r⌋ = x✝)","tactic":"rw [and_congr_right_iff]","premises":[{"full_name":"and_congr_right_iff","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[156,16],"def_end_pos":[156,35]}]},{"state_before":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\n⊢ k > 0 → (⌈↑k * r⌉ - 1 = x✝ ↔ ⌊↑k * r⌋ = x✝)","state_after":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\n⊢ ⌈↑k * r⌉ - 1 = x✝ ↔ ⌊↑k * r⌋ = x✝","tactic":"intro hk","premises":[]},{"state_before":"case h.e'_2.h.h.e'_2.h.a\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\n⊢ ⌈↑k * r⌉ - 1 = x✝ ↔ ⌊↑k * r⌋ = x✝","state_after":"case h.e'_2.h.h.e'_2.h.a.a.h.e'_2\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\n⊢ ⌈↑k * r⌉ - 1 = ⌊↑k * r⌋","tactic":"congr!","premises":[]},{"state_before":"case h.e'_2.h.h.e'_2.h.a.a.h.e'_2\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\n⊢ ⌈↑k * r⌉ - 1 = ⌊↑k * r⌋","state_after":"case h.e'_2.h.h.e'_2.h.a.a.h.e'_2\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\n⊢ ↑⌊↑k * r⌋ < ↑k * r ∧ ↑k * r ≤ ↑⌊↑k * r⌋ + 1","tactic":"rw [sub_eq_iff_eq_add, Int.ceil_eq_iff, Int.cast_add, Int.cast_one, add_sub_cancel_right]","premises":[{"full_name":"Int.cast_add","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[94,8],"def_end_pos":[94,16]},{"full_name":"Int.cast_one","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[70,8],"def_end_pos":[70,16]},{"full_name":"Int.ceil_eq_iff","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[1146,8],"def_end_pos":[1146,19]},{"full_name":"add_sub_cancel_right","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[698,2],"def_end_pos":[698,13]},{"full_name":"sub_eq_iff_eq_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[756,2],"def_end_pos":[756,13]}]},{"state_before":"case h.e'_2.h.h.e'_2.h.a.a.h.e'_2\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\n⊢ ↑⌊↑k * r⌋ < ↑k * r ∧ ↑k * r ≤ ↑⌊↑k * r⌋ + 1","state_after":"case h.e'_2.h.h.e'_2.h.a.a.h.e'_2\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\nh : ↑⌊↑k * r⌋ = ↑k * r\n⊢ False","tactic":"refine ⟨(Int.floor_le _).lt_of_ne fun h ↦ ?_, (Int.lt_floor_add_one _).le⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Int.floor_le","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[613,8],"def_end_pos":[613,16]},{"full_name":"Int.lt_floor_add_one","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[633,8],"def_end_pos":[633,24]}]},{"state_before":"case h.e'_2.h.h.e'_2.h.a.a.h.e'_2\nr : ℝ\nhr : Irrational r\nx✝ k : ℤ\nhk : k > 0\nh : ↑⌊↑k * r⌋ = ↑k * r\n⊢ False","state_after":"no goals","tactic":"exact (hr.int_mul hk.ne').ne_int ⌊k * r⌋ h.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Int.floor","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[563,4],"def_end_pos":[563,9]},{"full_name":"Irrational.int_mul","def_path":"Mathlib/Data/Real/Irrational.lean","def_pos":[350,8],"def_end_pos":[350,15]},{"full_name":"Irrational.ne_int","def_path":"Mathlib/Data/Real/Irrational.lean","def_pos":[181,8],"def_end_pos":[181,14]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]}]}]}
{"url":"Mathlib/Data/Finset/Image.lean","commit":"","full_name":"Finset.mem_subtype","start":[625,0],"end":[628,43],"file_path":"Mathlib/Data/Finset/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\na : α\nha : p a\n⊢ ⟨a, ha⟩ ∈ Finset.subtype p s ↔ ↑⟨a, ha⟩ ∈ s","state_after":"no goals","tactic":"simp [Finset.subtype, ha]","premises":[{"full_name":"Finset.subtype","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[620,14],"def_end_pos":[620,21]}]}]}
{"url":"Mathlib/Topology/VectorBundle/Basic.lean","commit":"","full_name":"Pretrivialization.linearMapAt_apply","start":[122,0],"end":[124,22],"file_path":"Mathlib/Topology/VectorBundle/Basic.lean","tactics":[{"state_before":"R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ne✝ : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb✝ : B\ny✝ : E b✝\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst✝ : Pretrivialization.IsLinear R e\nb : B\ny : E b\n⊢ (Pretrivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0","state_after":"no goals","tactic":"rw [coe_linearMapAt]","premises":[{"full_name":"Pretrivialization.coe_linearMapAt","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[113,8],"def_end_pos":[113,23]}]}]}
{"url":"Mathlib/Algebra/Module/Submodule/Ker.lean","commit":"","full_name":"LinearMap.injective_restrict_iff_disjoint","start":[201,0],"end":[204,89],"file_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nK : Type u_5\nM : Type u_6\nM₁ : Type u_7\nM₂ : Type u_8\nM₃ : Type u_9\nV : Type u_10\nV₂ : Type u_11\ninst✝¹¹ : Ring R\ninst✝¹⁰ : Ring R₂\ninst✝⁹ : Ring R₃\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_12\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F τ₁₂ M M₂\nf✝ : F\np : Submodule R M\nf : M →ₗ[R] M\nhf : ∀ x ∈ p, f x ∈ p\n⊢ Injective ⇑(f.restrict hf) ↔ Disjoint p (ker f)","state_after":"no goals","tactic":"rw [← ker_eq_bot, ker_restrict hf, ker_eq_bot, injective_domRestrict_iff, disjoint_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LinearMap.injective_domRestrict_iff","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[188,14],"def_end_pos":[188,39]},{"full_name":"LinearMap.ker_eq_bot","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[185,8],"def_end_pos":[185,18]},{"full_name":"LinearMap.ker_restrict","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[114,8],"def_end_pos":[114,20]},{"full_name":"disjoint_iff","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[116,8],"def_end_pos":[116,20]}]}]}
{"url":"Mathlib/Data/Set/Function.lean","commit":"","full_name":"Set.InjOn.image_subset_image_iff","start":[655,0],"end":[659,52],"file_path":"Mathlib/Data/Set/Function.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\nh₁ : s₁ ⊆ s\nh₂ : s₂ ⊆ s\n⊢ f '' s₁ ⊆ f '' s₂ ↔ s₁ ⊆ s₂","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\nh₁ : s₁ ⊆ s\nh₂ : s₂ ⊆ s\nh' : f '' s₁ ⊆ f '' s₂\n⊢ s₁ ⊆ s₂","tactic":"refine ⟨fun h' ↦ ?_, image_subset _⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Set.image_subset","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[250,8],"def_end_pos":[250,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\nh₁ : s₁ ⊆ s\nh₂ : s₂ ⊆ s\nh' : f '' s₁ ⊆ f '' s₂\n⊢ s₁ ⊆ s₂","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\nh₁ : s₁ ⊆ s\nh₂ : s₂ ⊆ s\nh' : f '' s₁ ⊆ f '' s₂\n⊢ f ⁻¹' (f '' s₁) ∩ s ⊆ f ⁻¹' (f '' s₂) ∩ s","tactic":"rw [← h.preimage_image_inter h₁, ← h.preimage_image_inter h₂]","premises":[{"full_name":"Set.InjOn.preimage_image_inter","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[628,8],"def_end_pos":[628,34]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\nh₁ : s₁ ⊆ s\nh₂ : s₂ ⊆ s\nh' : f '' s₁ ⊆ f '' s₂\n⊢ f ⁻¹' (f '' s₁) ∩ s ⊆ f ⁻¹' (f '' s₂) ∩ s","state_after":"no goals","tactic":"exact inter_subset_inter_left _ (preimage_mono h')","premises":[{"full_name":"Set.inter_subset_inter_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[810,8],"def_end_pos":[810,31]},{"full_name":"Set.preimage_mono","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[56,8],"def_end_pos":[56,21]}]}]}
{"url":"Mathlib/Analysis/Convex/Segment.lean","commit":"","full_name":"segment_inter_eq_endpoint_of_linearIndependent_sub","start":[254,0],"end":[269,6],"file_path":"Mathlib/Analysis/Convex/Segment.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] = {c}","state_after":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] ⊆ {c}\n\ncase h₂\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ {c} ⊆ [c-[𝕜]x] ∩ [c-[𝕜]y]","tactic":"apply Subset.antisymm","premises":[{"full_name":"Set.Subset.antisymm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[302,8],"def_end_pos":[302,23]}]},{"state_before":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] ⊆ {c}\n\ncase h₂\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ {c} ⊆ [c-[𝕜]x] ∩ [c-[𝕜]y]","state_after":"case h₂\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ {c} ⊆ [c-[𝕜]x] ∩ [c-[𝕜]y]\n\ncase h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] ⊆ {c}","tactic":"swap","premises":[]},{"state_before":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\n⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] ⊆ {c}","state_after":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\nz : E\nhzt : z ∈ [c-[𝕜]x]\nhzs : z ∈ [c-[𝕜]y]\n⊢ z ∈ {c}","tactic":"intro z ⟨hzt, hzs⟩","premises":[]},{"state_before":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\nz : E\nhzt : z ∈ [c-[𝕜]x]\nhzs : z ∈ [c-[𝕜]y]\n⊢ z ∈ {c}","state_after":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\nz : E\nhzt : ∃ x_1 ∈ Icc 0 1, (1 - x_1) • c + x_1 • x = z\nhzs : ∃ x ∈ Icc 0 1, (1 - x) • c + x • y = z\n⊢ z ∈ {c}","tactic":"rw [segment_eq_image, mem_image] at hzt hzs","premises":[{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"segment_eq_image","def_path":"Mathlib/Analysis/Convex/Segment.lean","def_pos":[172,8],"def_end_pos":[172,24]}]},{"state_before":"case h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\nz : E\nhzt : ∃ x_1 ∈ Icc 0 1, (1 - x_1) • c + x_1 • x = z\nhzs : ∃ x ∈ Icc 0 1, (1 - x) • c + x • y = z\n⊢ z ∈ {c}","state_after":"case h₁.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nhzs : ∃ x_1 ∈ Icc 0 1, (1 - x_1) • c + x_1 • y = (1 - p) • c + p • x\n⊢ (1 - p) • c + p • x ∈ {c}","tactic":"rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩","premises":[]},{"state_before":"case h₁.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nhzs : ∃ x_1 ∈ Icc 0 1, (1 - x_1) • c + x_1 • y = (1 - p) • c + p • x\n⊢ (1 - p) • c + p • x ∈ {c}","state_after":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q ≤ 1\n⊢ (1 - p) • c + p • x ∈ {c}","tactic":"rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩","premises":[]},{"state_before":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q ≤ 1\n⊢ (1 - p) • c + p • x ∈ {c}","state_after":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\n⊢ (1 - p) • c + p • x ∈ {c}","tactic":"have Hx : x = (x - c) + c := by abel","premises":[]},{"state_before":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\n⊢ (1 - p) • c + p • x ∈ {c}","state_after":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\nHy : y = y - c + c\n⊢ (1 - p) • c + p • x ∈ {c}","tactic":"have Hy : y = (y - c) + c := by abel","premises":[]},{"state_before":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\nHy : y = y - c + c\n⊢ (1 - p) • c + p • x ∈ {c}","state_after":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\nHy : y = y - c + c\n⊢ (1 - p) • c + p • x ∈ {c}","tactic":"rw [Hx, Hy, smul_add, smul_add] at H","premises":[{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]}]},{"state_before":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\nHy : y = y - c + c\n⊢ (1 - p) • c + p • x ∈ {c}","state_after":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\nHy : y = y - c + c\nthis : c + q • (y - c) = c + p • (x - c)\n⊢ (1 - p) • c + p • x ∈ {c}","tactic":"have : c + q • (y - c) = c + p • (x - c) := by\n    convert H using 1 <;> simp [sub_smul]","premises":[{"full_name":"sub_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[245,8],"def_end_pos":[245,16]}]},{"state_before":"case h₁.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)\nq0 : 0 ≤ q\nq1 : q ≤ 1\nHx : x = x - c + c\nHy : y = y - c + c\nthis : c + q • (y - c) = c + p • (x - c)\n⊢ (1 - p) • c + p • x ∈ {c}","state_after":"case h₁.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\nHx : x = x - c + c\nHy : y = y - c + c\np0 : 0 ≤ 0\np1 : 0 ≤ 1\nq0 : 0 ≤ 0\nq1 : 0 ≤ 1\nH : (1 - 0) • c + (0 • (y - c) + 0 • c) = (1 - 0) • c + (0 • (x - c) + 0 • c)\nthis : c + 0 • (y - c) = c + 0 • (x - c)\n⊢ (1 - 0) • c + 0 • x ∈ {c}","tactic":"obtain ⟨rfl, rfl⟩ : p = 0 ∧ q = 0 := h.eq_zero_of_pair' ((add_right_inj c).1 this).symm","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LinearIndependent.eq_zero_of_pair'","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[594,6],"def_end_pos":[594,40]},{"full_name":"add_right_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[63,2],"def_end_pos":[63,13]}]},{"state_before":"case h₁.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\nHx : x = x - c + c\nHy : y = y - c + c\np0 : 0 ≤ 0\np1 : 0 ≤ 1\nq0 : 0 ≤ 0\nq1 : 0 ≤ 1\nH : (1 - 0) • c + (0 • (y - c) + 0 • c) = (1 - 0) • c + (0 • (x - c) + 0 • c)\nthis : c + 0 • (y - c) = c + 0 • (x - c)\n⊢ (1 - 0) • c + 0 • x ∈ {c}","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","commit":"","full_name":"LinearPMap.IsFormalAdjoint.symm","start":[70,0],"end":[73,54],"file_path":"Mathlib/Analysis/InnerProductSpace/LinearPMap.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\nS : F →ₗ.[𝕜] E\nh : T.IsFormalAdjoint S\ny : ↥S.domain\nx✝ : ↥T.domain\n⊢ ⟪↑S y, ↑x✝⟫_𝕜 = ⟪↑y, ↑T x✝⟫_𝕜","state_after":"no goals","tactic":"rw [← inner_conj_symm, ← inner_conj_symm (y : F), h]","premises":[{"full_name":"inner_conj_symm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[389,8],"def_end_pos":[389,23]}]}]}
{"url":"Mathlib/Analysis/Convolution.lean","commit":"","full_name":"MeasureTheory.posConvolution_eq_convolution_indicator","start":[1343,0],"end":[1388,39],"file_path":"Mathlib/Analysis/Convolution.lean","tactics":[{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\n⊢ posConvolution f g L ν = (Ioi 0).indicator f ⋆[L, ν] (Ioi 0).indicator g","state_after":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\n⊢ posConvolution f g L ν x = ((Ioi 0).indicator f ⋆[L, ν] (Ioi 0).indicator g) x","tactic":"ext1 x","premises":[]},{"state_before":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\n⊢ posConvolution f g L ν x = ((Ioi 0).indicator f ⋆[L, ν] (Ioi 0).indicator g) x","state_after":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\n⊢ (if x ∈ Ioi 0 then (fun x => ∫ (t : ℝ) in 0 ..x, (L (f t)) (g (x - t)) ∂ν) x else 0) =\n    ∫ (t : ℝ), (L (if t ∈ Ioi 0 then f t else 0)) (if x - t ∈ Ioi 0 then g (x - t) else 0) ∂ν","tactic":"unfold convolution posConvolution indicator","premises":[{"full_name":"MeasureTheory.convolution","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[403,18],"def_end_pos":[403,29]},{"full_name":"MeasureTheory.posConvolution","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[1339,18],"def_end_pos":[1339,32]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]}]},{"state_before":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\n⊢ (if x ∈ Ioi 0 then (fun x => ∫ (t : ℝ) in 0 ..x, (L (f t)) (g (x - t)) ∂ν) x else 0) =\n    ∫ (t : ℝ), (L (if t ∈ Ioi 0 then f t else 0)) (if x - t ∈ Ioi 0 then g (x - t) else 0) ∂ν","state_after":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\n⊢ (if x ∈ Ioi 0 then ∫ (t : ℝ) in 0 ..x, (L (f t)) (g (x - t)) ∂ν else 0) =\n    ∫ (t : ℝ), (L (if t ∈ Ioi 0 then f t else 0)) (if x - t ∈ Ioi 0 then g (x - t) else 0) ∂ν","tactic":"simp only","premises":[]},{"state_before":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\n⊢ (if x ∈ Ioi 0 then ∫ (t : ℝ) in 0 ..x, (L (f t)) (g (x - t)) ∂ν else 0) =\n    ∫ (t : ℝ), (L (if t ∈ Ioi 0 then f t else 0)) (if x - t ∈ Ioi 0 then g (x - t) else 0) ∂ν","state_after":"case pos\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\nh : x ∈ Ioi 0\n⊢ ∫ (t : ℝ) in 0 ..x, (L (f t)) (g (x - t)) ∂ν =\n    ∫ (t : ℝ), (L (if t ∈ Ioi 0 then f t else 0)) (if x - t ∈ Ioi 0 then g (x - t) else 0) ∂ν\n\ncase neg\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F\nf✝ f' : G → E\ng✝ g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ E'\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : autoParam (Measure ℝ) _auto✝\ninst✝ : NoAtoms ν\nx : ℝ\nh : x ∉ Ioi 0\n⊢ 0 = ∫ (t : ℝ), (L (if t ∈ Ioi 0 then f t else 0)) (if x - t ∈ Ioi 0 then g (x - t) else 0) ∂ν","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Geometry/Manifold/ChartedSpace.lean","commit":"","full_name":"ChartedSpaceCore.open_target","start":[858,0],"end":[862,68],"file_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","tactics":[{"state_before":"H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne : PartialEquiv M H\nhe : e ∈ c.atlas\n⊢ IsOpen e.target","state_after":"H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne : PartialEquiv M H\nhe : e ∈ c.atlas\nE : e.target ∩ ↑e.symm ⁻¹' e.source = e.target\n⊢ IsOpen e.target","tactic":"have E : e.target ∩ e.symm ⁻¹' e.source = e.target :=\n    Subset.antisymm inter_subset_left fun x hx ↦\n      ⟨hx, PartialEquiv.target_subset_preimage_source _ hx⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"PartialEquiv.source","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[124,2],"def_end_pos":[124,8]},{"full_name":"PartialEquiv.symm","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[148,14],"def_end_pos":[148,18]},{"full_name":"PartialEquiv.target","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[126,2],"def_end_pos":[126,8]},{"full_name":"PartialEquiv.target_subset_preimage_source","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[463,8],"def_end_pos":[463,37]},{"full_name":"Set.Subset.antisymm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[302,8],"def_end_pos":[302,23]},{"full_name":"Set.inter_subset_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[761,8],"def_end_pos":[761,25]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]}]},{"state_before":"H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne : PartialEquiv M H\nhe : e ∈ c.atlas\nE : e.target ∩ ↑e.symm ⁻¹' e.source = e.target\n⊢ IsOpen e.target","state_after":"no goals","tactic":"simpa [PartialEquiv.trans_source, E] using c.open_source e e he he","premises":[{"full_name":"ChartedSpaceCore.open_source","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[838,2],"def_end_pos":[838,13]},{"full_name":"PartialEquiv.trans_source","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[601,8],"def_end_pos":[601,20]}]}]}
{"url":"Mathlib/Analysis/Complex/CauchyIntegral.lean","commit":"","full_name":"Complex.hasFPowerSeriesOnBall_of_differentiable_off_countable","start":[527,0],"end":[543,77],"file_path":"Mathlib/Analysis/Complex/CauchyIntegral.lean","tactics":[{"state_before":"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))","state_after":"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\nhw' : c + w ∈ ball c ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))","tactic":"have hw' : c + w ∈ ball c R := by\n      simpa only [add_mem_ball_iff_norm, ← coe_nnnorm, mem_emetric_ball_zero_iff,\n        NNReal.coe_lt_coe, ENNReal.coe_lt_coe] using hw","premises":[{"full_name":"ENNReal.coe_lt_coe","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[316,27],"def_end_pos":[316,37]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Metric.ball","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[350,4],"def_end_pos":[350,8]},{"full_name":"NNReal.coe_lt_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[327,25],"def_end_pos":[327,35]},{"full_name":"add_mem_ball_iff_norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[991,2],"def_end_pos":[991,13]},{"full_name":"coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[615,40],"def_end_pos":[615,50]},{"full_name":"mem_emetric_ball_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[704,2],"def_end_pos":[704,13]}]},{"state_before":"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\nhw' : c + w ∈ ball c ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))","state_after":"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\nhw' : c + w ∈ ball c ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w)\n    ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)","tactic":"rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable\n      hs hw' hc hd]","premises":[{"full_name":"Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable","def_path":"Mathlib/Analysis/Complex/CauchyIntegral.lean","def_pos":[443,8],"def_end_pos":[443,88]}]},{"state_before":"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\nhw' : c + w ∈ ball c ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w)\n    ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)","state_after":"no goals","tactic":"exact (hasFPowerSeriesOn_cauchy_integral\n      ((hc.mono sphere_subset_closedBall).circleIntegrable R.2) hR).hasSum hw","premises":[{"full_name":"ContinuousOn.circleIntegrable","def_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","def_pos":[257,8],"def_end_pos":[257,37]},{"full_name":"ContinuousOn.mono","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[798,8],"def_end_pos":[798,25]},{"full_name":"HasFPowerSeriesOnBall.hasSum","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[354,2],"def_end_pos":[354,8]},{"full_name":"Metric.sphere_subset_closedBall","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[450,8],"def_end_pos":[450,32]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"hasFPowerSeriesOn_cauchy_integral","def_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","def_pos":[565,8],"def_end_pos":[565,41]}]}]}
{"url":"Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean","commit":"","full_name":"is_bot_adic_iff","start":[193,0],"end":[205,35],"file_path":"Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : TopologicalRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalRing A\n⊢ IsAdic ⊥ ↔ DiscreteTopology A","state_after":"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : TopologicalRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalRing A\n⊢ ((∀ (n : ℕ), IsOpen ↑(⊥ ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s) ↔ DiscreteTopology A","tactic":"rw [isAdic_iff]","premises":[{"full_name":"isAdic_iff","def_path":"Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean","def_pos":[146,8],"def_end_pos":[146,18]}]},{"state_before":"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : TopologicalRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalRing A\n⊢ ((∀ (n : ℕ), IsOpen ↑(⊥ ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s) ↔ DiscreteTopology A","state_after":"case mp\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : TopologicalRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalRing A\n⊢ ((∀ (n : ℕ), IsOpen ↑(⊥ ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s) → DiscreteTopology A\n\ncase mpr\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : TopologicalRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalRing A\n⊢ DiscreteTopology A → (∀ (n : ℕ), IsOpen ↑(⊥ ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Deriv.lean","commit":"","full_name":"StrictMonoOn.exists_slope_lt_deriv_aux","start":[63,0],"end":[73,52],"file_path":"Mathlib/Analysis/Convex/Deriv.lean","tactics":[{"state_before":"x y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","state_after":"x y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","tactic":"have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>\n    (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt","premises":[{"full_name":"DifferentiableAt.differentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[576,8],"def_end_pos":[576,47]},{"full_name":"DifferentiableOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[188,4],"def_end_pos":[188,20]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Ioo","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[42,4],"def_end_pos":[42,7]},{"full_name":"differentiableAt_of_deriv_ne_zero","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[227,8],"def_end_pos":[227,41]}]},{"state_before":"x y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","state_after":"case intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","tactic":"obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=\n    exists_deriv_eq_slope f hxy hf A","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.Ioo","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[42,4],"def_end_pos":[42,7]},{"full_name":"deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[143,4],"def_end_pos":[143,9]},{"full_name":"exists_deriv_eq_slope","def_path":"Mathlib/Analysis/Calculus/MeanValue.lean","def_pos":[746,8],"def_end_pos":[746,29]}]},{"state_before":"case intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","state_after":"case intro.intro.intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhab : a < b\nhby : b < y\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","tactic":"rcases nonempty_Ioo.2 hay with ⟨b, ⟨hab, hby⟩⟩","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.nonempty_Ioo","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[234,8],"def_end_pos":[234,20]}]},{"state_before":"case intro.intro.intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhab : a < b\nhby : b < y\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a","state_after":"case intro.intro.intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhab : a < b\nhby : b < y\n⊢ (f y - f x) / (y - x) < deriv f b","tactic":"refine ⟨b, ⟨hxa.trans hab, hby⟩, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro.intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhab : a < b\nhby : b < y\n⊢ (f y - f x) / (y - x) < deriv f b","state_after":"case intro.intro.intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhab : a < b\nhby : b < y\n⊢ deriv f a < deriv f b","tactic":"rw [← ha]","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhab : a < b\nhby : b < y\n⊢ deriv f a < deriv f b","state_after":"no goals","tactic":"exact hf'_mono ⟨hxa, hay⟩ ⟨hxa.trans hab, hby⟩ hab","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]}]}
{"url":"Mathlib/Data/PFunctor/Univariate/Basic.lean","commit":"","full_name":"PFunctor.W.dest_mk","start":[109,0],"end":[110,72],"file_path":"Mathlib/Data/PFunctor/Univariate/Basic.lean","tactics":[{"state_before":"P : PFunctor.{u}\nα : Type v₁\nβ : Type v₂\nγ : Type v₃\np : ↑P P.W\n⊢ (mk p).dest = p","state_after":"case mk\nP : PFunctor.{u}\nα : Type v₁\nβ : Type v₂\nγ : Type v₃\nfst✝ : P.A\nsnd✝ : P.B fst✝ → P.W\n⊢ (mk ⟨fst✝, snd✝⟩).dest = ⟨fst✝, snd✝⟩","tactic":"cases p","premises":[]},{"state_before":"case mk\nP : PFunctor.{u}\nα : Type v₁\nβ : Type v₂\nγ : Type v₃\nfst✝ : P.A\nsnd✝ : P.B fst✝ → P.W\n⊢ (mk ⟨fst✝, snd✝⟩).dest = ⟨fst✝, snd✝⟩","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Combinatorics/Schnirelmann.lean","commit":"","full_name":"le_schnirelmannDensity_iff","start":[140,0],"end":[142,78],"file_path":"Mathlib/Combinatorics/Schnirelmann.lean","tactics":[{"state_before":"A : Set ℕ\ninst✝ : DecidablePred fun x => x ∈ A\nx x✝¹ : ℝ\nx✝ : x✝¹ ∈ Set.range fun n => ↑(filter (fun x => x ∈ A) (Ioc 0 ↑n)).card / ↑↑n\nw✝ : { n // 0 < n }\nhx : (fun n => ↑(filter (fun x => x ∈ A) (Ioc 0 ↑n)).card / ↑↑n) w✝ = x✝¹\n⊢ 0 ≤ (fun n => ↑(filter (fun x => x ∈ A) (Ioc 0 ↑n)).card / ↑↑n) w✝","state_after":"no goals","tactic":"positivity","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Real.sin_pi","start":[203,0],"end":[205,97],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"⊢ sin π = 0","state_after":"⊢ sin (π / 2) * 0 + 0 * sin (π / 2) = 0","tactic":"rw [← mul_div_cancel_left₀ π (two_ne_zero' ℝ), two_mul, add_div, sin_add, cos_pi_div_two]","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.cos_pi_div_two","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[126,8],"def_end_pos":[126,22]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"Real.sin_add","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[728,15],"def_end_pos":[728,22]},{"full_name":"add_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[27,8],"def_end_pos":[27,15]},{"full_name":"mul_div_cancel_left₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[172,14],"def_end_pos":[172,34]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]},{"full_name":"two_ne_zero'","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[68,6],"def_end_pos":[68,18]}]},{"state_before":"⊢ sin (π / 2) * 0 + 0 * sin (π / 2) = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Sym/Sym2.lean","commit":"","full_name":"Sym2.filter_image_mk_not_isDiag","start":[663,0],"end":[675,36],"file_path":"Mathlib/Data/Sym/Sym2.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\n⊢ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) = image Sym2.mk s.offDiag","state_after":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nz : Sym2 α\n⊢ z ∈ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) ↔ z ∈ image Sym2.mk s.offDiag","tactic":"ext z","premises":[]},{"state_before":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nz : Sym2 α\n⊢ z ∈ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) ↔ z ∈ image Sym2.mk s.offDiag","state_after":"case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ s(x✝, y✝) ∈ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) ↔ s(x✝, y✝) ∈ image Sym2.mk s.offDiag","tactic":"induction z","premises":[]},{"state_before":"case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ s(x✝, y✝) ∈ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) ↔ s(x✝, y✝) ∈ image Sym2.mk s.offDiag","state_after":"case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ (∃ a b, (a ∈ s ∧ b ∈ s) ∧ s(a, b) = s(x✝, y✝)) ∧ ¬s(x✝, y✝).IsDiag ↔\n    ∃ a b, (a ∈ s ∧ b ∈ s ∧ a ≠ b) ∧ s(a, b) = s(x✝, y✝)","tactic":"simp only [mem_image, mem_offDiag, mem_filter, Prod.exists, mem_product]","premises":[{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Finset.mem_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[303,8],"def_end_pos":[303,17]},{"full_name":"Finset.mem_offDiag","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[267,8],"def_end_pos":[267,19]},{"full_name":"Finset.mem_product","def_path":"Mathlib/Data/Finset/Prod.lean","def_pos":[50,8],"def_end_pos":[50,19]},{"full_name":"Prod.exists","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[32,8],"def_end_pos":[32,16]}]},{"state_before":"case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ (∃ a b, (a ∈ s ∧ b ∈ s) ∧ s(a, b) = s(x✝, y✝)) ∧ ¬s(x✝, y✝).IsDiag ↔\n    ∃ a b, (a ∈ s ∧ b ∈ s ∧ a ≠ b) ∧ s(a, b) = s(x✝, y✝)","state_after":"case a.h.mp\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ (∃ a b, (a ∈ s ∧ b ∈ s) ∧ s(a, b) = s(x✝, y✝)) ∧ ¬s(x✝, y✝).IsDiag →\n    ∃ a b, (a ∈ s ∧ b ∈ s ∧ a ≠ b) ∧ s(a, b) = s(x✝, y✝)\n\ncase a.h.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ (∃ a b, (a ∈ s ∧ b ∈ s ∧ a ≠ b) ∧ s(a, b) = s(x✝, y✝)) →\n    (∃ a b, (a ∈ s ∧ b ∈ s) ∧ s(a, b) = s(x✝, y✝)) ∧ ¬s(x✝, y✝).IsDiag","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Discrete.lean","commit":"","full_name":"CategoryTheory.Discrete.monoidal_associator","start":[26,0],"end":[35,56],"file_path":"Mathlib/CategoryTheory/Monoidal/Discrete.lean","tactics":[{"state_before":"M : Type u\ninst✝ : Monoid M\nX x✝¹ x✝ : Discrete M\nf : x✝¹ ⟶ x✝\n⊢ (fun X Y => { as := X.as * Y.as }) X x✝¹ = (fun X Y => { as := X.as * Y.as }) X x✝","state_after":"M : Type u\ninst✝ : Monoid M\nX x✝¹ x✝ : Discrete M\nf : x✝¹ ⟶ x✝\n⊢ { as := X.as * x✝¹.as } = { as := X.as * x✝.as }","tactic":"dsimp","premises":[]},{"state_before":"M : Type u\ninst✝ : Monoid M\nX x✝¹ x✝ : Discrete M\nf : x✝¹ ⟶ x✝\n⊢ { as := X.as * x✝¹.as } = { as := X.as * x✝.as }","state_after":"no goals","tactic":"rw [eq_of_hom f]","premises":[{"full_name":"CategoryTheory.Discrete.eq_of_hom","def_path":"Mathlib/CategoryTheory/DiscreteCategory.lean","def_pos":[126,8],"def_end_pos":[126,17]}]},{"state_before":"M : Type u\ninst✝ : Monoid M\nX₁✝ X₂✝ : Discrete M\nf : X₁✝ ⟶ X₂✝\nX : Discrete M\n⊢ (fun X Y => { as := X.as * Y.as }) X₁✝ X = (fun X Y => { as := X.as * Y.as }) X₂✝ X","state_after":"M : Type u\ninst✝ : Monoid M\nX₁✝ X₂✝ : Discrete M\nf : X₁✝ ⟶ X₂✝\nX : Discrete M\n⊢ { as := X₁✝.as * X.as } = { as := X₂✝.as * X.as }","tactic":"dsimp","premises":[]},{"state_before":"M : Type u\ninst✝ : Monoid M\nX₁✝ X₂✝ : Discrete M\nf : X₁✝ ⟶ X₂✝\nX : Discrete M\n⊢ { as := X₁✝.as * X.as } = { as := X₂✝.as * X.as }","state_after":"no goals","tactic":"rw [eq_of_hom f]","premises":[{"full_name":"CategoryTheory.Discrete.eq_of_hom","def_path":"Mathlib/CategoryTheory/DiscreteCategory.lean","def_pos":[126,8],"def_end_pos":[126,17]}]},{"state_before":"M : Type u\ninst✝ : Monoid M\nX₁✝ Y₁✝ X₂✝ Y₂✝ : Discrete M\nf : X₁✝ ⟶ Y₁✝\ng : X₂✝ ⟶ Y₂✝\n⊢ (fun X Y => { as := X.as * Y.as }) X₁✝ X₂✝ = (fun X Y => { as := X.as * Y.as }) Y₁✝ Y₂✝","state_after":"M : Type u\ninst✝ : Monoid M\nX₁✝ Y₁✝ X₂✝ Y₂✝ : Discrete M\nf : X₁✝ ⟶ Y₁✝\ng : X₂✝ ⟶ Y₂✝\n⊢ { as := X₁✝.as * X₂✝.as } = { as := Y₁✝.as * Y₂✝.as }","tactic":"dsimp","premises":[]},{"state_before":"M : Type u\ninst✝ : Monoid M\nX₁✝ Y₁✝ X₂✝ Y₂✝ : Discrete M\nf : X₁✝ ⟶ Y₁✝\ng : X₂✝ ⟶ Y₂✝\n⊢ { as := X₁✝.as * X₂✝.as } = { as := Y₁✝.as * Y₂✝.as }","state_after":"no goals","tactic":"rw [eq_of_hom f, eq_of_hom g]","premises":[{"full_name":"CategoryTheory.Discrete.eq_of_hom","def_path":"Mathlib/CategoryTheory/DiscreteCategory.lean","def_pos":[126,8],"def_end_pos":[126,17]}]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean","commit":"","full_name":"CstarRing.conjugate_le_norm_smul'","start":[162,0],"end":[167,30],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean","tactics":[{"state_before":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\n⊢ a * b * star a ≤ ‖b‖ • (a * star a)","state_after":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\n⊢ a * b * star a ≤ ‖b‖ • (a * star a)","tactic":"have h₁ : a * b * star a = star (star a) * b * star a := by simp","premises":[{"full_name":"Star.star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[46,2],"def_end_pos":[46,6]}]},{"state_before":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\n⊢ a * b * star a ≤ ‖b‖ • (a * star a)","state_after":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\nh₂ : a * star a = star (star a) * star a\n⊢ a * b * star a ≤ ‖b‖ • (a * star a)","tactic":"have h₂ : a * star a = star (star a) * star a := by simp","premises":[{"full_name":"Star.star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[46,2],"def_end_pos":[46,6]}]},{"state_before":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\nh₂ : a * star a = star (star a) * star a\n⊢ a * b * star a ≤ ‖b‖ • (a * star a)","state_after":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\nh₂ : a * star a = star (star a) * star a\n⊢ star (star a) * b * star a ≤ ‖b‖ • (star (star a) * star a)","tactic":"simp only [h₁, h₂]","premises":[]},{"state_before":"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\nh₂ : a * star a = star (star a) * star a\n⊢ star (star a) * b * star a ≤ ‖b‖ • (star (star a) * star a)","state_after":"no goals","tactic":"exact conjugate_le_norm_smul","premises":[{"full_name":"CstarRing.conjugate_le_norm_smul","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean","def_pos":[151,6],"def_end_pos":[151,38]}]}]}
{"url":"Mathlib/Analysis/Convex/Body.lean","commit":"","full_name":"ConvexBody.iInter_smul_eq_self","start":[197,0],"end":[219,48],"file_path":"Mathlib/Analysis/Convex/Body.lean","tactics":[{"state_before":"V : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\nK✝ L : ConvexBody V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\n⊢ ⋂ n, (1 + ↑(u n)) • ↑K = ↑K","state_after":"case h\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\nK✝ L : ConvexBody V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\n⊢ x ∈ ⋂ n, (1 + ↑(u n)) • ↑K ↔ x ∈ ↑K","tactic":"ext x","premises":[]},{"state_before":"case h\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\nK✝ L : ConvexBody V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\n⊢ x ∈ ⋂ n, (1 + ↑(u n)) • ↑K ↔ x ∈ ↑K","state_after":"case h.refine_1\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\nK✝ L : ConvexBody V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\nh : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K\n⊢ x ∈ ↑K\n\ncase h.refine_2\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\nK✝ L : ConvexBody V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\nh : x ∈ ↑K\n⊢ x ∈ ⋂ n, (1 + ↑(u n)) • ↑K","tactic":"refine ⟨fun h => ?_, fun h => ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Data/Set/MulAntidiagonal.lean","commit":"","full_name":"Set.MulAntidiagonal.finite_of_isPWO","start":[98,0],"end":[109,63],"file_path":"Mathlib/Data/Set/MulAntidiagonal.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\n⊢ (s.mulAntidiagonal t a).Finite","state_after":"α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\n⊢ False","tactic":"refine not_infinite.1 fun h => ?_","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.not_infinite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[124,8],"def_end_pos":[124,20]}]},{"state_before":"α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\n⊢ False","state_after":"case intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ False","tactic":"obtain ⟨g, hg⟩ :=\n    h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2","premises":[{"full_name":"Set.Infinite.natEmbedding","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[1181,18],"def_end_pos":[1181,39]},{"full_name":"Set.PartiallyWellOrderedOn.exists_monotone_subseq","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[341,8],"def_end_pos":[341,53]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"case intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ False","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : ℕ\nmn : m < n\nh2' :\n  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ False","tactic":"obtain ⟨m, n, mn, h2'⟩ := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2","premises":[{"full_name":"Set.Infinite.natEmbedding","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[1181,18],"def_end_pos":[1181,39]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : ℕ\nmn : m < n\nh2' :\n  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ False","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : ℕ\nmn : m < n\nh2' :\n  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)","tactic":"refine mn.ne (g.injective <| (h.natEmbedding _).injective ?_)","premises":[{"full_name":"Function.Embedding.injective","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[124,18],"def_end_pos":[124,27]},{"full_name":"RelEmbedding.injective","def_path":"Mathlib/Order/RelIso/Basic.lean","def_pos":[227,8],"def_end_pos":[227,17]},{"full_name":"Set.Infinite.natEmbedding","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[1181,18],"def_end_pos":[1181,39]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : ℕ\nmn : m < n\nh2' :\n  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)","state_after":"no goals","tactic":"exact eq_of_fst_le_fst_of_snd_le_snd _ _ _ (hg _ _ mn.le) h2'","premises":[{"full_name":"Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd","def_path":"Mathlib/Data/Set/MulAntidiagonal.lean","def_pos":[89,8],"def_end_pos":[89,38]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Average.lean","commit":"","full_name":"MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub","start":[723,0],"end":[780,29],"file_path":"Mathlib/MeasureTheory/Integral/Average.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","tactic":"have g_int : ∀ᶠ i in l, Integrable (g i) μ := by\n    filter_upwards [(tendsto_order.1 hg).1 _ zero_lt_one] with i hi\n    contrapose hi\n    simp only [integral_undef hi, lt_self_iff_false, not_false_eq_true]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]},{"full_name":"MeasureTheory.Integrable","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[389,4],"def_end_pos":[389,14]},{"full_name":"MeasureTheory.integral_undef","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[754,8],"def_end_pos":[754,22]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"lt_self_iff_false","def_path":"Mathlib/Order/Basic.lean","def_pos":[155,8],"def_end_pos":[155,25]},{"full_name":"not_false_eq_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[134,16],"def_end_pos":[134,33]},{"full_name":"tendsto_order","def_path":"Mathlib/Topology/Order/Basic.lean","def_pos":[115,8],"def_end_pos":[115,21]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\nthis : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 (0 + 1 • c))\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","tactic":"have := L0.add (hg.smul_const c)","premises":[{"full_name":"Filter.Tendsto.add","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[115,2],"def_end_pos":[115,13]},{"full_name":"Filter.Tendsto.smul_const","def_path":"Mathlib/Topology/Algebra/MulAction.lean","def_pos":[98,8],"def_end_pos":[98,33]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\nthis : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 (0 + 1 • c))\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\nthis : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 c)\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","tactic":"simp only [one_smul, zero_add] at this","premises":[{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\nthis : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 c)\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)","state_after":"no goals","tactic":"exact Tendsto.congr' I this","premises":[{"full_name":"Filter.Tendsto.congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2634,8],"def_end_pos":[2634,22]}]}]}
{"url":"Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean","commit":"","full_name":"CategoryTheory.Adjunction.faithful_R_of_epi_counit_app","start":[133,0],"end":[138,19],"file_path":"Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ f = g","state_after":"case a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ h.counit.app X ≫ f = h.counit.app X ≫ g","tactic":"apply Epi.left_cancellation (f := h.counit.app X)","premises":[{"full_name":"CategoryTheory.Adjunction.counit","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[65,2],"def_end_pos":[65,8]},{"full_name":"CategoryTheory.Epi.left_cancellation","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[243,2],"def_end_pos":[243,19]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]}]},{"state_before":"case a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ h.counit.app X ≫ f = h.counit.app X ≫ g","state_after":"case a.a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ (h.homEquiv (R.obj X) Y) (h.counit.app X ≫ f) = (h.homEquiv (R.obj X) Y) (h.counit.app X ≫ g)","tactic":"apply (h.homEquiv (R.obj X) Y).injective","premises":[{"full_name":"CategoryTheory.Adjunction.homEquiv","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[61,2],"def_end_pos":[61,10]},{"full_name":"Equiv.injective","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[175,18],"def_end_pos":[175,27]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]}]},{"state_before":"case a.a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ (h.homEquiv (R.obj X) Y) (h.counit.app X ≫ f) = (h.homEquiv (R.obj X) Y) (h.counit.app X ≫ g)","state_after":"no goals","tactic":"simpa using hfg","premises":[]}]}
{"url":"Mathlib/Dynamics/PeriodicPts.lean","commit":"","full_name":"Function.IsPeriodicPt.const_mul","start":[115,0],"end":[116,40],"file_path":"Mathlib/Dynamics/PeriodicPts.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhm : IsPeriodicPt f m x\nn : ℕ\n⊢ IsPeriodicPt f (n * m) x","state_after":"no goals","tactic":"simp only [mul_comm n, hm.mul_const n]","premises":[{"full_name":"Function.IsPeriodicPt.mul_const","def_path":"Mathlib/Dynamics/PeriodicPts.lean","def_pos":[112,18],"def_end_pos":[112,27]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Restrict.lean","commit":"","full_name":"MeasureTheory.Measure.restrict_union_le","start":[254,0],"end":[256,80],"file_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","tactics":[{"state_before":"R : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s'✝ t✝ s s' t : Set α\nht : MeasurableSet t\n⊢ (μ.restrict (s ∪ s')) t ≤ (μ.restrict s + μ.restrict s') t","state_after":"no goals","tactic":"simpa [ht, inter_union_distrib_left] using measure_union_le (t ∩ s) (t ∩ s')","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"MeasureTheory.measure_union_le","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[80,8],"def_end_pos":[80,24]},{"full_name":"Set.inter_union_distrib_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[831,8],"def_end_pos":[831,32]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","commit":"","full_name":"HurwitzZeta.hasSum_int_oddKernel","start":[213,0],"end":[226,9],"file_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","tactics":[{"state_before":"a x : ℝ\nhx : 0 < x\n⊢ HasSum (fun n => (↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) (oddKernel (↑a) x)","state_after":"a x : ℝ\nhx : 0 < x\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))","tactic":"rw [← hasSum_ofReal, oddKernel_def' a x]","premises":[{"full_name":"Complex.hasSum_ofReal","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[574,8],"def_end_pos":[574,21]},{"full_name":"HurwitzZeta.oddKernel_def'","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[116,6],"def_end_pos":[116,20]}]},{"state_before":"a x : ℝ\nhx : 0 < x\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))","tactic":"have h1 := hasSum_jacobiTheta₂_term (a * I * x) (by rwa [I_mul_im, ofReal_re])","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.I_mul_im","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[264,8],"def_end_pos":[264,16]},{"full_name":"Complex.ofReal_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"hasSum_jacobiTheta₂_term","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[260,6],"def_end_pos":[260,30]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))","tactic":"have h2 := hasSum_jacobiTheta₂'_term (a * I * x) (by rwa [I_mul_im, ofReal_re])","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.I_mul_im","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[264,8],"def_end_pos":[264,16]},{"full_name":"Complex.ofReal_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"hasSum_jacobiTheta₂'_term","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[268,6],"def_end_pos":[268,31]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ ↑((↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) =\n    cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x))","tactic":"refine (((h2.div_const (2 * π * I)).add (h1.mul_left ↑a)).mul_left\n    (cexp (-π * a ^ 2 * x))).congr_fun (fun n ↦ ?_)","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[50,4],"def_end_pos":[50,7]},{"full_name":"HasSum.add","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[245,2],"def_end_pos":[245,13]},{"full_name":"HasSum.congr_fun","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[57,2],"def_end_pos":[57,13]},{"full_name":"HasSum.div_const","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Ring.lean","def_pos":[71,8],"def_end_pos":[71,24]},{"full_name":"HasSum.mul_left","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Ring.lean","def_pos":[32,8],"def_end_pos":[32,23]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ ↑((↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) =\n    cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x))","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ ↑((↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) =\n    (↑n + ↑a) * cexp (-↑π * ↑a ^ 2 * ↑x + (2 * ↑π * I * ↑n * (↑a * I * ↑x) + ↑π * I * ↑n ^ 2 * (I * ↑x)))","tactic":"rw [jacobiTheta₂'_term, mul_assoc (2 * π * I), mul_div_cancel_left₀ _ two_pi_I_ne_zero, ← add_mul,\n    mul_left_comm, jacobiTheta₂_term, ← Complex.exp_add]","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.exp_add","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[165,8],"def_end_pos":[165,15]},{"full_name":"Complex.two_pi_I_ne_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Complex/Log.lean","def_pos":[126,8],"def_end_pos":[126,24]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]},{"full_name":"jacobiTheta₂'_term","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[57,4],"def_end_pos":[57,22]},{"full_name":"jacobiTheta₂_term","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[40,4],"def_end_pos":[40,21]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_div_cancel_left₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[172,14],"def_end_pos":[172,34]},{"full_name":"mul_left_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[152,8],"def_end_pos":[152,21]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ ↑((↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) =\n    (↑n + ↑a) * cexp (-↑π * ↑a ^ 2 * ↑x + (2 * ↑π * I * ↑n * (↑a * I * ↑x) + ↑π * I * ↑n ^ 2 * (I * ↑x)))","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ (↑n + ↑a) * cexp (-↑π * (↑n + ↑a) ^ 2 * ↑x) =\n    (↑n + ↑a) * cexp (-↑π * ↑a ^ 2 * ↑x + (2 * ↑π * I * ↑n * (↑a * I * ↑x) + ↑π * I * ↑n ^ 2 * (I * ↑x)))","tactic":"push_cast","premises":[]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ (↑n + ↑a) * cexp (-↑π * (↑n + ↑a) ^ 2 * ↑x) =\n    (↑n + ↑a) * cexp (-↑π * ↑a ^ 2 * ↑x + (2 * ↑π * I * ↑n * (↑a * I * ↑x) + ↑π * I * ↑n ^ 2 * (I * ↑x)))","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ (↑n + ↑a) * cexp (-↑π * (↑n + ↑a) ^ 2 * ↑x) =\n    (↑n + ↑a) * cexp ((-↑π * ↑a ^ 2 + (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2) * I) * ↑x)","tactic":"simp only [← mul_assoc, ← add_mul]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ (↑n + ↑a) * cexp (-↑π * (↑n + ↑a) ^ 2 * ↑x) =\n    (↑n + ↑a) * cexp ((-↑π * ↑a ^ 2 + (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2) * I) * ↑x)","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ -↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2) * I","tactic":"congrm _ * cexp (?_ * x)","premises":[{"full_name":"Complex.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[50,4],"def_end_pos":[50,7]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ -↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2) * I","state_after":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ -↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * ↑n * ↑a * -1 + ↑π * ↑n ^ 2 * -1)","tactic":"simp only [mul_right_comm _ I, add_mul, mul_assoc _ I, I_mul_I]","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"Complex.I_mul_I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[243,8],"def_end_pos":[243,15]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[156,8],"def_end_pos":[156,22]}]},{"state_before":"a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ -↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * ↑n * ↑a * -1 + ↑π * ↑n ^ 2 * -1)","state_after":"no goals","tactic":"ring_nf","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/StrictConvexSpace.lean","commit":"","full_name":"norm_add_lt_of_not_sameRay","start":[163,0],"end":[176,43],"file_path":"Mathlib/Analysis/Convex/StrictConvexSpace.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : ¬SameRay ℝ x y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","state_after":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : x ≠ 0 ∧ y ≠ 0 ∧ ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","tactic":"simp only [sameRay_iff_inv_norm_smul_eq, not_or, ← Ne.eq_def] at h","premises":[{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]},{"full_name":"sameRay_iff_inv_norm_smul_eq","def_path":"Mathlib/Analysis/Normed/Module/Ray.lean","def_pos":[81,8],"def_end_pos":[81,36]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : x ≠ 0 ∧ y ≠ 0 ∧ ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","state_after":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","tactic":"rcases h with ⟨hx, hy, hne⟩","premises":[]},{"state_before":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","state_after":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","tactic":"rw [← norm_pos_iff] at hx hy","premises":[{"full_name":"norm_pos_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1216,29],"def_end_pos":[1216,41]}]},{"state_before":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","state_after":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","tactic":"have hxy : 0 < ‖x‖ + ‖y‖ := add_pos hx hy","premises":[{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]}]},{"state_before":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","state_after":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","tactic":"have :=\n    combo_mem_ball_of_ne (inv_norm_smul_mem_closed_unit_ball x)\n      (inv_norm_smul_mem_closed_unit_ball y) hne (div_pos hx hxy) (div_pos hy hxy)\n      (by rw [← add_div, div_self hxy.ne'])","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"add_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[27,8],"def_end_pos":[27,15]},{"full_name":"combo_mem_ball_of_ne","def_path":"Mathlib/Analysis/Convex/StrictConvexSpace.lean","def_pos":[142,8],"def_end_pos":[142,28]},{"full_name":"div_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[45,6],"def_end_pos":[45,13]},{"full_name":"div_self","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[251,14],"def_end_pos":[251,22]},{"full_name":"inv_norm_smul_mem_closed_unit_ball","def_path":"Mathlib/Analysis/NormedSpace/Real.lean","def_pos":[37,8],"def_end_pos":[37,42]}]},{"state_before":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖","state_after":"no goals","tactic":"rwa [mem_ball_zero_iff, div_eq_inv_mul, div_eq_inv_mul, mul_smul, mul_smul, smul_inv_smul₀ hx.ne',\n    smul_inv_smul₀ hy.ne', ← smul_add, norm_smul, Real.norm_of_nonneg (inv_pos.2 hxy).le, ←\n    div_eq_inv_mul, div_lt_one hxy] at this","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"Real.norm_of_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1138,8],"def_end_pos":[1138,22]},{"full_name":"div_eq_inv_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[522,8],"def_end_pos":[522,22]},{"full_name":"div_lt_one","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[303,8],"def_end_pos":[303,18]},{"full_name":"inv_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[23,14],"def_end_pos":[23,21]},{"full_name":"mem_ball_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[513,2],"def_end_pos":[513,13]},{"full_name":"norm_smul","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[79,8],"def_end_pos":[79,17]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]},{"full_name":"smul_inv_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[39,8],"def_end_pos":[39,22]}]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Supported.lean","commit":"","full_name":"MvPolynomial.supported_univ","start":[87,0],"end":[89,42],"file_path":"Mathlib/Algebra/MvPolynomial/Supported.lean","tactics":[{"state_before":"σ : Type u_1\nτ : Type u_2\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ supported R Set.univ = ⊤","state_after":"no goals","tactic":"simp [Algebra.eq_top_iff, mem_supported]","premises":[{"full_name":"Algebra.eq_top_iff","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[737,8],"def_end_pos":[737,18]},{"full_name":"MvPolynomial.mem_supported","def_path":"Mathlib/Algebra/MvPolynomial/Supported.lean","def_pos":[66,8],"def_end_pos":[66,21]}]}]}
{"url":"Mathlib/Topology/GDelta.lean","commit":"","full_name":"IsGδ.sInter","start":[117,0],"end":[119,56],"file_path":"Mathlib/Topology/GDelta.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nι : Type u_3\nι' : Sort u_4\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nh : ∀ s ∈ S, IsGδ s\nhS : S.Countable\n⊢ IsGδ (⋂₀ S)","state_after":"no goals","tactic":"simpa only [sInter_eq_biInter] using IsGδ.biInter hS h","premises":[{"full_name":"IsGδ.biInter","def_path":"Mathlib/Topology/GDelta.lean","def_pos":[109,8],"def_end_pos":[109,20]},{"full_name":"Set.sInter_eq_biInter","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1093,8],"def_end_pos":[1093,25]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","commit":"","full_name":"Real.contDiffOn_log","start":[69,0],"end":[72,45],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","tactics":[{"state_before":"x : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ n log {0}ᶜ","state_after":"x : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log {0}ᶜ","tactic":"suffices ContDiffOn ℝ ⊤ log {0}ᶜ from this.of_le le_top","premises":[{"full_name":"ContDiffOn","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[596,4],"def_end_pos":[596,14]},{"full_name":"ContDiffOn.of_le","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[633,8],"def_end_pos":[633,24]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Real.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[39,18],"def_end_pos":[39,21]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[62,8],"def_end_pos":[62,14]}]},{"state_before":"x : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log {0}ᶜ","state_after":"x : ℝ\nn : ℕ∞\n⊢ DifferentiableOn ℝ log {0}ᶜ ∧ ContDiffOn ℝ ⊤ (deriv log) {0}ᶜ","tactic":"refine (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"contDiffOn_top_iff_deriv_of_isOpen","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1850,8],"def_end_pos":[1850,42]},{"full_name":"isOpen_compl_singleton","def_path":"Mathlib/Topology/Separation.lean","def_pos":[527,8],"def_end_pos":[527,30]}]},{"state_before":"x : ℝ\nn : ℕ∞\n⊢ DifferentiableOn ℝ log {0}ᶜ ∧ ContDiffOn ℝ ⊤ (deriv log) {0}ᶜ","state_after":"no goals","tactic":"simp [differentiableOn_log, contDiffOn_inv]","premises":[{"full_name":"Real.differentiableOn_log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean","def_pos":[53,8],"def_end_pos":[53,28]},{"full_name":"contDiffOn_inv","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1597,8],"def_end_pos":[1597,22]}]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieIdeal.incl_idealRange","start":[1208,0],"end":[1213,7],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ I.incl.idealRange = I","state_after":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ∃ N, ↑N = ↑I","tactic":"rw [LieHom.idealRange_eq_lieSpan_range, ← LieSubalgebra.coe_to_submodule, ←\n    LieSubmodule.coe_toSubmodule_eq_iff, incl_range, coe_to_lieSubalgebra_to_submodule,\n    LieSubmodule.coe_lieSpan_submodule_eq_iff]","premises":[{"full_name":"LieHom.idealRange_eq_lieSpan_range","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1002,8],"def_end_pos":[1002,35]},{"full_name":"LieIdeal.coe_to_lieSubalgebra_to_submodule","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[229,8],"def_end_pos":[229,50]},{"full_name":"LieIdeal.incl_range","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1182,8],"def_end_pos":[1182,18]},{"full_name":"LieSubalgebra.coe_to_submodule","def_path":"Mathlib/Algebra/Lie/Subalgebra.lean","def_pos":[198,8],"def_end_pos":[198,24]},{"full_name":"LieSubmodule.coe_lieSpan_submodule_eq_iff","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[675,8],"def_end_pos":[675,36]},{"full_name":"LieSubmodule.coe_toSubmodule_eq_iff","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[135,8],"def_end_pos":[135,30]}]},{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ∃ N, ↑N = ↑I","state_after":"no goals","tactic":"use I","premises":[]}]}
{"url":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","commit":"","full_name":"dvd_sub_left","start":[110,0],"end":[114,75],"file_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalRing α\na b c : α\nh : a ∣ c\n⊢ a ∣ b - c ↔ a ∣ b","state_after":"no goals","tactic":"simpa only [← sub_eq_add_neg] using dvd_add_left ((dvd_neg (α := α)).2 h)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"dvd_add_left","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[103,8],"def_end_pos":[103,20]},{"full_name":"dvd_neg","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[75,8],"def_end_pos":[75,15]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/Algebra/GeomSum.lean","commit":"","full_name":"zero_geom_sum","start":[61,0],"end":[67,24],"file_path":"Mathlib/Algebra/GeomSum.lean","tactics":[{"state_before":"α : Type u\ninst✝ : Semiring α\n⊢ ∑ i ∈ range 0, 0 ^ i = if 0 = 0 then 0 else 1","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u\ninst✝ : Semiring α\n⊢ ∑ i ∈ range 1, 0 ^ i = if 1 = 0 then 0 else 1","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u\ninst✝ : Semiring α\nn : ℕ\n⊢ ∑ i ∈ range (n + 2), 0 ^ i = if n + 2 = 0 then 0 else 1","state_after":"α : Type u\ninst✝ : Semiring α\nn : ℕ\n⊢ 0 ^ (n + 1) + ∑ i ∈ range (n + 1), 0 ^ i = if n + 2 = 0 then 0 else 1","tactic":"rw [geom_sum_succ']","premises":[{"full_name":"geom_sum_succ'","def_path":"Mathlib/Algebra/GeomSum.lean","def_pos":[49,8],"def_end_pos":[49,22]}]},{"state_before":"α : Type u\ninst✝ : Semiring α\nn : ℕ\n⊢ 0 ^ (n + 1) + ∑ i ∈ range (n + 1), 0 ^ i = if n + 2 = 0 then 0 else 1","state_after":"no goals","tactic":"simp [zero_geom_sum]","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Complex.sin_pi","start":[958,0],"end":[959,69],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"⊢ sin ↑π = 0","state_after":"⊢ ↑0 = 0","tactic":"rw [← ofReal_sin, Real.sin_pi]","premises":[{"full_name":"Complex.ofReal_sin","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[523,8],"def_end_pos":[523,18]},{"full_name":"Real.sin_pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[204,8],"def_end_pos":[204,14]}]},{"state_before":"⊢ ↑0 = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Add.lean","commit":"","full_name":"derivWithin_const_sub","start":[314,0],"end":[316,48],"file_path":"Mathlib/Analysis/Calculus/Deriv/Add.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhxs : UniqueDiffWithinAt 𝕜 s x\nc : F\n⊢ derivWithin (fun y => c - f y) s x = -derivWithin f s x","state_after":"no goals","tactic":"simp [derivWithin, fderivWithin_const_sub hxs]","premises":[{"full_name":"derivWithin","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[135,4],"def_end_pos":[135,15]},{"full_name":"fderivWithin_const_sub","def_path":"Mathlib/Analysis/Calculus/FDeriv/Add.lean","def_pos":[663,8],"def_end_pos":[663,30]}]}]}
{"url":"Mathlib/LinearAlgebra/Dimension/Basic.lean","commit":"","full_name":"LinearMap.rank_le_of_surjective","start":[330,0],"end":[333,21],"file_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","tactics":[{"state_before":"R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M₁\nh : Surjective ⇑f\n⊢ Module.rank R M₁ ≤ Module.rank R M","state_after":"R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M₁\nh : Surjective ⇑f\n⊢ Module.rank R ↥(range f) ≤ Module.rank R M","tactic":"rw [← rank_range_of_surjective f h]","premises":[{"full_name":"rank_range_of_surjective","def_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","def_pos":[317,8],"def_end_pos":[317,32]}]},{"state_before":"R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M₁\nh : Surjective ⇑f\n⊢ Module.rank R ↥(range f) ≤ Module.rank R M","state_after":"no goals","tactic":"apply rank_range_le","premises":[{"full_name":"rank_range_le","def_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","def_pos":[264,8],"def_end_pos":[264,21]}]}]}
{"url":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","commit":"","full_name":"MeasureTheory.AEStronglyMeasurable.piecewise","start":[1552,0],"end":[1571,81],"file_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhs : MeasurableSet s\nhf : AEStronglyMeasurable f (μ.restrict s)\nhg : AEStronglyMeasurable g (μ.restrict sᶜ)\n⊢ AEStronglyMeasurable (s.piecewise f g) μ","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhs : MeasurableSet s\nhf : AEStronglyMeasurable f (μ.restrict s)\nhg : AEStronglyMeasurable g (μ.restrict sᶜ)\n⊢ s.piecewise f g =ᶠ[ae μ] s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg)","tactic":"refine ⟨s.piecewise (hf.mk f) (hg.mk g),\n    StronglyMeasurable.piecewise hs hf.stronglyMeasurable_mk hg.stronglyMeasurable_mk, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"MeasureTheory.AEStronglyMeasurable.mk","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1111,28],"def_end_pos":[1111,30]},{"full_name":"MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1114,8],"def_end_pos":[1114,29]},{"full_name":"MeasureTheory.StronglyMeasurable.piecewise","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[721,18],"def_end_pos":[721,27]},{"full_name":"Set.piecewise","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[884,4],"def_end_pos":[884,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhs : MeasurableSet s\nhf : AEStronglyMeasurable f (μ.restrict s)\nhg : AEStronglyMeasurable g (μ.restrict sᶜ)\n⊢ s.piecewise f g =ᶠ[ae μ] s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg)","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhs : MeasurableSet s\nhf : AEStronglyMeasurable f (μ.restrict s)\nhg : AEStronglyMeasurable g (μ.restrict sᶜ)\n⊢ ∀ᵐ (x : α) ∂μ.restrict s,\n    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhs : MeasurableSet s\nhf : AEStronglyMeasurable f (μ.restrict s)\nhg : AEStronglyMeasurable g (μ.restrict sᶜ)\n⊢ ∀ᵐ (x : α) ∂μ.restrict sᶜ,\n    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x","tactic":"refine ae_of_ae_restrict_of_ae_restrict_compl s ?_ ?_","premises":[{"full_name":"MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[571,8],"def_end_pos":[571,46]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Uniform.lean","commit":"","full_name":"dist_self_add_right","start":[186,0],"end":[188,54],"file_path":"Mathlib/Analysis/Normed/Group/Uniform.lean","tactics":[{"state_before":"𝓕 : Type u_1\nα : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ dist a (a * b) = ‖b‖","state_after":"no goals","tactic":"rw [← dist_one_left, ← dist_mul_left a 1 b, mul_one]","premises":[{"full_name":"dist_mul_left","def_path":"Mathlib/Topology/MetricSpace/IsometricSMul.lean","def_pos":[258,8],"def_end_pos":[258,21]},{"full_name":"dist_one_left","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[368,8],"def_end_pos":[368,21]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean","commit":"","full_name":"Real.Angle.arg_expMapCircle","start":[125,0],"end":[130,78],"file_path":"Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean","tactics":[{"state_before":"θ : Angle\n⊢ ↑(↑θ.expMapCircle).arg = θ","state_after":"case h\nx✝ : ℝ\n⊢ ↑(↑(↑x✝).expMapCircle).arg = ↑x✝","tactic":"induction θ using Real.Angle.induction_on","premises":[{"full_name":"Real.Angle.induction_on","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[68,18],"def_end_pos":[68,30]}]},{"state_before":"case h\nx✝ : ℝ\n⊢ ↑(↑(↑x✝).expMapCircle).arg = ↑x✝","state_after":"no goals","tactic":"rw [Real.Angle.expMapCircle_coe, expMapCircle_apply, exp_mul_I, ← ofReal_cos, ← ofReal_sin, ←\n    Real.Angle.cos_coe, ← Real.Angle.sin_coe, arg_cos_add_sin_mul_I_coe_angle]","premises":[{"full_name":"Complex.arg_cos_add_sin_mul_I_coe_angle","def_path":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","def_pos":[464,8],"def_end_pos":[464,39]},{"full_name":"Complex.exp_mul_I","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[644,8],"def_end_pos":[644,17]},{"full_name":"Complex.ofReal_cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[540,8],"def_end_pos":[540,18]},{"full_name":"Complex.ofReal_sin","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[523,8],"def_end_pos":[523,18]},{"full_name":"Real.Angle.cos_coe","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[278,8],"def_end_pos":[278,15]},{"full_name":"Real.Angle.expMapCircle_coe","def_path":"Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean","def_pos":[100,8],"def_end_pos":[100,35]},{"full_name":"Real.Angle.sin_coe","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[266,8],"def_end_pos":[266,15]},{"full_name":"expMapCircle_apply","def_path":"Mathlib/Analysis/Complex/Circle.lean","def_pos":[109,8],"def_end_pos":[109,26]}]}]}
{"url":"Mathlib/RingTheory/Trace/Basic.lean","commit":"","full_name":"Algebra.trace_eq_of_equiv_equiv","start":[186,0],"end":[194,5],"file_path":"Mathlib/RingTheory/Trace/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\n⊢ (trace A₁ B₁) x = e₁.symm ((trace A₂ B₂) (e₂ x))","state_after":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\nthis : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra\n⊢ (trace A₁ B₁) x = e₁.symm ((trace A₂ B₂) (e₂ x))","tactic":"letI := (RingHom.comp (e₂ : B₁ →+* B₂) (algebraMap A₁ B₁)).toAlgebra","premises":[{"full_name":"RingHom","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[297,10],"def_end_pos":[297,17]},{"full_name":"RingHom.comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[563,4],"def_end_pos":[563,8]},{"full_name":"RingHom.toAlgebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[198,4],"def_end_pos":[198,21]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\nthis : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra\n⊢ (trace A₁ B₁) x = e₁.symm ((trace A₂ B₂) (e₂ x))","state_after":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\nthis : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra\ne' : B₁ ≃ₐ[A₁] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }\n⊢ (trace A₁ B₁) x = e₁.symm ((trace A₂ B₂) (e₂ x))","tactic":"let e' : B₁ ≃ₐ[A₁] B₂ := { e₂ with commutes' := fun _ ↦ rfl }","premises":[{"full_name":"AlgEquiv","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[26,10],"def_end_pos":[26,18]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\nthis : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra\ne' : B₁ ≃ₐ[A₁] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }\n⊢ (trace A₁ B₁) x = e₁.symm ((trace A₂ B₂) (e₂ x))","state_after":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\nthis : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra\ne' : B₁ ≃ₐ[A₁] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }\n⊢ (trace A₁ B₂) (e' x) = (trace A₁ B₂) (e₂ x)","tactic":"rw [← Algebra.trace_eq_of_ringEquiv e₁ he, ← Algebra.trace_eq_of_algEquiv e',\n    RingEquiv.symm_apply_apply]","premises":[{"full_name":"Algebra.trace_eq_of_algEquiv","def_path":"Mathlib/RingTheory/Trace/Basic.lean","def_pos":[160,6],"def_end_pos":[160,34]},{"full_name":"Algebra.trace_eq_of_ringEquiv","def_path":"Mathlib/RingTheory/Trace/Basic.lean","def_pos":[166,6],"def_end_pos":[166,35]},{"full_name":"RingEquiv.symm_apply_apply","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[312,8],"def_end_pos":[312,24]}]},{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : CommRing T\ninst✝¹⁶ : Algebra R S\ninst✝¹⁵ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\nι κ : Type w\ninst✝¹¹ : Fintype ι\nF : Type u_6\ninst✝¹⁰ : Field F\ninst✝⁹ : Algebra R L\ninst✝⁸ : Algebra L F\ninst✝⁷ : Algebra R F\ninst✝⁶ : IsScalarTower R L F\nA₁ : Type u_7\nB₁ : Type u_8\nA₂ : Type u_9\nB₂ : Type u_10\ninst✝⁵ : CommRing A₁\ninst✝⁴ : CommRing B₁\ninst✝³ : CommRing A₂\ninst✝² : CommRing B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx : B₁\nthis : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra\ne' : B₁ ≃ₐ[A₁] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }\n⊢ (trace A₁ B₂) (e' x) = (trace A₁ B₂) (e₂ x)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","commit":"","full_name":"lp.single_smul","start":[925,0],"end":[934,38],"file_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\n⊢ lp.single p i (c • a) = c • lp.single p i a","state_after":"α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j","tactic":"refine ext (funext (fun (j : α) => ?_))","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]},{"full_name":"lp.ext","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[311,8],"def_end_pos":[311,11]}]},{"state_before":"α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j","state_after":"case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : j = i\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j\n\ncase neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j","tactic":"by_cases hi : j = i","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.rightShiftAddEquiv_apply","start":[169,0],"end":[177,26],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn a n' : ℤ\nhn' : n' + a = n\nγ : Cochain K L n\n⊢ (fun γ => γ.rightUnshift n hn') ((fun γ => γ.rightShift a n' hn') γ) = γ","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn a n' : ℤ\nhn' : n' + a = n\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'\n⊢ (fun γ => γ.rightShift a n' hn') ((fun γ => γ.rightUnshift n hn') γ) = γ","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\nn a n' : ℤ\nhn' : n' + a = n\nγ γ' : Cochain K L n\n⊢ { toFun := fun γ => γ.rightShift a n' hn', invFun := fun γ => γ.rightUnshift n hn', left_inv := ⋯,\n          right_inv := ⋯ }.toFun\n      (γ + γ') =\n    { toFun := fun γ => γ.rightShift a n' hn', invFun := fun γ => γ.rightUnshift n hn', left_inv := ⋯,\n            right_inv := ⋯ }.toFun\n        γ +\n      { toFun := fun γ => γ.rightShift a n' hn', invFun := fun γ => γ.rightUnshift n hn', left_inv := ⋯,\n            right_inv := ⋯ }.toFun\n        γ'","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Geometry/Manifold/PartitionOfUnity.lean","commit":"","full_name":"SmoothBumpCovering.mem_extChartAt_source_of_eq_one","start":[413,0],"end":[415,65],"file_path":"Mathlib/Geometry/Manifold/PartitionOfUnity.lean","tactics":[{"state_before":"ι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\ns : Set M\nU : M → Set M\nfs : SmoothBumpCovering ι I M s\ni : ι\nx : M\nh : ↑(fs.toFun i) x = 1\n⊢ x ∈ (extChartAt I (fs.c i)).source","state_after":"ι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\ns : Set M\nU : M → Set M\nfs : SmoothBumpCovering ι I M s\ni : ι\nx : M\nh : ↑(fs.toFun i) x = 1\n⊢ x ∈ (chartAt H (fs.c i)).source","tactic":"rw [extChartAt_source]","premises":[{"full_name":"extChartAt_source","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[1042,8],"def_end_pos":[1042,25]}]},{"state_before":"ι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\ns : Set M\nU : M → Set M\nfs : SmoothBumpCovering ι I M s\ni : ι\nx : M\nh : ↑(fs.toFun i) x = 1\n⊢ x ∈ (chartAt H (fs.c i)).source","state_after":"no goals","tactic":"exact fs.mem_chartAt_source_of_eq_one h","premises":[{"full_name":"SmoothBumpCovering.mem_chartAt_source_of_eq_one","def_path":"Mathlib/Geometry/Manifold/PartitionOfUnity.lean","def_pos":[409,8],"def_end_pos":[409,36]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Add.lean","commit":"","full_name":"differentiable_add_const_iff","start":[206,0],"end":[209,68],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Add.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nh : Differentiable 𝕜 fun y => f y + c\n⊢ Differentiable 𝕜 f","state_after":"no goals","tactic":"simpa using h.add_const (-c)","premises":[{"full_name":"Differentiable.add_const","def_path":"Mathlib/Analysis/Calculus/FDeriv/Add.lean","def_pos":[203,8],"def_end_pos":[203,32]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","commit":"","full_name":"HurwitzZeta.sinKernel_undef","start":[138,0],"end":[141,13],"file_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","tactics":[{"state_before":"a : UnitAddCircle\nx : ℝ\nhx : x ≤ 0\n⊢ sinKernel a x = 0","state_after":"case H\nx : ℝ\nhx : x ≤ 0\na' : ℝ\n⊢ sinKernel (↑a') x = 0","tactic":"induction' a using QuotientAddGroup.induction_on' with a'","premises":[{"full_name":"QuotientAddGroup.induction_on'","def_path":"Mathlib/GroupTheory/Coset.lean","def_pos":[374,2],"def_end_pos":[374,13]}]},{"state_before":"case H\nx : ℝ\nhx : x ≤ 0\na' : ℝ\n⊢ sinKernel (↑a') x = 0","state_after":"no goals","tactic":"rw [← ofReal_eq_zero, sinKernel_def, jacobiTheta₂'_undef _ (by rwa [I_mul_im, ofReal_re]),\n    zero_div]","premises":[{"full_name":"Complex.I_mul_im","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[264,8],"def_end_pos":[264,16]},{"full_name":"Complex.ofReal_eq_zero","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[136,8],"def_end_pos":[136,22]},{"full_name":"Complex.ofReal_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"HurwitzZeta.sinKernel_def","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[134,6],"def_end_pos":[134,19]},{"full_name":"jacobiTheta₂'_undef","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[282,6],"def_end_pos":[282,25]},{"full_name":"zero_div","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[298,8],"def_end_pos":[298,16]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","commit":"","full_name":"equicontinuous_iff_continuous","start":[499,0],"end":[505,90],"file_path":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\nα : Type u_7\nα' : Type u_8\nβ : Type u_9\nβ' : Type u_10\nγ : Type u_11\n𝓕 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nuα : UniformSpace α\nuβ : UniformSpace β\nuγ : UniformSpace γ\nF : ι → X → α\n⊢ Equicontinuous F ↔ Continuous (⇑UniformFun.ofFun ∘ swap F)","state_after":"no goals","tactic":"simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Equicontinuous","def_path":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","def_pos":[113,4],"def_end_pos":[113,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"continuous_iff_continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1427,8],"def_end_pos":[1427,35]},{"full_name":"equicontinuousAt_iff_continuousAt","def_path":"Mathlib/Topology/UniformSpace/Equicontinuity.lean","def_pos":[484,8],"def_end_pos":[484,41]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Basic.lean","commit":"","full_name":"Cardinal.mk_sUnion_le","start":[1748,0],"end":[1750,20],"file_path":"Mathlib/SetTheory/Cardinal/Basic.lean","tactics":[{"state_before":"α✝ β : Type u\nc : Cardinal.{?u.202561}\nα : Type u\nA : Set (Set α)\n⊢ #↑(⋃₀ A) ≤ #↑A * ⨆ s, #↑↑s","state_after":"α✝ β : Type u\nc : Cardinal.{?u.202561}\nα : Type u\nA : Set (Set α)\n⊢ #↑(⋃ i, ↑i) ≤ #↑A * ⨆ s, #↑↑s","tactic":"rw [sUnion_eq_iUnion]","premises":[{"full_name":"Set.sUnion_eq_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1096,8],"def_end_pos":[1096,24]}]},{"state_before":"α✝ β : Type u\nc : Cardinal.{?u.202561}\nα : Type u\nA : Set (Set α)\n⊢ #↑(⋃ i, ↑i) ≤ #↑A * ⨆ s, #↑↑s","state_after":"no goals","tactic":"apply mk_iUnion_le","premises":[{"full_name":"Cardinal.mk_iUnion_le","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1740,8],"def_end_pos":[1740,20]}]}]}
{"url":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","commit":"","full_name":"Measurable.exists","start":[1046,0],"end":[1048,96],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np✝ q : α → Prop\ninst✝ : Countable ι\np : ι → α → Prop\nhp : ∀ (i : ι), Measurable (p i)\n⊢ MeasurableSet {a | ∃ i, p i a}","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np✝ q : α → Prop\ninst✝ : Countable ι\np : ι → α → Prop\nhp : ∀ (i : ι), Measurable (p i)\n⊢ MeasurableSet (⋃ i, {x | p i x})","tactic":"rw [setOf_exists]","premises":[{"full_name":"Set.setOf_exists","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[207,8],"def_end_pos":[207,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np✝ q : α → Prop\ninst✝ : Countable ι\np : ι → α → Prop\nhp : ∀ (i : ι), Measurable (p i)\n⊢ MeasurableSet (⋃ i, {x | p i x})","state_after":"no goals","tactic":"exact MeasurableSet.iUnion fun i ↦ (hp i).setOf","premises":[{"full_name":"MeasurableSet.iUnion","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[106,18],"def_end_pos":[106,38]}]}]}
{"url":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","commit":"","full_name":"CategoryTheory.MorphismProperty.LeftFraction.Localization.homMk_comp_homMk","start":[495,0],"end":[499,40],"file_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.63018, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁ : W.LeftFraction X Y\nz₂ : W.LeftFraction Y Z\nz₃ : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\n⊢ homMk z₁ ≫ homMk z₂ = homMk (z₁.comp₀ z₂ z₃)","state_after":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.63018, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁ : W.LeftFraction X Y\nz₂ : W.LeftFraction Y Z\nz₃ : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\n⊢ Hom.comp (homMk z₁) (homMk z₂) = homMk (z₁.comp₀ z₂ z₃)","tactic":"change Hom.comp _ _ = _","premises":[{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.comp","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[339,18],"def_end_pos":[339,26]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{?u.63018, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁ : W.LeftFraction X Y\nz₂ : W.LeftFraction Y Z\nz₃ : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\n⊢ Hom.comp (homMk z₁) (homMk z₂) = homMk (z₁.comp₀ z₂ z₃)","state_after":"no goals","tactic":"erw [Hom.comp_eq, comp_eq z₁ z₂ z₃ h₃]","premises":[{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.comp_eq","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[418,6],"def_end_pos":[418,17]},{"full_name":"CategoryTheory.MorphismProperty.LeftFraction.comp_eq","def_path":"Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean","def_pos":[329,6],"def_end_pos":[329,13]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
{"url":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","commit":"","full_name":"ContMDiff.sub","start":[162,0],"end":[164,91],"file_path":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁸ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nF : Type u_4\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\nJ : ModelWithCorners 𝕜 F F\nG : Type u_5\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : ChartedSpace H G\ninst✝¹¹ : Group G\ninst✝¹⁰ : LieGroup I G\nE' : Type u_6\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H' M\nE'' : Type u_9\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_10\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM' : Type u_11\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H'' M'\nn : ℕ∞\nf g : M → G\nhf : ContMDiff I' I n f\nhg : ContMDiff I' I n g\n⊢ ContMDiff I' I n fun x => f x / g x","state_after":"𝕜 : Type u_1\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁸ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nF : Type u_4\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\nJ : ModelWithCorners 𝕜 F F\nG : Type u_5\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : ChartedSpace H G\ninst✝¹¹ : Group G\ninst✝¹⁰ : LieGroup I G\nE' : Type u_6\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H' M\nE'' : Type u_9\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_10\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM' : Type u_11\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H'' M'\nn : ℕ∞\nf g : M → G\nhf : ContMDiff I' I n f\nhg : ContMDiff I' I n g\n⊢ ContMDiff I' I n fun x => f x * (g x)⁻¹","tactic":"simp_rw [div_eq_mul_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁸ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nF : Type u_4\ninst✝¹⁵ : NormedAddCommGroup F\ninst✝¹⁴ : NormedSpace 𝕜 F\nJ : ModelWithCorners 𝕜 F F\nG : Type u_5\ninst✝¹³ : TopologicalSpace G\ninst✝¹² : ChartedSpace H G\ninst✝¹¹ : Group G\ninst✝¹⁰ : LieGroup I G\nE' : Type u_6\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝⁷ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H' M\nE'' : Type u_9\ninst✝⁴ : NormedAddCommGroup E''\ninst✝³ : NormedSpace 𝕜 E''\nH'' : Type u_10\ninst✝² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM' : Type u_11\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H'' M'\nn : ℕ∞\nf g : M → G\nhf : ContMDiff I' I n f\nhg : ContMDiff I' I n g\n⊢ ContMDiff I' I n fun x => f x * (g x)⁻¹","state_after":"no goals","tactic":"exact hf.mul hg.inv","premises":[{"full_name":"ContMDiff.inv","def_path":"Mathlib/Geometry/Manifold/Algebra/LieGroup.lean","def_pos":[124,8],"def_end_pos":[124,21]},{"full_name":"ContMDiff.mul","def_path":"Mathlib/Geometry/Manifold/Algebra/Monoid.lean","def_pos":[104,8],"def_end_pos":[104,21]}]}]}
{"url":"Mathlib/Algebra/Homology/Embedding/Restriction.lean","commit":"","full_name":"HomologicalComplex.restriction_d","start":[28,0],"end":[34,70],"file_path":"Mathlib/Algebra/Homology/Embedding/Restriction.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{?u.313, u_3} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nK L M : HomologicalComplex C c'\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij : ¬c.Rel i j\n⊢ ¬c'.Rel (e.f i) (e.f j)","state_after":"no goals","tactic":"simpa only [← e.rel_iff] using hij","premises":[{"full_name":"ComplexShape.Embedding.rel_iff","def_path":"Mathlib/Algebra/Homology/Embedding/Basic.lean","def_pos":[79,6],"def_end_pos":[79,13]}]}]}
{"url":"Mathlib/FieldTheory/Separable.lean","commit":"","full_name":"Associated.separable","start":[119,0],"end":[130,33],"file_path":"Mathlib/FieldTheory/Separable.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nha : Associated f g\nh : f.Separable\n⊢ g.Separable","state_after":"case intro.mk\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : f.Separable\nu v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\n⊢ g.Separable","tactic":"obtain ⟨⟨u, v, h1, h2⟩, ha⟩ := ha","premises":[]},{"state_before":"case intro.mk\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : f.Separable\nu v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\n⊢ g.Separable","state_after":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : a * f + b * derivative f = 1\n⊢ g.Separable","tactic":"obtain ⟨a, b, h⟩ := h","premises":[]},{"state_before":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : a * f + b * derivative f = 1\n⊢ g.Separable","state_after":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : a * f + b * derivative f = 1\n⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1","tactic":"refine ⟨a * v + b * derivative v, b * v, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Polynomial.derivative","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[38,4],"def_end_pos":[38,14]}]},{"state_before":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : a * f + b * derivative f = 1\n⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1","state_after":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : (a * f + b * derivative f) * (u * v) = 1 * 1\n⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1","tactic":"replace h := congr($h * $(h1))","premises":[]},{"state_before":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : (a * f + b * derivative f) * (u * v) = 1 * 1\n⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1","state_after":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : (a * f + b * derivative f) * (u * v) = 1 * 1\nh3 : derivative (u * v) = derivative 1\n⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1","tactic":"have h3 := congr(derivative $(h1))","premises":[{"full_name":"Polynomial.derivative","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[38,4],"def_end_pos":[38,14]}]},{"state_before":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : (a * f + b * derivative f) * (u * v) = 1 * 1\nh3 : derivative (u * v) = derivative 1\n⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1","state_after":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : (a * f + b * derivative f) * (u * v) = 1 * 1\nh3 : derivative u * v + u * derivative v = 0\n⊢ (a * v + b * derivative v) * (f * u) + b * v * (derivative f * u + f * derivative u) = 1","tactic":"simp only [← ha, derivative_mul, derivative_one] at h3 ⊢","premises":[{"full_name":"Polynomial.derivative_mul","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[238,8],"def_end_pos":[238,22]},{"full_name":"Polynomial.derivative_one","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[113,8],"def_end_pos":[113,22]}]},{"state_before":"case intro.mk.intro.intro\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g u v : R[X]\nh1 : u * v = 1\nh2 : v * u = 1\nha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g\na b : R[X]\nh : (a * f + b * derivative f) * (u * v) = 1 * 1\nh3 : derivative u * v + u * derivative v = 0\n⊢ (a * v + b * derivative v) * (f * u) + b * v * (derivative f * u + f * derivative u) = 1","state_after":"no goals","tactic":"calc\n    _ = (a * f + b * derivative f) * (u * v)\n      + (b * f) * (derivative u * v + u * derivative v) := by ring1\n    _ = 1 := by rw [h, h3]; ring1","premises":[{"full_name":"Polynomial.derivative","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[38,4],"def_end_pos":[38,14]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","commit":"","full_name":"Polynomial.degree_add_le","start":[549,0],"end":[551,51],"file_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q✝ : R[X]\nι : Type u_1\np q : R[X]\n⊢ (p + q).degree ≤ max p.degree q.degree","state_after":"no goals","tactic":"simpa only [degree, ← support_toFinsupp, toFinsupp_add]\n    using AddMonoidAlgebra.sup_support_add_le _ _ _","premises":[{"full_name":"AddMonoidAlgebra.sup_support_add_le","def_path":"Mathlib/Algebra/MonoidAlgebra/Degree.lean","def_pos":[85,8],"def_end_pos":[85,26]},{"full_name":"Polynomial.degree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[48,4],"def_end_pos":[48,10]},{"full_name":"Polynomial.support_toFinsupp","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[354,8],"def_end_pos":[354,25]},{"full_name":"Polynomial.toFinsupp_add","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[186,8],"def_end_pos":[186,21]}]}]}
{"url":"Mathlib/Data/Set/Prod.lean","commit":"","full_name":"Set.empty_prod","start":[77,0],"end":[80,23],"file_path":"Mathlib/Data/Set/Prod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\n⊢ ∅ ×ˢ t = ∅","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx✝ : α × β\n⊢ x✝ ∈ ∅ ×ˢ t ↔ x✝ ∈ ∅","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx✝ : α × β\n⊢ x✝ ∈ ∅ ×ˢ t ↔ x✝ ∈ ∅","state_after":"no goals","tactic":"exact false_and_iff _","premises":[{"full_name":"false_and_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[96,8],"def_end_pos":[96,21]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.rightShift_rightUnshift","start":[123,0],"end":[129,56],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'\nn : ℤ\nhn' : n' + a = n\n⊢ (γ.rightUnshift n hn').rightShift a n' hn' = γ","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'\nn : ℤ\nhn' : n' + a = n\np q : ℤ\nhpq : p + n' = q\n⊢ ((γ.rightUnshift n hn').rightShift a n' hn').v p q hpq = γ.v p q hpq","tactic":"ext p q hpq","premises":[]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'\nn : ℤ\nhn' : n' + a = n\np q : ℤ\nhpq : p + n' = q\n⊢ ((γ.rightUnshift n hn').rightShift a n' hn').v p q hpq = γ.v p q hpq","state_after":"no goals","tactic":"simp only [(γ.rightUnshift n hn').rightShift_v a n' hn' p q hpq (p + n) rfl,\n    γ.rightUnshift_v n hn' p (p + n) rfl q hpq,\n    shiftFunctorObjXIso, assoc, Iso.hom_inv_id, comp_id]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]},{"full_name":"CochainComplex.HomComplex.Cochain.rightShift_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[48,6],"def_end_pos":[48,18]},{"full_name":"CochainComplex.HomComplex.Cochain.rightUnshift","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[71,4],"def_end_pos":[71,16]},{"full_name":"CochainComplex.HomComplex.Cochain.rightUnshift_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[76,6],"def_end_pos":[76,20]},{"full_name":"CochainComplex.shiftFunctorObjXIso","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean","def_pos":[69,4],"def_end_pos":[69,23]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Topology/Bornology/Basic.lean","commit":"","full_name":"Bornology.isBounded_biUnion","start":[243,0],"end":[245,75],"file_path":"Mathlib/Topology/Bornology/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ns✝ : Set α\ninst✝ : Bornology α\ns : Set ι\nf : ι → Set α\nhs : s.Finite\n⊢ IsBounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, IsBounded (f i)","state_after":"no goals","tactic":"simp only [← isCobounded_compl_iff, compl_iUnion, isCobounded_biInter hs]","premises":[{"full_name":"Bornology.isCobounded_biInter","def_path":"Mathlib/Topology/Bornology/Basic.lean","def_pos":[225,8],"def_end_pos":[225,27]},{"full_name":"Bornology.isCobounded_compl_iff","def_path":"Mathlib/Topology/Bornology/Basic.lean","def_pos":[132,8],"def_end_pos":[132,29]},{"full_name":"Set.compl_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[389,8],"def_end_pos":[389,20]}]}]}
{"url":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","commit":"","full_name":"padicNormE.norm_rat_le_one","start":[819,0],"end":[834,20],"file_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","tactics":[{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : n = 0\n⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1","state_after":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : n = 0\nthis : { num := n, den := d, den_nz := hn, reduced := hd } = 0\n⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1","tactic":"have : (⟨n, d, hn, hd⟩ : ℚ) = 0 := Rat.zero_iff_num_zero.mpr hnz","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Rat.mk'","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[20,2],"def_end_pos":[20,5]},{"full_name":"Rat.zero_iff_num_zero","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[399,8],"def_end_pos":[399,25]}]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : n = 0\nthis : { num := n, den := d, den_nz := hn, reduced := hd } = 0\n⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1","state_after":"no goals","tactic":"norm_num [this]","premises":[]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\n⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1","state_after":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1","tactic":"have hnz' : (⟨n, d, hn, hd⟩ : ℚ) ≠ 0 := mt Rat.zero_iff_num_zero.1 hnz","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Rat.mk'","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[20,2],"def_end_pos":[20,5]},{"full_name":"Rat.zero_iff_num_zero","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[399,8],"def_end_pos":[399,25]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1","state_after":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ ↑(padicNorm p { num := n, den := d, den_nz := hn, reduced := hd }) ≤ 1","tactic":"rw [padicNormE.eq_padicNorm]","premises":[{"full_name":"padicNormE.eq_padicNorm","def_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","def_pos":[765,8],"def_end_pos":[765,20]}]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ ↑(padicNorm p { num := n, den := d, den_nz := hn, reduced := hd }) ≤ 1","state_after":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ padicNorm p { num := n, den := d, den_nz := hn, reduced := hd } ≤ 1","tactic":"norm_cast","premises":[]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ padicNorm p { num := n, den := d, den_nz := hn, reduced := hd } ≤ 1","state_after":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ (↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num)⁻¹ ≤ 1","tactic":"rw [padicNorm.eq_zpow_of_nonzero hnz', padicValRat, neg_sub,\n        padicValNat.eq_zero_of_not_dvd hq, Nat.cast_zero, zero_sub, zpow_neg, zpow_natCast]","premises":[{"full_name":"Nat.cast_zero","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[110,8],"def_end_pos":[110,17]},{"full_name":"neg_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]},{"full_name":"padicNorm.eq_zpow_of_nonzero","def_path":"Mathlib/NumberTheory/Padics/PadicNorm.lean","def_pos":[51,18],"def_end_pos":[51,36]},{"full_name":"padicValNat.eq_zero_of_not_dvd","def_path":"Mathlib/NumberTheory/Padics/PadicVal.lean","def_pos":[106,8],"def_end_pos":[106,26]},{"full_name":"padicValRat","def_path":"Mathlib/NumberTheory/Padics/PadicVal.lean","def_pos":[185,4],"def_end_pos":[185,15]},{"full_name":"zero_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[337,2],"def_end_pos":[337,13]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]},{"full_name":"zpow_neg","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[429,6],"def_end_pos":[429,14]}]},{"state_before":"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ (↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num)⁻¹ ≤ 1","state_after":"case ha\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ 1 ≤ ↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num","tactic":"apply inv_le_one","premises":[{"full_name":"inv_le_one","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[191,8],"def_end_pos":[191,18]}]},{"state_before":"case ha\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ 1 ≤ ↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num","state_after":"case ha\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ 1 ≤ p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num","tactic":"norm_cast","premises":[]},{"state_before":"case ha\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ 1 ≤ p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num","state_after":"case ha.h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ 0 < p","tactic":"apply one_le_pow","premises":[{"full_name":"Nat.one_le_pow","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[626,6],"def_end_pos":[626,16]}]},{"state_before":"case ha.h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ 0 < p","state_after":"no goals","tactic":"exact hp.1.pos","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Nat.Prime.pos","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[51,8],"def_end_pos":[51,17]}]}]}
{"url":"Mathlib/Algebra/Polynomial/EraseLead.lean","commit":"","full_name":"Polynomial.eraseLead_ne_zero","start":[75,0],"end":[78,100],"file_path":"Mathlib/Algebra/Polynomial/EraseLead.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nf0 : 2 ≤ f.support.card\n⊢ f.eraseLead ≠ 0","state_after":"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nf0 : 2 ≤ f.support.card\n⊢ ¬(f.support.erase f.natDegree).card = 0","tactic":"rw [Ne, ← card_support_eq_zero, eraseLead_support]","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.card_support_eq_zero","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[368,8],"def_end_pos":[368,28]},{"full_name":"Polynomial.eraseLead_support","def_path":"Mathlib/Algebra/Polynomial/EraseLead.lean","def_pos":[39,8],"def_end_pos":[39,25]}]},{"state_before":"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nf0 : 2 ≤ f.support.card\n⊢ ¬(f.support.erase f.natDegree).card = 0","state_after":"no goals","tactic":"exact\n    (zero_lt_one.trans_le <| (tsub_le_tsub_right f0 1).trans Finset.pred_card_le_card_erase).ne.symm","premises":[{"full_name":"Finset.pred_card_le_card_erase","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[166,8],"def_end_pos":[166,31]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"tsub_le_tsub_right","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[97,18],"def_end_pos":[97,36]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]}]}
{"url":"Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean","commit":"","full_name":"AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_eq","start":[121,0],"end":[126,59],"file_path":"Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean","tactics":[{"state_before":"A : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{u_6, u_1} A\ninst✝² : Category.{u_7, u_2} A'\ninst✝¹ : Category.{u_5, u_3} B\ninst✝ : Category.{u_8, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\n⊢ (equivalence₂ eB hF).counitIso = equivalence₂CounitIso eB hF","state_after":"case w.w.h\nA : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{u_6, u_1} A\ninst✝² : Category.{u_7, u_2} A'\ninst✝¹ : Category.{u_5, u_3} B\ninst✝ : Category.{u_8, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ (equivalence₂ eB hF).counitIso.hom.app Y' = (equivalence₂CounitIso eB hF).hom.app Y'","tactic":"ext Y'","premises":[]},{"state_before":"case w.w.h\nA : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{u_6, u_1} A\ninst✝² : Category.{u_7, u_2} A'\ninst✝¹ : Category.{u_5, u_3} B\ninst✝ : Category.{u_8, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ (equivalence₂ eB hF).counitIso.hom.app Y' = (equivalence₂CounitIso eB hF).hom.app Y'","state_after":"case w.w.h\nA : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{u_6, u_1} A\ninst✝² : Category.{u_7, u_2} A'\ninst✝¹ : Category.{u_5, u_3} B\ninst✝ : Category.{u_8, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ eB.inverse.map ((equivalence₁ hF).counitIso.hom.app (eB.functor.obj Y')) ≫ eB.unitIso.inv.app Y' =\n    (equivalence₂CounitIso eB hF).hom.app Y'","tactic":"dsimp [equivalence₂, Iso.refl]","premises":[{"full_name":"AlgebraicTopology.DoldKan.Compatibility.equivalence₂","def_path":"Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean","def_pos":[102,4],"def_end_pos":[102,16]},{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]}]},{"state_before":"case w.w.h\nA : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{u_6, u_1} A\ninst✝² : Category.{u_7, u_2} A'\ninst✝¹ : Category.{u_5, u_3} B\ninst✝ : Category.{u_8, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ eB.inverse.map ((equivalence₁ hF).counitIso.hom.app (eB.functor.obj Y')) ≫ eB.unitIso.inv.app Y' =\n    (equivalence₂CounitIso eB hF).hom.app Y'","state_after":"no goals","tactic":"simp only [equivalence₁CounitIso_eq, equivalence₂CounitIso_hom_app,\n    equivalence₁CounitIso_hom_app, Functor.map_comp, assoc]","premises":[{"full_name":"AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_eq","def_path":"Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean","def_pos":[80,8],"def_end_pos":[80,32]},{"full_name":"AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_hom_app","def_path":"Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean","def_pos":[70,2],"def_end_pos":[70,8]},{"full_name":"AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_hom_app","def_path":"Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean","def_pos":[110,2],"def_end_pos":[110,8]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]}]}]}
{"url":"Mathlib/Algebra/Homology/ExactSequence.lean","commit":"","full_name":"CategoryTheory.ComposableArrows.Exact.δ₀","start":[236,0],"end":[239,12],"file_path":"Mathlib/Algebra/Homology/ExactSequence.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C (n + 2)\nhS : S.Exact\n⊢ S.δ₀.Exact","state_after":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C (n + 2)\nhS : (mk₂ (S.map' 0 1 ⋯ ⋯) (S.map' 1 2 ⋯ ⋯)).Exact ∧ S.δ₀.Exact\n⊢ S.δ₀.Exact","tactic":"rw [exact_iff_δ₀] at hS","premises":[{"full_name":"CategoryTheory.ComposableArrows.exact_iff_δ₀","def_path":"Mathlib/Algebra/Homology/ExactSequence.lean","def_pos":[216,6],"def_end_pos":[216,18]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C (n + 2)\nhS : (mk₂ (S.map' 0 1 ⋯ ⋯) (S.map' 1 2 ⋯ ⋯)).Exact ∧ S.δ₀.Exact\n⊢ S.δ₀.Exact","state_after":"no goals","tactic":"exact hS.2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
{"url":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","commit":"","full_name":"StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source","start":[288,0],"end":[297,56],"file_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","tactics":[{"state_before":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G' P\ninst✝ : HasGroupoid M G\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\n⊢ LiftPropWithinAt P g s x ↔ LiftPropWithinAt P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)","state_after":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G' P\ninst✝ : HasGroupoid M G\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\n⊢ ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧\n      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔\n    ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧\n      P (↑(chartAt H' ((g ∘ ↑e.symm) (↑e x))) ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)","tactic":"rw [liftPropWithinAt_self_source, liftPropWithinAt_iff',\n    e.symm.continuousWithinAt_iff_continuousWithinAt_comp_right xe, e.symm_symm]","premises":[{"full_name":"ChartedSpace.liftPropWithinAt_iff'","def_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","def_pos":[151,9],"def_end_pos":[151,30]},{"full_name":"PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[991,8],"def_end_pos":[991,60]},{"full_name":"PartialHomeomorph.symm","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[78,14],"def_end_pos":[78,18]},{"full_name":"PartialHomeomorph.symm_symm","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[309,28],"def_end_pos":[309,37]},{"full_name":"StructureGroupoid.liftPropWithinAt_self_source","def_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","def_pos":[204,8],"def_end_pos":[204,36]}]},{"state_before":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G' P\ninst✝ : HasGroupoid M G\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\n⊢ ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧\n      P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔\n    ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧\n      P (↑(chartAt H' ((g ∘ ↑e.symm) (↑e x))) ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)","state_after":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G' P\ninst✝ : HasGroupoid M G\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\n⊢ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔\n    P (↑(chartAt H' ((g ∘ ↑e.symm) (↑e x))) ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)","tactic":"refine and_congr Iff.rfl ?_","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"and_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[43,8],"def_end_pos":[43,17]}]},{"state_before":"H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G' P\ninst✝ : HasGroupoid M G\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\n⊢ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔\n    P (↑(chartAt H' ((g ∘ ↑e.symm) (↑e x))) ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)","state_after":"no goals","tactic":"rw [Function.comp_apply, e.left_inv xe, ← Function.comp.assoc,\n    hG.liftPropWithinAt_indep_chart_source_aux (chartAt _ (g x) ∘ g) (chart_mem_maximalAtlas G x)\n      (mem_chart_source _ x) he xe, Function.comp.assoc]","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.comp.assoc","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[91,8],"def_end_pos":[91,18]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"PartialHomeomorph.left_inv","def_path":"Mathlib/Topology/PartialHomeomorph.lean","def_pos":[144,8],"def_end_pos":[144,16]},{"full_name":"StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux","def_path":"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean","def_pos":[230,8],"def_end_pos":[230,47]},{"full_name":"StructureGroupoid.chart_mem_maximalAtlas","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[985,8],"def_end_pos":[985,48]},{"full_name":"chartAt","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[563,7],"def_end_pos":[563,14]},{"full_name":"mem_chart_source","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[568,6],"def_end_pos":[568,22]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/Subobject.lean","commit":"","full_name":"CategoryTheory.Abelian.subobjectIsoSubobjectOp_apply","start":[25,0],"end":[58,31],"file_path":"Mathlib/CategoryTheory/Abelian/Subobject.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ Subobject X ≃o (Subobject (op X))ᵒᵈ","state_after":"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ (↑(cokernelOrderHom X)).comp ↑(kernelOrderHom X) = OrderHom.id\n\ncase refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ (↑(kernelOrderHom X)).comp ↑(cokernelOrderHom X) = OrderHom.id","tactic":"refine OrderIso.ofHomInv (cokernelOrderHom X) (kernelOrderHom X) ?_ ?_","premises":[{"full_name":"CategoryTheory.Limits.cokernelOrderHom","def_path":"Mathlib/CategoryTheory/Subobject/Limits.lean","def_pos":[213,4],"def_end_pos":[213,20]},{"full_name":"CategoryTheory.Limits.kernelOrderHom","def_path":"Mathlib/CategoryTheory/Subobject/Limits.lean","def_pos":[235,4],"def_end_pos":[235,18]},{"full_name":"OrderIso.ofHomInv","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[982,4],"def_end_pos":[982,12]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/Density.lean","commit":"","full_name":"ProbabilityTheory.Kernel.tendsto_m_density","start":[477,0],"end":[482,30],"file_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\na : α\ninst✝ : IsFiniteKernel ν\ns : Set β\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : γ) ∂ν a, Tendsto (fun n => κ.densityProcess ν n a x s) atTop (𝓝 (κ.density ν a x s))","state_after":"no goals","tactic":"filter_upwards [tendsto_densityProcess_limitProcess hκν a hs, density_ae_eq_limitProcess hκν a hs]\n    with t h1 h2 using h2 ▸ h1","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"ProbabilityTheory.Kernel.density_ae_eq_limitProcess","def_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","def_pos":[470,6],"def_end_pos":[470,32]},{"full_name":"ProbabilityTheory.Kernel.tendsto_densityProcess_limitProcess","def_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","def_pos":[403,6],"def_end_pos":[403,41]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]}]}
{"url":"Mathlib/CategoryTheory/WithTerminal.lean","commit":"","full_name":"CategoryTheory.WithInitial.mapComp_inv_app","start":[443,0],"end":[449,40],"file_path":"Mathlib/CategoryTheory/WithTerminal.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nD : Type u_1\nE : Type u_2\ninst✝¹ : Category.{?u.258557, u_1} D\ninst✝ : Category.{?u.258561, u_2} E\nF : C ⥤ D\nG : D ⥤ E\n⊢ ∀ {X Y : WithInitial C} (f : X ⟶ Y),\n    (map (F ⋙ G)).map f ≫\n        ((fun X =>\n              match X with\n              | of x => Iso.refl ((map (F ⋙ G)).obj (of x))\n              | star => Iso.refl ((map (F ⋙ G)).obj star))\n            Y).hom =\n      ((fun X =>\n              match X with\n              | of x => Iso.refl ((map (F ⋙ G)).obj (of x))\n              | star => Iso.refl ((map (F ⋙ G)).obj star))\n            X).hom ≫\n        (map F ⋙ map G).map f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Algebra/Category/AlgebraCat/Basic.lean","commit":"","full_name":"AlgebraCat.free_map","start":[137,0],"end":[154,7],"file_path":"Mathlib/Algebra/Category/AlgebraCat/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\n⊢ ∀ (X : Type u),\n    { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n        (𝟙 X) =\n      𝟙\n        ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj\n          X)","state_after":"R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X) =\n    𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)","tactic":"intro X","premises":[]},{"state_before":"R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X) =\n    𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)","state_after":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n          (𝟙 X)) ∘\n      FreeAlgebra.ι R =\n    ⇑(𝟙\n          ({ obj := fun S => mk (FreeAlgebra R S),\n                map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj\n            X)) ∘\n      FreeAlgebra.ι R","tactic":"apply FreeAlgebra.hom_ext","premises":[{"full_name":"FreeAlgebra.hom_ext","def_path":"Mathlib/Algebra/FreeAlgebra.lean","def_pos":[424,8],"def_end_pos":[424,15]}]},{"state_before":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n          (𝟙 X)) ∘\n      FreeAlgebra.ι R =\n    ⇑(𝟙\n          ({ obj := fun S => mk (FreeAlgebra R S),\n                map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj\n            X)) ∘\n      FreeAlgebra.ι R","state_after":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ FreeAlgebra.ι R ∘ 𝟙 X = ⇑(𝟙 (mk (FreeAlgebra R X))) ∘ FreeAlgebra.ι R","tactic":"simp only [FreeAlgebra.ι_comp_lift]","premises":[{"full_name":"FreeAlgebra.ι_comp_lift","def_path":"Mathlib/Algebra/FreeAlgebra.lean","def_pos":[391,8],"def_end_pos":[391,19]}]},{"state_before":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ FreeAlgebra.ι R ∘ 𝟙 X = ⇑(𝟙 (mk (FreeAlgebra R X))) ∘ FreeAlgebra.ι R","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\n⊢ ∀ {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z),\n    { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n        (f ≫ g) =\n      { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f ≫\n        { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g","state_after":"R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n      (f✝ ≫ g✝) =\n    { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫\n      { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝","tactic":"intros","premises":[]},{"state_before":"R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n      (f✝ ≫ g✝) =\n    { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫\n      { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝","state_after":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n          (f✝ ≫ g✝)) ∘\n      FreeAlgebra.ι R =\n    ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n            f✝ ≫\n          { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n            g✝) ∘\n      FreeAlgebra.ι R","tactic":"apply FreeAlgebra.hom_ext","premises":[{"full_name":"FreeAlgebra.hom_ext","def_path":"Mathlib/Algebra/FreeAlgebra.lean","def_pos":[424,8],"def_end_pos":[424,15]}]},{"state_before":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n          (f✝ ≫ g✝)) ∘\n      FreeAlgebra.ι R =\n    ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n            f✝ ≫\n          { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n            g✝) ∘\n      FreeAlgebra.ι R","state_after":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ FreeAlgebra.ι R ∘ (f✝ ≫ g✝) =\n    ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R","tactic":"simp only [FreeAlgebra.ι_comp_lift]","premises":[{"full_name":"FreeAlgebra.ι_comp_lift","def_path":"Mathlib/Algebra/FreeAlgebra.lean","def_pos":[391,8],"def_end_pos":[391,19]}]},{"state_before":"case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ FreeAlgebra.ι R ∘ (f✝ ≫ g✝) =\n    ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R","state_after":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ =\n    (⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R) x✝","tactic":"ext1","premises":[]},{"state_before":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ =\n    (⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R) x✝","state_after":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ =\n    ((⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝))) ∘\n        FreeAlgebra.ι R)\n      x✝","tactic":"erw [CategoryTheory.coe_comp]","premises":[{"full_name":"CategoryTheory.coe_comp","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[114,8],"def_end_pos":[114,16]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ =\n    ((⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝))) ∘\n        FreeAlgebra.ι R)\n      x✝","state_after":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ FreeAlgebra.ι R (g✝ (f✝ x✝)) =\n    ((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) (((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝)) (FreeAlgebra.ι R x✝))","tactic":"simp only [CategoryTheory.coe_comp, Function.comp_apply, types_comp_apply]","premises":[{"full_name":"CategoryTheory.coe_comp","def_path":"Mathlib/CategoryTheory/ConcreteCategory/Basic.lean","def_pos":[114,8],"def_end_pos":[114,16]},{"full_name":"CategoryTheory.types_comp_apply","def_path":"Mathlib/CategoryTheory/Types.lean","def_pos":[68,8],"def_end_pos":[68,24]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]}]},{"state_before":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ FreeAlgebra.ι R (g✝ (f✝ x✝)) =\n    ((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) (((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝)) (FreeAlgebra.ι R x✝))","state_after":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ FreeAlgebra.ι R (g✝ (f✝ x✝)) = (FreeAlgebra.ι R ∘ g✝) (f✝ x✝)","tactic":"erw [FreeAlgebra.lift_ι_apply, FreeAlgebra.lift_ι_apply]","premises":[{"full_name":"FreeAlgebra.lift_ι_apply","def_path":"Mathlib/Algebra/FreeAlgebra.lean","def_pos":[397,8],"def_end_pos":[397,20]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ FreeAlgebra.ι R (g✝ (f✝ x✝)) = (FreeAlgebra.ι R ∘ g✝) (f✝ x✝)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Ring/Int.lean","commit":"","full_name":"Int.even_mul_succ_self","start":[161,0],"end":[162,52],"file_path":"Mathlib/Algebra/Ring/Int.lean","tactics":[{"state_before":"m n✝ n : ℤ\n⊢ Even (n * (n + 1))","state_after":"no goals","tactic":"simpa [even_mul, parity_simps] using n.even_or_odd","premises":[{"full_name":"Int.even_mul","def_path":"Mathlib/Algebra/Group/Int.lean","def_pos":[204,22],"def_end_pos":[204,30]},{"full_name":"Int.even_or_odd","def_path":"Mathlib/Algebra/Ring/Int.lean","def_pos":[104,6],"def_end_pos":[104,17]}]}]}
{"url":"Mathlib/LinearAlgebra/FreeAlgebra.lean","commit":"","full_name":"FreeAlgebra.rank_eq","start":[41,0],"end":[44,42],"file_path":"Mathlib/LinearAlgebra/FreeAlgebra.lean","tactics":[{"state_before":"R : Type u\nX : Type v\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ Module.rank R (FreeAlgebra R X) = Cardinal.lift.{u, v} (Cardinal.mk (List X))","state_after":"no goals","tactic":"rw [← (Basis.mk_eq_rank'.{_,_,_,u} (basisFreeMonoid R X)).trans (Cardinal.lift_id _),\n    Cardinal.lift_umax'.{v,u}, FreeMonoid]","premises":[{"full_name":"Basis.mk_eq_rank'","def_path":"Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean","def_pos":[333,8],"def_end_pos":[333,25]},{"full_name":"Cardinal.lift_id","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[203,8],"def_end_pos":[203,15]},{"full_name":"Cardinal.lift_umax'","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[191,8],"def_end_pos":[191,18]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"FreeAlgebra.basisFreeMonoid","def_path":"Mathlib/LinearAlgebra/FreeAlgebra.lean","def_pos":[32,18],"def_end_pos":[32,33]},{"full_name":"FreeMonoid","def_path":"Mathlib/Algebra/FreeMonoid/Basic.lean","def_pos":[27,4],"def_end_pos":[27,14]}]}]}
{"url":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","commit":"","full_name":"LipschitzWith.prod","start":[231,0],"end":[236,36],"file_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx y : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\n⊢ LipschitzWith (max Kf Kg) fun x => (f x, g x)","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ edist ((fun x => (f x, g x)) x) ((fun x => (f x, g x)) y) ≤ ↑(max Kf Kg) * edist x y","tactic":"intro x y","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ edist ((fun x => (f x, g x)) x) ((fun x => (f x, g x)) y) ≤ ↑(max Kf Kg) * edist x y","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ max (edist ((fun x => (f x, g x)) x).1 ((fun x => (f x, g x)) y).1)\n      (edist ((fun x => (f x, g x)) x).2 ((fun x => (f x, g x)) y).2) ≤\n    max (↑Kf * edist x y) (↑Kg * edist x y)","tactic":"rw [ENNReal.coe_mono.map_max, Prod.edist_eq, ENNReal.max_mul]","premises":[{"full_name":"ENNReal.coe_mono","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[326,8],"def_end_pos":[326,16]},{"full_name":"ENNReal.max_mul","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[58,8],"def_end_pos":[58,15]},{"full_name":"Monotone.map_max","def_path":"Mathlib/Order/MinMax.lean","def_pos":[214,8],"def_end_pos":[214,24]},{"full_name":"Prod.edist_eq","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[426,8],"def_end_pos":[426,21]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ max (edist ((fun x => (f x, g x)) x).1 ((fun x => (f x, g x)) y).1)\n      (edist ((fun x => (f x, g x)) x).2 ((fun x => (f x, g x)) y).2) ≤\n    max (↑Kf * edist x y) (↑Kg * edist x y)","state_after":"no goals","tactic":"exact max_le_max (hf x y) (hg x y)","premises":[{"full_name":"max_le_max","def_path":"Mathlib/Order/MinMax.lean","def_pos":[63,8],"def_end_pos":[63,18]}]}]}
{"url":"Mathlib/GroupTheory/Coset.lean","commit":"","full_name":"QuotientAddGroup.rightRel_eq","start":[291,0],"end":[295,24],"file_path":"Mathlib/GroupTheory/Coset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b = (b * a⁻¹ ∈ s)","state_after":"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s","tactic":"simp only [eq_iff_iff]","premises":[{"full_name":"eq_iff_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1378,16],"def_end_pos":[1378,26]}]},{"state_before":"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s","state_after":"no goals","tactic":"apply rightRel_apply","premises":[{"full_name":"QuotientGroup.rightRel_apply","def_path":"Mathlib/GroupTheory/Coset.lean","def_pos":[283,8],"def_end_pos":[283,22]}]}]}
{"url":"Mathlib/Order/Interval/Set/Basic.lean","commit":"","full_name":"Set.Icc_diff_right","start":[641,0],"end":[643,52],"file_path":"Mathlib/Order/Interval/Set/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c x : α\n⊢ x ∈ Icc a b \\ {b} ↔ x ∈ Ico a b","state_after":"no goals","tactic":"simp [lt_iff_le_and_ne, and_assoc]","premises":[{"full_name":"and_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[148,8],"def_end_pos":[148,17]},{"full_name":"lt_iff_le_and_ne","def_path":"Mathlib/Order/Basic.lean","def_pos":[309,8],"def_end_pos":[309,24]}]}]}
{"url":"Mathlib/Logic/Embedding/Basic.lean","commit":"","full_name":"Function.Embedding.equiv_symm_toEmbedding_trans_toEmbedding","start":[148,0],"end":[152,6],"file_path":"Mathlib/Logic/Embedding/Basic.lean","tactics":[{"state_before":"α : Sort u_1\nβ : Sort u_2\ne : α ≃ β\n⊢ e.symm.toEmbedding.trans e.toEmbedding = Embedding.refl β","state_after":"case h\nα : Sort u_1\nβ : Sort u_2\ne : α ≃ β\nx✝ : β\n⊢ (e.symm.toEmbedding.trans e.toEmbedding) x✝ = (Embedding.refl β) x✝","tactic":"ext","premises":[]},{"state_before":"case h\nα : Sort u_1\nβ : Sort u_2\ne : α ≃ β\nx✝ : β\n⊢ (e.symm.toEmbedding.trans e.toEmbedding) x✝ = (Embedding.refl β) x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","commit":"","full_name":"Orientation.rotation_pi","start":[131,0],"end":[135,17],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\n⊢ o.rotation ↑π = LinearIsometryEquiv.neg ℝ","state_after":"case h\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx : V\n⊢ (o.rotation ↑π) x = (LinearIsometryEquiv.neg ℝ) x","tactic":"ext x","premises":[]},{"state_before":"case h\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx : V\n⊢ (o.rotation ↑π) x = (LinearIsometryEquiv.neg ℝ) x","state_after":"no goals","tactic":"simp [rotation]","premises":[{"full_name":"Orientation.rotation","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","def_pos":[58,4],"def_end_pos":[58,12]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/Density.lean","commit":"","full_name":"ProbabilityTheory.Kernel.densityProcess_fst_univ","start":[703,0],"end":[720,30],"file_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\n⊢ κ.densityProcess κ.fst n a x univ = if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\n⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ.fst a) (countablePartitionSet n x)).toReal =\n    if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1","tactic":"rw [densityProcess]","premises":[{"full_name":"ProbabilityTheory.Kernel.densityProcess","def_path":"Mathlib/Probability/Kernel/Disintegration/Density.lean","def_pos":[93,4],"def_end_pos":[93,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\n⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ.fst a) (countablePartitionSet n x)).toReal =\n    if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1","state_after":"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nh : (κ.fst a) (countablePartitionSet n x) = 0\n⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ.fst a) (countablePartitionSet n x)).toReal = 0\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nh : ¬(κ.fst a) (countablePartitionSet n x) = 0\n⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ.fst a) (countablePartitionSet n x)).toReal = 1","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/CategoryTheory/Conj.lean","commit":"","full_name":"CategoryTheory.Iso.refl_conj","start":[50,0],"end":[52,81],"file_path":"Mathlib/CategoryTheory/Conj.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nα : X ≅ Y\nf : End X\n⊢ (refl X).conj f = f","state_after":"no goals","tactic":"rw [conj_apply, Iso.refl_inv, Iso.refl_hom, Category.id_comp, Category.comp_id]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Iso.conj_apply","def_path":"Mathlib/CategoryTheory/Conj.lean","def_pos":[39,8],"def_end_pos":[39,18]},{"full_name":"CategoryTheory.Iso.refl_hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[113,8],"def_end_pos":[113,13]},{"full_name":"CategoryTheory.Iso.refl_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[113,8],"def_end_pos":[113,13]}]}]}
{"url":"Mathlib/GroupTheory/PushoutI.lean","commit":"","full_name":"Monoid.PushoutI.homEquiv_symm_apply","start":[144,0],"end":[155,85],"file_path":"Mathlib/GroupTheory/PushoutI.lean","tactics":[{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝² : Monoid K\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nf : PushoutI φ →* K\ni : ι\n⊢ ((fun i => f.comp (of i), f.comp (base φ)).1 i).comp (φ i) = (fun i => f.comp (of i), f.comp (base φ)).2","state_after":"no goals","tactic":"rw [MonoidHom.comp_assoc, of_comp_eq_base]","premises":[{"full_name":"Monoid.PushoutI.of_comp_eq_base","def_path":"Mathlib/GroupTheory/PushoutI.lean","def_pos":[87,8],"def_end_pos":[87,23]},{"full_name":"MonoidHom.comp_assoc","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[732,8],"def_end_pos":[732,28]}]},{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝² : Monoid K\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nx✝ : PushoutI φ →* K\n⊢ ∀ (i : ι),\n    ((fun f => lift (↑f).1 (↑f).2 ⋯) ((fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), ⋯⟩) x✝)).comp (of i) =\n      x✝.comp (of i)","state_after":"no goals","tactic":"simp [DFunLike.ext_iff]","premises":[{"full_name":"DFunLike.ext_iff","def_path":"Mathlib/Data/FunLike/Basic.lean","def_pos":[196,8],"def_end_pos":[196,15]}]},{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝² : Monoid K\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nx✝ : PushoutI φ →* K\n⊢ ((fun f => lift (↑f).1 (↑f).2 ⋯) ((fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), ⋯⟩) x✝)).comp (base φ) =\n    x✝.comp (base φ)","state_after":"no goals","tactic":"simp [DFunLike.ext_iff]","premises":[{"full_name":"DFunLike.ext_iff","def_path":"Mathlib/Data/FunLike/Basic.lean","def_pos":[196,8],"def_end_pos":[196,15]}]},{"state_before":"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝² : Monoid K\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nx✝ : { f // ∀ (i : ι), (f.1 i).comp (φ i) = f.2 }\nfst✝ : (i : ι) → G i →* K\nsnd✝ : H →* K\nproperty✝ : ∀ (i : ι), ((fst✝, snd✝).1 i).comp (φ i) = (fst✝, snd✝).2\n⊢ (fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), ⋯⟩)\n      ((fun f => lift (↑f).1 (↑f).2 ⋯) ⟨(fst✝, snd✝), property✝⟩) =\n    ⟨(fst✝, snd✝), property✝⟩","state_after":"no goals","tactic":"simp [DFunLike.ext_iff, Function.funext_iff]","premises":[{"full_name":"DFunLike.ext_iff","def_path":"Mathlib/Data/FunLike/Basic.lean","def_pos":[196,8],"def_end_pos":[196,15]},{"full_name":"Function.funext_iff","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[63,8],"def_end_pos":[63,27]}]}]}
{"url":"Mathlib/Algebra/GroupWithZero/Conj.lean","commit":"","full_name":"isConj_iff₀","start":[19,0],"end":[22,58],"file_path":"Mathlib/Algebra/GroupWithZero/Conj.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : GroupWithZero α\na b : α\n⊢ (∃ x, x ≠ 0 ∧ SemiconjBy x a b) ↔ ∃ c, c ≠ 0 ∧ c * a * c⁻¹ = b","state_after":"case a.h.e'_2.h.a\nα : Type u_1\ninst✝ : GroupWithZero α\na b c : α\n⊢ c ≠ 0 ∧ SemiconjBy c a b ↔ c ≠ 0 ∧ c * a * c⁻¹ = b","tactic":"congr! 2 with c","premises":[]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.replicate_le_replicate","start":[863,0],"end":[864,95],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\na : α\nk n : ℕ\n⊢ replicate k a ≤ replicate n a ↔ List.replicate k a <+ List.replicate n a","state_after":"no goals","tactic":"rw [← replicate_le_coe, coe_replicate]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Multiset.coe_replicate","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[803,8],"def_end_pos":[803,21]},{"full_name":"Multiset.replicate_le_coe","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[854,8],"def_end_pos":[854,24]}]}]}
{"url":"Mathlib/Data/Finset/Option.lean","commit":"","full_name":"Finset.eraseNone_union","start":[110,0],"end":[114,6],"file_path":"Mathlib/Data/Finset/Option.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq (Option α)\ninst✝ : DecidableEq α\ns t : Finset (Option α)\n⊢ eraseNone (s ∪ t) = eraseNone s ∪ eraseNone t","state_after":"case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq (Option α)\ninst✝ : DecidableEq α\ns t : Finset (Option α)\na✝ : α\n⊢ a✝ ∈ eraseNone (s ∪ t) ↔ a✝ ∈ eraseNone s ∪ eraseNone t","tactic":"ext","premises":[]},{"state_before":"case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq (Option α)\ninst✝ : DecidableEq α\ns t : Finset (Option α)\na✝ : α\n⊢ a✝ ∈ eraseNone (s ∪ t) ↔ a✝ ∈ eraseNone s ∪ eraseNone t","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/ZMod/Units.lean","commit":"","full_name":"ZMod.eq_unit_mul_divisor","start":[72,0],"end":[106,83],"file_path":"Mathlib/Data/ZMod/Units.lean","tactics":[{"state_before":"n m N : ℕ\na : ZMod N\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inl\nn m : ℕ\na : ZMod 0\n⊢ ∃ d, d ∣ 0 ∧ ∃ u, IsUnit u ∧ a = u * ↑d\n\ncase inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"rcases eq_or_ne N 0 with rfl | hN","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"have : NeZero N := ⟨hN⟩","premises":[{"full_name":"NeZero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[23,6],"def_end_pos":[23,12]}]},{"state_before":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"let d := a.val.gcd N","premises":[{"full_name":"Nat.gcd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[32,4],"def_end_pos":[32,7]},{"full_name":"ZMod.val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[45,4],"def_end_pos":[45,7]}]},{"state_before":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"have hd : d ≠ 0 := Nat.gcd_ne_zero_right hN","premises":[{"full_name":"Nat.gcd_ne_zero_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[160,8],"def_end_pos":[160,25]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"case inr\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"obtain ⟨a₀, (ha₀ : _ = d * _)⟩ := a.val.gcd_dvd_left N","premises":[{"full_name":"Nat.gcd_dvd_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[86,8],"def_end_pos":[86,20]},{"full_name":"ZMod.val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[45,4],"def_end_pos":[45,7]}]},{"state_before":"case inr.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"obtain ⟨N₀, (hN₀ : _ = d * _)⟩ := a.val.gcd_dvd_right N","premises":[{"full_name":"Nat.gcd_dvd_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[88,8],"def_end_pos":[88,21]},{"full_name":"ZMod.val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[45,4],"def_end_pos":[45,7]}]},{"state_before":"case inr.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"refine ⟨d, ⟨N₀, hN₀⟩, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case inr.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"have hu₀ : IsUnit (a₀ : ZMod N₀) := by\n    refine (isUnit_iff_coprime _ _).mpr (Nat.isCoprime_iff_coprime.mp ?_)\n    obtain ⟨p, q, hpq⟩ : ∃ (p q : ℤ), d = a.val * p + N * q := ⟨_, _, Nat.gcd_eq_gcd_ab _ _⟩\n    rw [ha₀, hN₀, Nat.cast_mul, Nat.cast_mul, mul_assoc, mul_assoc, ← mul_add, eq_comm,\n      mul_comm _ p, mul_comm _ q] at hpq\n    exact ⟨p, q, Int.eq_one_of_mul_eq_self_right (Nat.cast_ne_zero.mpr hd) hpq⟩","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.eq_one_of_mul_eq_self_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[508,8],"def_end_pos":[508,35]},{"full_name":"IsUnit","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[523,4],"def_end_pos":[523,10]},{"full_name":"Nat.cast_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[56,25],"def_end_pos":[56,33]},{"full_name":"Nat.cast_ne_zero","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[76,8],"def_end_pos":[76,20]},{"full_name":"Nat.gcd_eq_gcd_ab","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[122,8],"def_end_pos":[122,21]},{"full_name":"Nat.isCoprime_iff_coprime","def_path":"Mathlib/RingTheory/Coprime/Lemmas.lean","def_pos":[40,8],"def_end_pos":[40,33]},{"full_name":"ZMod","def_path":"Mathlib/Data/ZMod/Defs.lean","def_pos":[89,4],"def_end_pos":[89,8]},{"full_name":"ZMod.isUnit_iff_coprime","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[843,6],"def_end_pos":[843,24]},{"full_name":"ZMod.val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[45,4],"def_end_pos":[45,7]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case inr.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (unitsMap ⋯) u = hu₀.unit\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"obtain ⟨u, hu⟩ := (ZMod.unitsMap_surjective (⟨d, mul_comm d N₀ ▸ hN₀⟩ : N₀ ∣ N)) hu₀.unit","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsUnit.unit","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[648,28],"def_end_pos":[648,32]},{"full_name":"ZMod.unitsMap_surjective","def_path":"Mathlib/Data/ZMod/Units.lean","def_pos":[38,8],"def_end_pos":[38,27]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (unitsMap ⋯) u = hu₀.unit\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","tactic":"rw [unitsMap_def, ← Units.eq_iff, Units.coe_map, IsUnit.unit_spec, MonoidHom.coe_coe] at hu","premises":[{"full_name":"IsUnit.unit_spec","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[665,8],"def_end_pos":[665,17]},{"full_name":"MonoidHom.coe_coe","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[375,8],"def_end_pos":[375,25]},{"full_name":"Units.coe_map","def_path":"Mathlib/Algebra/Group/Units/Hom.lean","def_pos":[65,8],"def_end_pos":[65,15]},{"full_name":"Units.eq_iff","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[133,8],"def_end_pos":[133,14]},{"full_name":"ZMod.unitsMap_def","def_path":"Mathlib/Data/ZMod/Units.lean","def_pos":[23,6],"def_end_pos":[23,18]}]},{"state_before":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ ∃ u, IsUnit u ∧ a = u * ↑d","state_after":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ a = ↑u * ↑d","tactic":"refine ⟨u.val, u.isUnit, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Units.isUnit","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[555,18],"def_end_pos":[555,30]},{"full_name":"Units.val","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ a = ↑u * ↑d","state_after":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ d * a₀ % N = d * (↑u).val % N","tactic":"rw [← ZMod.natCast_zmod_val a, ← ZMod.natCast_zmod_val u.1, ha₀, ← Nat.cast_mul,\n    ZMod.natCast_eq_natCast_iff, mul_comm _ d, Nat.ModEq]","premises":[{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"Nat.cast_mul","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[56,25],"def_end_pos":[56,33]},{"full_name":"Units.val","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"ZMod.natCast_eq_natCast_iff","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[525,8],"def_end_pos":[525,30]},{"full_name":"ZMod.natCast_zmod_val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[188,8],"def_end_pos":[188,24]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ d * a₀ % N = d * (↑u).val % N","state_after":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ a₀ % N₀ = (↑u).val % N₀","tactic":"simp only [hN₀, Nat.mul_mod_mul_left, Nat.mul_right_inj hd]","premises":[{"full_name":"Nat.mul_mod_mul_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[327,8],"def_end_pos":[327,24]},{"full_name":"Nat.mul_right_inj","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[315,16],"def_end_pos":[315,29]}]},{"state_before":"case inr.intro.intro.intro\nn m N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\nhu₀ : IsUnit ↑a₀\nu : (ZMod N)ˣ\nhu : (castHom ⋯ (ZMod N₀)) ↑u = ↑a₀\n⊢ a₀ % N₀ = (↑u).val % N₀","state_after":"no goals","tactic":"rw [← Nat.ModEq, ← ZMod.natCast_eq_natCast_iff, ← hu, natCast_val, castHom_apply]","premises":[{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"ZMod.castHom_apply","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[343,8],"def_end_pos":[343,21]},{"full_name":"ZMod.natCast_eq_natCast_iff","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[525,8],"def_end_pos":[525,30]},{"full_name":"ZMod.natCast_val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[266,8],"def_end_pos":[266,19]}]}]}
{"url":"Mathlib/Algebra/Star/SelfAdjoint.lean","commit":"","full_name":"IsSelfAdjoint.smul","start":[269,0],"end":[271,96],"file_path":"Mathlib/Algebra/Star/SelfAdjoint.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝⁴ : Star R\ninst✝³ : AddMonoid A\ninst✝² : StarAddMonoid A\ninst✝¹ : SMul R A\ninst✝ : StarModule R A\nr : R\nhr : IsSelfAdjoint r\nx : A\nhx : IsSelfAdjoint x\n⊢ IsSelfAdjoint (r • x)","state_after":"no goals","tactic":"simp only [isSelfAdjoint_iff, star_smul, hr.star_eq, hx.star_eq]","premises":[{"full_name":"IsSelfAdjoint.star_eq","def_path":"Mathlib/Algebra/Star/SelfAdjoint.lean","def_pos":[68,8],"def_end_pos":[68,15]},{"full_name":"StarModule.star_smul","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[408,2],"def_end_pos":[408,11]},{"full_name":"isSelfAdjoint_iff","def_path":"Mathlib/Algebra/Star/SelfAdjoint.lean","def_pos":[71,8],"def_end_pos":[71,32]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","commit":"","full_name":"CategoryTheory.Presheaf.imageSieve_app","start":[60,0],"end":[65,30],"file_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\n⊢ imageSieve f ((f.app (op U)) s) = ⊤","state_after":"case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V ⟶ U\n⊢ (imageSieve f ((f.app (op U)) s)).arrows i ↔ ⊤.arrows i","tactic":"ext V i","premises":[]},{"state_before":"case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V ⟶ U\n⊢ (imageSieve f ((f.app (op U)) s)).arrows i ↔ ⊤.arrows i","state_after":"case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V ⟶ U\n⊢ ∃ t, (f.app (op V)) t = (G.map i.op) ((f.app (op U)) s)","tactic":"simp only [Sieve.top_apply, iff_true_iff, imageSieve_apply]","premises":[{"full_name":"CategoryTheory.Presheaf.imageSieve_apply","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[44,2],"def_end_pos":[44,7]},{"full_name":"CategoryTheory.Sieve.top_apply","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[356,8],"def_end_pos":[356,17]},{"full_name":"iff_true_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[105,8],"def_end_pos":[105,20]}]},{"state_before":"case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V ⟶ U\n⊢ ∃ t, (f.app (op V)) t = (G.map i.op) ((f.app (op U)) s)","state_after":"case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V ⟶ U\nthis : ∀ (x : (forget A).obj (F.obj (op U))), (f.app (op V)) ((F.map i.op) x) = (G.map i.op) ((f.app (op U)) x)\n⊢ ∃ t, (f.app (op V)) t = (G.map i.op) ((f.app (op U)) s)","tactic":"have := elementwise_of% (f.naturality i.op)","premises":[{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : ConcreteCategory A\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V ⟶ U\nthis : ∀ (x : (forget A).obj (F.obj (op U))), (f.app (op V)) ((F.map i.op) x) = (G.map i.op) ((f.app (op U)) x)\n⊢ ∃ t, (f.app (op V)) t = (G.map i.op) ((f.app (op U)) s)","state_after":"no goals","tactic":"exact ⟨F.map i.op s, this s⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]}]}
{"url":"Mathlib/Data/Fin/Basic.lean","commit":"","full_name":"Fin.predAbove_castSucc_of_lt","start":[1239,0],"end":[1241,66],"file_path":"Mathlib/Data/Fin/Basic.lean","tactics":[{"state_before":"n m : ℕ\np i : Fin n\nh : p < i\nhi : optParam (i.castSucc ≠ 0) ⋯\n⊢ p.predAbove i.castSucc = i.castSucc.pred hi","state_after":"no goals","tactic":"rw [predAbove_of_castSucc_lt _ _ (castSucc_lt_castSucc_iff.2 h)]","premises":[{"full_name":"Fin.castSucc_lt_castSucc_iff","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[397,16],"def_end_pos":[397,40]},{"full_name":"Fin.predAbove_of_castSucc_lt","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[1222,6],"def_end_pos":[1222,30]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]}]}
{"url":"Mathlib/RingTheory/FinitePresentation.lean","commit":"","full_name":"RingHom.FinitePresentation.comp","start":[437,0],"end":[447,40],"file_path":"Mathlib/RingTheory/FinitePresentation.lean","tactics":[{"state_before":"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\nins1 : Algebra A B := f.toAlgebra\nins2 : Algebra B C := g.toAlgebra\nins3 : Algebra A C := (g.comp f).toAlgebra\na : A\nb : B\nc : C\n⊢ (a • b) • c = a • b • c","state_after":"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\nins1 : Algebra A B := f.toAlgebra\nins2 : Algebra B C := g.toAlgebra\nins3 : Algebra A C := (g.comp f).toAlgebra\na : A\nb : B\nc : C\n⊢ (algebraMap B C) ((algebraMap A B) a) * ((algebraMap B C) b * c) = (algebraMap A C) a * ((algebraMap B C) b * c)","tactic":"simp [Algebra.smul_def, mul_assoc]","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\nins1 : Algebra A B := f.toAlgebra\nins2 : Algebra B C := g.toAlgebra\nins3 : Algebra A C := (g.comp f).toAlgebra\na : A\nb : B\nc : C\n⊢ (algebraMap B C) ((algebraMap A B) a) * ((algebraMap B C) b * c) = (algebraMap A C) a * ((algebraMap B C) b * c)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Set/Pointwise/Basic.lean","commit":"","full_name":"Set.univ_sub_univ","start":[826,0],"end":[827,78],"file_path":"Mathlib/Data/Set/Pointwise/Basic.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ univ / univ = univ","state_after":"no goals","tactic":"simp [div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/Order/CompleteLattice.lean","commit":"","full_name":"iSup_range","start":[1139,0],"end":[1140,36],"file_path":"Mathlib/Order/CompleteLattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nβ₂ : Type u_3\nγ : Type u_4\nι : Sort u_5\nι' : Sort u_6\nκ : ι → Sort u_7\nκ' : ι' → Sort u_8\ninst✝ : CompleteLattice α\nf✝ g✝ s t : ι → α\na b : α\ng : β → α\nf : ι → β\n⊢ ⨆ b ∈ range f, g b = ⨆ i, g (f i)","state_after":"no goals","tactic":"rw [← iSup_subtype'', iSup_range']","premises":[{"full_name":"iSup_range'","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[548,8],"def_end_pos":[548,19]},{"full_name":"iSup_subtype''","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[987,8],"def_end_pos":[987,22]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","commit":"","full_name":"Polynomial.scaleRoots_ne_zero","start":[47,0],"end":[53,15],"file_path":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\n⊢ p.scaleRoots s ≠ 0","state_after":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\n⊢ False","tactic":"intro h","premises":[]},{"state_before":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\n⊢ False","state_after":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis : p.coeff p.natDegree ≠ 0\n⊢ False","tactic":"have : p.coeff p.natDegree ≠ 0 := mt leadingCoeff_eq_zero.mp hp","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.leadingCoeff_eq_zero","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[577,8],"def_end_pos":[577,28]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]},{"state_before":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis : p.coeff p.natDegree ≠ 0\n⊢ False","state_after":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis✝ : p.coeff p.natDegree ≠ 0\nthis : (p.scaleRoots s).coeff p.natDegree = 0\n⊢ False","tactic":"have : (scaleRoots p s).coeff p.natDegree = 0 :=\n    congr_fun (congr_arg (coeff : R[X] → ℕ → R) h) p.natDegree","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"Polynomial.scaleRoots","def_path":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","def_pos":[30,18],"def_end_pos":[30,28]}]},{"state_before":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis✝ : p.coeff p.natDegree ≠ 0\nthis : (p.scaleRoots s).coeff p.natDegree = 0\n⊢ False","state_after":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis✝ : p.coeff p.natDegree ≠ 0\nthis : p.leadingCoeff = 0\n⊢ False","tactic":"rw [coeff_scaleRoots_natDegree] at this","premises":[{"full_name":"Polynomial.coeff_scaleRoots_natDegree","def_path":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","def_pos":[38,8],"def_end_pos":[38,34]}]},{"state_before":"R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis✝ : p.coeff p.natDegree ≠ 0\nthis : p.leadingCoeff = 0\n⊢ False","state_after":"no goals","tactic":"contradiction","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean","commit":"","full_name":"CategoryTheory.HomOrthogonal.matrixDecomposition_comp","start":[139,0],"end":[158,31],"file_path":"Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean","tactics":[{"state_before":"C : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\ni : ι\n⊢ o.matrixDecomposition (z ≫ w) i = o.matrixDecomposition w i * o.matrixDecomposition z i","state_after":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property : a ∈ f ⁻¹' {h c}\n⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property⟩ =\n    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property⟩","tactic":"ext ⟨c, ⟨⟩⟩ ⟨a, j_property⟩","premises":[]},{"state_before":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property : a ∈ f ⁻¹' {h c}\n⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property⟩ =\n    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property⟩","state_after":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property✝⟩ =\n    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property✝⟩","tactic":"simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property","premises":[{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]},{"full_name":"Set.mem_singleton_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[999,8],"def_end_pos":[999,25]}]},{"state_before":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property✝⟩ =\n    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property✝⟩","state_after":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ w ≫ biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c","tactic":"simp only [Matrix.mul_apply, Limits.biproduct.components,\n    HomOrthogonal.matrixDecomposition_apply, Category.comp_id, Category.id_comp, Category.assoc,\n    End.mul_def, eqToHom_refl, eqToHom_trans_assoc, Finset.sum_congr]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.End.mul_def","def_path":"Mathlib/CategoryTheory/Endomorphism.lean","def_pos":[60,8],"def_end_pos":[60,15]},{"full_name":"CategoryTheory.HomOrthogonal.matrixDecomposition_apply","def_path":"Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean","def_pos":[71,2],"def_end_pos":[71,7]},{"full_name":"CategoryTheory.Limits.biproduct.components","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[933,4],"def_end_pos":[933,24]},{"full_name":"CategoryTheory.eqToHom_refl","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[44,8],"def_end_pos":[44,20]},{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"Matrix.mul_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[849,8],"def_end_pos":[849,17]}]},{"state_before":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ w ≫ biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c","state_after":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ eqToHom ⋯ ≫\n      biproduct.ι (fun a => s (f a)) a ≫\n        z ≫\n          ((∑ j : β, biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫\n            biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c","tactic":"conv_lhs => rw [← Category.id_comp w, ← biproduct.total]","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Limits.biproduct.total","def_path":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","def_pos":[202,8],"def_end_pos":[202,23]}]},{"state_before":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ eqToHom ⋯ ≫\n      biproduct.ι (fun a => s (f a)) a ≫\n        z ≫\n          ((∑ j : β, biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫\n            biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c","state_after":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ ∑ j : β,\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            ((biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫\n              biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c","tactic":"simp only [Preadditive.sum_comp, Preadditive.comp_sum]","premises":[{"full_name":"CategoryTheory.Preadditive.comp_sum","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[162,8],"def_end_pos":[162,16]},{"full_name":"CategoryTheory.Preadditive.sum_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ ∑ j : β,\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            ((biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫\n              biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c","state_after":"case a.mk.refl.mk.w\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ ∀ (x : β) (h_1 : x ∈ g ⁻¹' {h c}),\n    eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            ((biproduct.π (fun b => s (g b)) x ≫ biproduct.ι (fun b => s (g b)) x) ≫ w) ≫\n              biproduct.π (fun b => s (h b)) c =\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑⟨x, h_1⟩ ≫\n              biproduct.ι (fun a => s (g a)) ↑⟨x, h_1⟩ ≫ w ≫ biproduct.π (fun b => s (h b)) c\n\ncase a.mk.refl.mk.w'\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ ∀ x ∉ g ⁻¹' {h c},\n    eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            ((biproduct.π (fun b => s (g b)) x ≫ biproduct.ι (fun b => s (g b)) x) ≫ w) ≫\n              biproduct.π (fun b => s (h b)) c =\n      0","tactic":"apply Finset.sum_congr_set","premises":[{"full_name":"Finset.sum_congr_set","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1002,2],"def_end_pos":[1002,13]}]}]}
{"url":"Mathlib/Topology/Homotopy/Path.lean","commit":"","full_name":"Path.Homotopy.map_apply","start":[217,0],"end":[231,15],"file_path":"Mathlib/Topology/Homotopy/Path.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nf : C(X, Y)\n⊢ ∀ (x : ↑I), { toFun := ⇑f ∘ ⇑F, continuous_toFun := ⋯ }.toFun (0, x) = (p.map ⋯).toContinuousMap x","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nf : C(X, Y)\n⊢ ∀ (x : ↑I), { toFun := ⇑f ∘ ⇑F, continuous_toFun := ⋯ }.toFun (1, x) = (q.map ⋯).toContinuousMap x","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nf : C(X, Y)\nt x : ↑I\nhx : x ∈ {0, 1}\n⊢ { toFun := fun x => { toFun := ⇑f ∘ ⇑F, continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }.toFun (t, x),\n        continuous_toFun := ⋯ }\n      x =\n    (p.map ⋯).toContinuousMap x","state_after":"case inl\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nf : C(X, Y)\nt x : ↑I\nhx : x = 0\n⊢ { toFun := fun x => { toFun := ⇑f ∘ ⇑F, continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }.toFun (t, x),\n        continuous_toFun := ⋯ }\n      x =\n    (p.map ⋯).toContinuousMap x\n\ncase inr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nf : C(X, Y)\nt x : ↑I\nhx : x ∈ {1}\n⊢ { toFun := fun x => { toFun := ⇑f ∘ ⇑F, continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }.toFun (t, x),\n        continuous_toFun := ⋯ }\n      x =\n    (p.map ⋯).toContinuousMap x","tactic":"cases' hx with hx hx","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Group/Prod.lean","commit":"","full_name":"MeasureTheory.measurePreserving_add_prod","start":[354,0],"end":[359,83],"file_path":"Mathlib/MeasureTheory/Group/Prod.lean","tactics":[{"state_before":"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ns : Set G\ninst✝ : μ.IsMulRightInvariant\n⊢ MeasurePreserving ?m.119417 ?m.119413 (?m.119343.prod ?m.119344)","state_after":"no goals","tactic":"apply measurePreserving_prod_mul_swap_right μ ν","premises":[{"full_name":"MeasureTheory.measurePreserving_prod_mul_swap_right","def_path":"Mathlib/MeasureTheory/Group/Prod.lean","def_pos":[350,8],"def_end_pos":[350,45]}]}]}
{"url":"Mathlib/Algebra/Group/Equiv/TypeTags.lean","commit":"","full_name":"AddEquiv.toMultiplicative_symm_apply_symm_apply","start":[16,0],"end":[33,28],"file_path":"Mathlib/Algebra/Group/Equiv/TypeTags.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : AddZeroClass H\nx : G ≃+ H\n⊢ (fun f =>\n        { toFun := ⇑(AddMonoidHom.toMultiplicative.symm f.toMonoidHom),\n          invFun := ⇑(AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n          map_add' := ⋯ })\n      ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative f.toAddMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_mul' := ⋯ })\n        x) =\n    x","state_after":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : AddZeroClass H\nx : G ≃+ H\nx✝ : G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative.symm f.toMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_add' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative f.toAddMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_mul' := ⋯ })\n          x))\n      x✝ =\n    x x✝","tactic":"ext","premises":[]},{"state_before":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : AddZeroClass H\nx : G ≃+ H\nx✝ : G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative.symm f.toMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_add' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative f.toAddMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_mul' := ⋯ })\n          x))\n      x✝ =\n    x x✝","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"G : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : AddZeroClass H\nx : Multiplicative G ≃* Multiplicative H\n⊢ (fun f =>\n        { toFun := ⇑(AddMonoidHom.toMultiplicative f.toAddMonoidHom),\n          invFun := ⇑(AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n          map_mul' := ⋯ })\n      ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative.symm f.toMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_add' := ⋯ })\n        x) =\n    x","state_after":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : AddZeroClass H\nx : Multiplicative G ≃* Multiplicative H\nx✝ : Multiplicative G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative f.toAddMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_mul' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative.symm f.toMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_add' := ⋯ })\n          x))\n      x✝ =\n    x x✝","tactic":"ext","premises":[]},{"state_before":"case h\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddZeroClass G\ninst✝ : AddZeroClass H\nx : Multiplicative G ≃* Multiplicative H\nx✝ : Multiplicative G\n⊢ ((fun f =>\n          { toFun := ⇑(AddMonoidHom.toMultiplicative f.toAddMonoidHom),\n            invFun := ⇑(AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom), left_inv := ⋯, right_inv := ⋯,\n            map_mul' := ⋯ })\n        ((fun f =>\n            { toFun := ⇑(AddMonoidHom.toMultiplicative.symm f.toMonoidHom),\n              invFun := ⇑(AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯,\n              map_add' := ⋯ })\n          x))\n      x✝ =\n    x x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Order/Concept.lean","commit":"","full_name":"Concept.snd_ssubset_snd_iff","start":[213,0],"end":[215,94],"file_path":"Mathlib/Order/Concept.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nκ : ι → Sort u_5\nr : α → β → Prop\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nc d : Concept α β r\n⊢ c.toProd.2 ⊂ d.toProd.2 ↔ d < c","state_after":"no goals","tactic":"rw [ssubset_iff_subset_not_subset, lt_iff_le_not_le, snd_subset_snd_iff, snd_subset_snd_iff]","premises":[{"full_name":"Concept.snd_subset_snd_iff","def_path":"Mathlib/Order/Concept.lean","def_pos":[206,8],"def_end_pos":[206,26]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"lt_iff_le_not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[51,8],"def_end_pos":[51,24]},{"full_name":"ssubset_iff_subset_not_subset","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[614,8],"def_end_pos":[614,37]}]}]}
{"url":"Mathlib/Data/Finset/Lattice.lean","commit":"","full_name":"Finset.sup_inf_distrib_left","start":[451,0],"end":[455,60],"file_path":"Mathlib/Data/Finset/Lattice.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\ns : Finset ι\nf : ι → α\na : α\n⊢ a ⊓ s.sup f = s.sup fun i => a ⊓ f i","state_after":"no goals","tactic":"induction s using Finset.cons_induction with\n  | empty => simp_rw [Finset.sup_empty, inf_bot_eq]\n  | cons _ _ _ h => rw [sup_cons, sup_cons, inf_sup_left, h]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.cons","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[739,4],"def_end_pos":[739,8]},{"full_name":"Finset.cons_induction","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1051,8],"def_end_pos":[1051,22]},{"full_name":"Finset.empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[463,14],"def_end_pos":[463,19]},{"full_name":"Finset.sup_cons","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[53,8],"def_end_pos":[53,16]},{"full_name":"Finset.sup_empty","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[49,8],"def_end_pos":[49,17]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"inf_bot_eq","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[393,6],"def_end_pos":[393,16]},{"full_name":"inf_sup_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[611,8],"def_end_pos":[611,20]}]}]}
{"url":"Mathlib/Data/Complex/Module.lean","commit":"","full_name":"skewAdjoint.I_smul_neg_I","start":[354,0],"end":[356,12],"file_path":"Mathlib/Data/Complex/Module.lean","tactics":[{"state_before":"A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ↥(skewAdjoint A)\n⊢ I • ↑(negISMul a) = ↑a","state_after":"no goals","tactic":"simp only [smul_smul, skewAdjoint.negISMul_apply_coe, neg_smul, smul_neg, I_mul_I, one_smul,\n    neg_neg]","premises":[{"full_name":"Complex.I_mul_I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[243,8],"def_end_pos":[243,15]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"skewAdjoint.negISMul_apply_coe","def_path":"Mathlib/Data/Complex/Module.lean","def_pos":[339,2],"def_end_pos":[339,7]},{"full_name":"smul_neg","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[276,8],"def_end_pos":[276,16]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Ordinal.lean","commit":"","full_name":"Cardinal.mk_compl_eq_mk_compl_finite_lift","start":[1193,0],"end":[1205,41],"file_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","state_after":"case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","tactic":"cases nonempty_fintype α","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","state_after":"case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","tactic":"rcases lift_mk_eq.{u, v, w}.1 h1 with ⟨e⟩","premises":[{"full_name":"Cardinal.lift_mk_eq","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[290,8],"def_end_pos":[290,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","state_after":"case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\nthis : Fintype β := Fintype.ofEquiv α e\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","tactic":"letI : Fintype β := Fintype.ofEquiv α e","premises":[{"full_name":"Fintype","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[55,6],"def_end_pos":[55,13]},{"full_name":"Fintype.ofEquiv","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[500,4],"def_end_pos":[500,11]}]},{"state_before":"case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\nthis : Fintype β := Fintype.ofEquiv α e\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","state_after":"case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\nthis : Fintype β := Fintype.ofEquiv α e\nh1 : Fintype.card α = Fintype.card β\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","tactic":"replace h1 : Fintype.card α = Fintype.card β := (Fintype.ofEquiv_card _).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Fintype.ofEquiv_card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[137,8],"def_end_pos":[137,20]}]},{"state_before":"case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\nthis : Fintype β := Fintype.ofEquiv α e\nh1 : Fintype.card α = Fintype.card β\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ","state_after":"no goals","tactic":"classical\n    lift s to Finset α using s.toFinite\n    lift t to Finset β using t.toFinite\n    simp only [Finset.coe_sort_coe, mk_fintype, Fintype.card_coe, lift_natCast, Nat.cast_inj] at h2\n    simp only [← Finset.coe_compl, Finset.coe_sort_coe, mk_coe_finset, Finset.card_compl,\n      lift_natCast, Nat.cast_inj, h1, h2]","premises":[{"full_name":"Cardinal.lift_natCast","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1114,8],"def_end_pos":[1114,20]},{"full_name":"Cardinal.mk_coe_finset","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1215,8],"def_end_pos":[1215,21]},{"full_name":"Cardinal.mk_fintype","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[404,8],"def_end_pos":[404,18]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card_compl","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[260,8],"def_end_pos":[260,25]},{"full_name":"Finset.coe_compl","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[163,8],"def_end_pos":[163,17]},{"full_name":"Finset.coe_sort_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[258,8],"def_end_pos":[258,20]},{"full_name":"Fintype.card_coe","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[651,8],"def_end_pos":[651,24]},{"full_name":"Nat.cast_inj","def_path":"Mathlib/Algebra/CharZero/Defs.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"Set.toFinite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[71,8],"def_end_pos":[71,16]}]}]}
{"url":"Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean","commit":"","full_name":"CategoryTheory.TwoSquare.GuitartExact.whiskerVertical_iff","start":[55,0],"end":[70,31],"file_path":"Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean","tactics":[{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁵ : Category.{u_8, u_1} C₁\ninst✝⁴ : Category.{u_7, u_2} C₂\ninst✝³ : Category.{?u.17017, u_3} C₃\ninst✝² : Category.{u_10, u_4} D₁\ninst✝¹ : Category.{u_9, u_5} D₂\ninst✝ : Category.{?u.17029, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\nα : L ≅ L'\nβ : R ≅ R'\n⊢ (w.whiskerVertical α.hom β.inv).GuitartExact ↔ w.GuitartExact","state_after":"case mp\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁵ : Category.{u_8, u_1} C₁\ninst✝⁴ : Category.{u_7, u_2} C₂\ninst✝³ : Category.{?u.17017, u_3} C₃\ninst✝² : Category.{u_10, u_4} D₁\ninst✝¹ : Category.{u_9, u_5} D₂\ninst✝ : Category.{?u.17029, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\nα : L ≅ L'\nβ : R ≅ R'\n⊢ (w.whiskerVertical α.hom β.inv).GuitartExact → w.GuitartExact\n\ncase mpr\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁵ : Category.{u_8, u_1} C₁\ninst✝⁴ : Category.{u_7, u_2} C₂\ninst✝³ : Category.{?u.17017, u_3} C₃\ninst✝² : Category.{u_10, u_4} D₁\ninst✝¹ : Category.{u_9, u_5} D₂\ninst✝ : Category.{?u.17029, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\nα : L ≅ L'\nβ : R ≅ R'\n⊢ w.GuitartExact → (w.whiskerVertical α.hom β.inv).GuitartExact","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Data/Complex/Module.lean","commit":"","full_name":"Complex.toMatrix_conjAe","start":[250,0],"end":[256,65],"file_path":"Mathlib/Data/Complex/Module.lean","tactics":[{"state_before":"⊢ (LinearMap.toMatrix basisOneI basisOneI) conjAe.toLinearMap = !![1, 0; 0, -1]","state_after":"case a\ni j : Fin 2\n⊢ (LinearMap.toMatrix basisOneI basisOneI) conjAe.toLinearMap i j = !![1, 0; 0, -1] i j","tactic":"ext i j","premises":[]},{"state_before":"case a\ni j : Fin 2\n⊢ (LinearMap.toMatrix basisOneI basisOneI) conjAe.toLinearMap i j = !![1, 0; 0, -1] i j","state_after":"no goals","tactic":"fin_cases i <;> fin_cases j <;> simp [LinearMap.toMatrix_apply]","premises":[{"full_name":"LinearMap.toMatrix_apply","def_path":"Mathlib/LinearAlgebra/Matrix/ToLin.lean","def_pos":[527,8],"def_end_pos":[527,32]}]}]}
{"url":"Mathlib/Data/Finset/Union.lean","commit":"","full_name":"Finset.disjiUnion_empty","start":[48,0],"end":[48,87],"file_path":"Mathlib/Data/Finset/Union.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns s₁ s₂ : Finset α\nt✝ t₁ t₂ t : α → Finset β\n⊢ (↑∅).PairwiseDisjoint t","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":".lake/packages/batteries/Batteries/Data/UInt.lean","commit":"","full_name":"USize.size_le","start":[220,0],"end":[223,53],"file_path":".lake/packages/batteries/Batteries/Data/UInt.lean","tactics":[{"state_before":"⊢ size ≤ 2 ^ 64","state_after":"⊢ 2 ^ System.Platform.numBits ≤ 2 ^ 64","tactic":"rw [size_eq]","premises":[{"full_name":"USize.size_eq","def_path":".lake/packages/batteries/Batteries/Data/UInt.lean","def_pos":[211,8],"def_end_pos":[211,21]}]},{"state_before":"⊢ 2 ^ System.Platform.numBits ≤ 2 ^ 64","state_after":"⊢ System.Platform.numBits ≤ 64","tactic":"apply Nat.pow_le_pow_of_le_right (by decide)","premises":[{"full_name":"Nat.pow_le_pow_of_le_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[761,8],"def_end_pos":[761,30]}]},{"state_before":"⊢ System.Platform.numBits ≤ 64","state_after":"no goals","tactic":"cases System.Platform.numBits_eq <;> simp_arith [*]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Lean.Meta.Simp.Config.autoUnfold","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[187,2],"def_end_pos":[187,12]},{"full_name":"Lean.Meta.Simp.Config.beta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[157,2],"def_end_pos":[157,6]},{"full_name":"Lean.Meta.Simp.Config.contextual","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[136,2],"def_end_pos":[136,12]},{"full_name":"Lean.Meta.Simp.Config.dsimp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[193,2],"def_end_pos":[193,7]},{"full_name":"Lean.Meta.Simp.Config.eta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[162,2],"def_end_pos":[162,5]},{"full_name":"Lean.Meta.Simp.Config.etaStruct","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[167,2],"def_end_pos":[167,11]},{"full_name":"Lean.Meta.Simp.Config.failIfUnchanged","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[198,2],"def_end_pos":[198,17]},{"full_name":"Lean.Meta.Simp.Config.ground","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[205,2],"def_end_pos":[205,8]},{"full_name":"Lean.Meta.Simp.Config.implicitDefEqProofs","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[228,2],"def_end_pos":[228,21]},{"full_name":"Lean.Meta.Simp.Config.index","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[220,2],"def_end_pos":[220,7]},{"full_name":"Lean.Meta.Simp.Config.iota","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[171,2],"def_end_pos":[171,6]},{"full_name":"Lean.Meta.Simp.Config.maxDischargeDepth","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[131,2],"def_end_pos":[131,19]},{"full_name":"Lean.Meta.Simp.Config.maxSteps","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[126,2],"def_end_pos":[126,10]},{"full_name":"Lean.Meta.Simp.Config.memoize","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[140,2],"def_end_pos":[140,9]},{"full_name":"Lean.Meta.Simp.Config.proj","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[175,2],"def_end_pos":[175,6]},{"full_name":"Lean.Meta.Simp.Config.singlePass","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[146,2],"def_end_pos":[146,12]},{"full_name":"Lean.Meta.Simp.Config.unfoldPartialApp","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[210,2],"def_end_pos":[210,18]},{"full_name":"Lean.Meta.Simp.Config.zeta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[152,2],"def_end_pos":[152,6]},{"full_name":"Lean.Meta.Simp.Config.zetaDelta","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[215,2],"def_end_pos":[215,11]},{"full_name":"System.Platform.numBits_eq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1823,8],"def_end_pos":[1823,34]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/RiemannZeta.lean","commit":"","full_name":"riemannZeta_residue_one","start":[201,0],"end":[203,37],"file_path":"Mathlib/NumberTheory/LSeries/RiemannZeta.lean","tactics":[{"state_before":"⊢ Tendsto (fun s => (s - 1) * riemannZeta s) (𝓝[≠] 1) (𝓝 1)","state_after":"no goals","tactic":"exact hurwitzZetaEven_residue_one 0","premises":[{"full_name":"HurwitzZeta.hurwitzZetaEven_residue_one","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","def_pos":[641,6],"def_end_pos":[641,33]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/Polynomial.lean","commit":"","full_name":"Polynomial.natDegree_det_X_add_C_le","start":[36,0],"end":[56,16],"file_path":"Mathlib/LinearAlgebra/Matrix/Polynomial.lean","tactics":[{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ (X • A.map ⇑C + B.map ⇑C).det.natDegree ≤ Fintype.card n","state_after":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ (∑ σ : Equiv.Perm n, sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i).natDegree ≤ Fintype.card n","tactic":"rw [det_apply]","premises":[{"full_name":"Matrix.det_apply","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[59,8],"def_end_pos":[59,17]}]},{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ (∑ σ : Equiv.Perm n, sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i).natDegree ≤ Fintype.card n","state_after":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ Finset.fold max 0 (natDegree ∘ fun σ => sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i) Finset.univ ≤\n    Fintype.card n","tactic":"refine (natDegree_sum_le _ _).trans ?_","premises":[{"full_name":"Polynomial.natDegree_sum_le","def_path":"Mathlib/Algebra/Polynomial/BigOperators.lean","def_pos":[53,8],"def_end_pos":[53,24]}]},{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ Finset.fold max 0 (natDegree ∘ fun σ => sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i) Finset.univ ≤\n    Fintype.card n","state_after":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∀ x ∈ Multiset.map (natDegree ∘ fun σ => sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i) Finset.univ.val,\n    x ≤ Fintype.card n","tactic":"refine Multiset.max_le_of_forall_le _ _ ?_","premises":[{"full_name":"Multiset.max_le_of_forall_le","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean","def_pos":[153,6],"def_end_pos":[153,25]}]},{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∀ x ∈ Multiset.map (natDegree ∘ fun σ => sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i) Finset.univ.val,\n    x ≤ Fintype.card n","state_after":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∀ (a : Equiv.Perm n), (sign a • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (a i) i).natDegree ≤ Fintype.card n","tactic":"simp only [forall_apply_eq_imp_iff, true_and_iff, Function.comp_apply, Multiset.map_map,\n    Multiset.mem_map, exists_imp, Finset.mem_univ_val]","premises":[{"full_name":"Finset.mem_univ_val","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[75,8],"def_end_pos":[75,20]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Multiset.map_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1129,8],"def_end_pos":[1129,15]},{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]},{"full_name":"exists_imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[200,8],"def_end_pos":[200,18]},{"full_name":"forall_apply_eq_imp_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[321,16],"def_end_pos":[321,39]},{"full_name":"true_and_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[94,8],"def_end_pos":[94,20]}]},{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∀ (a : Equiv.Perm n), (sign a • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (a i) i).natDegree ≤ Fintype.card n","state_after":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ (sign g • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (g i) i).natDegree ≤ Fintype.card n","tactic":"intro g","premises":[]},{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\ni : n\nx✝ : i ∈ Finset.univ\n⊢ ((X • A.map ⇑C + B.map ⇑C) (g i) i).natDegree ≤ 1","state_after":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\ni : n\nx✝ : i ∈ Finset.univ\n⊢ (X * C (A (g i) i) + C (B (g i) i)).natDegree ≤ 1","tactic":"dsimp only [add_apply, smul_apply, map_apply, smul_eq_mul]","premises":[{"full_name":"Matrix.add_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[249,8],"def_end_pos":[249,17]},{"full_name":"Matrix.map_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Matrix.smul_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[253,8],"def_end_pos":[253,18]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\ni : n\nx✝ : i ∈ Finset.univ\n⊢ (X * C (A (g i) i) + C (B (g i) i)).natDegree ≤ 1","state_after":"no goals","tactic":"compute_degree","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Nat.cast_withBot","def_path":"Mathlib/Data/Nat/Cast/WithTop.lean","def_pos":[23,8],"def_end_pos":[23,24]}]}]}
{"url":"Mathlib/Data/Nat/Size.lean","commit":"","full_name":"Nat.shiftLeft'_ne_zero_left","start":[27,0],"end":[28,51],"file_path":"Mathlib/Data/Nat/Size.lean","tactics":[{"state_before":"b : Bool\nm : ℕ\nh : m ≠ 0\nn : ℕ\n⊢ shiftLeft' b m n ≠ 0","state_after":"no goals","tactic":"induction n <;> simp [bit_ne_zero, shiftLeft', *]","premises":[{"full_name":"Nat.bit_ne_zero","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[287,8],"def_end_pos":[287,19]},{"full_name":"Nat.shiftLeft'","def_path":"Mathlib/Data/Nat/Bits.lean","def_pos":[135,4],"def_end_pos":[135,14]}]}]}
{"url":"Mathlib/Topology/ContinuousOn.lean","commit":"","full_name":"Inducing.continuousOn_iff","start":[990,0],"end":[992,51],"file_path":"Mathlib/Topology/ContinuousOn.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nι : Type u_5\nπ : ι → Type u_6\ninst✝³ : (i : ι) → TopologicalSpace (π i)\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ng : β → γ\nhg : Inducing g\ns : Set α\n⊢ ContinuousOn f s ↔ ContinuousOn (g ∘ f) s","state_after":"no goals","tactic":"simp_rw [ContinuousOn, hg.continuousWithinAt_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ContinuousOn","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[164,4],"def_end_pos":[164,16]},{"full_name":"Inducing.continuousWithinAt_iff","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[986,8],"def_end_pos":[986,39]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.foldr_cons_eq_toList","start":[461,0],"end":[465,59],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\nsize✝ : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\nr' : List α\n⊢ foldr List.cons (node size✝ l x r) r' = (node size✝ l x r).toList ++ r'","state_after":"α : Type u_1\nsize✝ : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\nr' : List α\n⊢ foldr List.cons l (x :: r.toList) ++ r' = (node size✝ l x r).toList ++ r'","tactic":"rw [foldr, foldr_cons_eq_toList l, foldr_cons_eq_toList r, ← List.cons_append,\n        ← List.append_assoc, ← foldr_cons_eq_toList l]","premises":[{"full_name":"List.append_assoc","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[501,16],"def_end_pos":[501,28]},{"full_name":"List.cons_append","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[484,16],"def_end_pos":[484,27]},{"full_name":"Ordnode.foldr","def_path":"Mathlib/Data/Ordmap/Ordnode.lean","def_pos":[575,4],"def_end_pos":[575,9]}]},{"state_before":"α : Type u_1\nsize✝ : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\nr' : List α\n⊢ foldr List.cons l (x :: r.toList) ++ r' = (node size✝ l x r).toList ++ r'","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Nat/Factorization/Induction.lean","commit":"","full_name":"Nat.multiplicative_factorization'","start":[116,0],"end":[123,54],"file_path":"Mathlib/Data/Nat/Factorization/Induction.lean","tactics":[{"state_before":"a b m n p : ℕ\nβ : Type u_1\ninst✝ : CommMonoid β\nf : ℕ → β\nh_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n⊢ f n = n.factorization.prod fun p k => f (p ^ k)","state_after":"case inl\na b m p : ℕ\nβ : Type u_1\ninst✝ : CommMonoid β\nf : ℕ → β\nh_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n⊢ f 0 = (factorization 0).prod fun p k => f (p ^ k)\n\ncase inr\na b m n p : ℕ\nβ : Type u_1\ninst✝ : CommMonoid β\nf : ℕ → β\nh_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\nhn : n ≠ 0\n⊢ f n = n.factorization.prod fun p k => f (p ^ k)","tactic":"obtain rfl | hn := eq_or_ne n 0","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
{"url":"Mathlib/MeasureTheory/Function/LpSpace.lean","commit":"","full_name":"ContinuousMap.toLp_norm_le","start":[1897,0],"end":[1901,48],"file_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : CompactSpace α\ninst✝³ : IsFiniteMeasure μ\n𝕜 : Type u_5\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\n⊢ ‖toLp p μ 𝕜‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : CompactSpace α\ninst✝³ : IsFiniteMeasure μ\n𝕜 : Type u_5\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\n⊢ ‖BoundedContinuousFunction.toLp p μ 𝕜‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹","tactic":"rw [toLp_norm_eq_toLp_norm_coe]","premises":[{"full_name":"ContinuousMap.toLp_norm_eq_toLp_norm_coe","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[1892,8],"def_end_pos":[1892,34]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : CompactSpace α\ninst✝³ : IsFiniteMeasure μ\n𝕜 : Type u_5\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\n⊢ ‖BoundedContinuousFunction.toLp p μ 𝕜‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹","state_after":"no goals","tactic":"exact BoundedContinuousFunction.toLp_norm_le μ","premises":[{"full_name":"BoundedContinuousFunction.toLp_norm_le","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[1810,8],"def_end_pos":[1810,20]}]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Normed.lean","commit":"","full_name":"summable_powerSeries_of_norm_lt","start":[590,0],"end":[601,72],"file_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\n⊢ Summable fun n => f n * z ^ n","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\n⊢ Summable fun n => f n * z ^ n","tactic":"have hw : 0 < ‖w‖ := (norm_nonneg z).trans_lt hz","premises":[{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\n⊢ Summable fun n => f n * z ^ n","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\n⊢ Summable fun n => f n * z ^ n","tactic":"obtain ⟨C, hC⟩ := exists_norm_le_of_cauchySeq h","premises":[{"full_name":"exists_norm_le_of_cauchySeq","def_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","def_pos":[454,6],"def_end_pos":[454,33]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\n⊢ Summable fun n => f n * z ^ n","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\n⊢ CauchySeq fun s => ∑ b ∈ s, f b * z ^ b","tactic":"rw [summable_iff_cauchySeq_finset]","premises":[{"full_name":"summable_iff_cauchySeq_finset","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Group.lean","def_pos":[168,2],"def_end_pos":[168,13]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\n⊢ CauchySeq fun s => ∑ b ∈ s, f b * z ^ b","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\nn : ℕ\n⊢ ‖f n * z ^ n‖ ≤ C * (‖z‖ / ‖w‖) ^ n","tactic":"refine cauchySeq_finset_of_geometric_bound (r := ‖z‖ / ‖w‖) (C := C) ((div_lt_one hw).mpr hz)\n    (fun n ↦ ?_)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"cauchySeq_finset_of_geometric_bound","def_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","def_pos":[401,8],"def_end_pos":[401,43]},{"full_name":"div_lt_one","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[303,8],"def_end_pos":[303,18]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\nn : ℕ\n⊢ ‖f n * z ^ n‖ ≤ C * (‖z‖ / ‖w‖) ^ n","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\nn : ℕ\n⊢ ‖f n‖ * ‖z‖ ^ n ≤ C / ‖w‖ ^ n * ‖z‖ ^ n","tactic":"rw [norm_mul, norm_pow, div_pow, ← mul_comm_div]","premises":[{"full_name":"div_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[589,6],"def_end_pos":[589,13]},{"full_name":"mul_comm_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[564,8],"def_end_pos":[564,20]},{"full_name":"norm_mul","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[661,8],"def_end_pos":[661,16]},{"full_name":"norm_pow","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[694,8],"def_end_pos":[694,16]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C\nn : ℕ\n⊢ ‖f n‖ * ‖z‖ ^ n ≤ C / ‖w‖ ^ n * ‖z‖ ^ n","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n‖ ≤ C / ‖w‖ ^ n\nn : ℕ\n⊢ ‖f n‖ * ‖z‖ ^ n ≤ C / ‖w‖ ^ n * ‖z‖ ^ n","tactic":"conv at hC => enter [n]; rw [norm_mul, norm_pow, ← _root_.le_div_iff (by positivity)]","premises":[{"full_name":"le_div_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[41,8],"def_end_pos":[41,18]},{"full_name":"norm_mul","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[661,8],"def_end_pos":[661,16]},{"full_name":"norm_pow","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[694,8],"def_end_pos":[694,16]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : NormedDivisionRing α\ninst✝ : CompleteSpace α\nf : ℕ → α\nw z : α\nh : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i\nhz : ‖z‖ < ‖w‖\nhw : 0 < ‖w‖\nC : ℝ\nhC : ∀ (n : ℕ), ‖f n‖ ≤ C / ‖w‖ ^ n\nn : ℕ\n⊢ ‖f n‖ * ‖z‖ ^ n ≤ C / ‖w‖ ^ n * ‖z‖ ^ n","state_after":"no goals","tactic":"exact mul_le_mul_of_nonneg_right (hC n) (pow_nonneg (norm_nonneg z) n)","premises":[{"full_name":"mul_le_mul_of_nonneg_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[194,8],"def_end_pos":[194,34]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]},{"full_name":"pow_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[141,8],"def_end_pos":[141,18]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.Products.isGood_nil","start":[782,0],"end":[785,62],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\n⊢ isGood {fun x => false} nil","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nh : eval {fun x => false} nil ∈ Submodule.span ℤ (eval {fun x => false} '' {m | m < nil})\n⊢ False","tactic":"intro h","premises":[]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nh : eval {fun x => false} nil ∈ Submodule.span ℤ (eval {fun x => false} '' {m | m < nil})\n⊢ False","state_after":"no goals","tactic":"simp only [Products.lt_nil_empty, Products.eval, List.map, List.prod_nil, Set.image_empty,\n    Submodule.span_empty, Submodule.mem_bot, one_ne_zero] at h","premises":[{"full_name":"List.map","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[361,18],"def_end_pos":[361,21]},{"full_name":"List.prod_nil","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"Profinite.NobelingProof.Products.eval","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[322,4],"def_end_pos":[322,8]},{"full_name":"Profinite.NobelingProof.Products.lt_nil_empty","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[774,8],"def_end_pos":[774,29]},{"full_name":"Set.image_empty","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[265,8],"def_end_pos":[265,19]},{"full_name":"Submodule.mem_bot","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[68,8],"def_end_pos":[68,15]},{"full_name":"Submodule.span_empty","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[277,8],"def_end_pos":[277,18]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]}]}]}
{"url":"Mathlib/Order/Hom/Lattice.lean","commit":"","full_name":"Disjoint.map","start":[263,0],"end":[264,49],"file_path":"Mathlib/Order/Hom/Lattice.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nδ : Type u_6\ninst✝⁵ : Lattice α\ninst✝⁴ : BoundedOrder α\ninst✝³ : Lattice β\ninst✝² : BoundedOrder β\ninst✝¹ : FunLike F α β\ninst✝ : BoundedLatticeHomClass F α β\nf : F\na b : α\nh : Disjoint a b\n⊢ Disjoint (f a) (f b)","state_after":"no goals","tactic":"rw [disjoint_iff, ← map_inf, h.eq_bot, map_bot]","premises":[{"full_name":"BotHomClass.map_bot","def_path":"Mathlib/Order/Hom/Bounded.lean","def_pos":[71,2],"def_end_pos":[71,9]},{"full_name":"Disjoint.eq_bot","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[122,8],"def_end_pos":[122,23]},{"full_name":"InfHomClass.map_inf","def_path":"Mathlib/Order/Hom/Lattice.lean","def_pos":[103,2],"def_end_pos":[103,9]},{"full_name":"disjoint_iff","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[116,8],"def_end_pos":[116,20]}]}]}
{"url":"Mathlib/Algebra/GeomSum.lean","commit":"","full_name":"mul_geom_sum","start":[208,0],"end":[209,54],"file_path":"Mathlib/Algebra/GeomSum.lean","tactics":[{"state_before":"α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\n⊢ op ((x - 1) * ∑ i ∈ range n, x ^ i) = op (x ^ n - 1)","state_after":"no goals","tactic":"simpa using geom_sum_mul (op x) n","premises":[{"full_name":"MulOpposite.op","def_path":"Mathlib/Algebra/Opposites.lean","def_pos":[74,4],"def_end_pos":[74,6]},{"full_name":"geom_sum_mul","def_path":"Mathlib/Algebra/GeomSum.lean","def_pos":[203,8],"def_end_pos":[203,20]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Associated.lean","commit":"","full_name":"Associates.finset_prod_mk","start":[116,0],"end":[122,35],"file_path":"Mathlib/Algebra/BigOperators/Associated.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CommMonoid α\np : Finset β\nf : β → α\n⊢ ∏ i ∈ p, Associates.mk (f i) = Associates.mk (∏ i ∈ p, f i)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CommMonoid α\np : Finset β\nf : β → α\nthis : (fun i => Associates.mk (f i)) = Associates.mk ∘ f\n⊢ ∏ i ∈ p, Associates.mk (f i) = Associates.mk (∏ i ∈ p, f i)","tactic":"have : (fun i => Associates.mk (f i)) = Associates.mk ∘ f :=\n    funext fun x => Function.comp_apply","premises":[{"full_name":"Associates.mk","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[737,17],"def_end_pos":[737,19]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CommMonoid α\np : Finset β\nf : β → α\nthis : (fun i => Associates.mk (f i)) = Associates.mk ∘ f\n⊢ ∏ i ∈ p, Associates.mk (f i) = Associates.mk (∏ i ∈ p, f i)","state_after":"no goals","tactic":"rw [Finset.prod_eq_multiset_prod, this, ← Multiset.map_map, prod_mk,\n    ← Finset.prod_eq_multiset_prod]","premises":[{"full_name":"Associates.prod_mk","def_path":"Mathlib/Algebra/BigOperators/Associated.lean","def_pos":[113,8],"def_end_pos":[113,15]},{"full_name":"Finset.prod_eq_multiset_prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[267,8],"def_end_pos":[267,29]},{"full_name":"Multiset.map_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1129,8],"def_end_pos":[1129,15]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Mul.lean","commit":"","full_name":"HasStrictFDerivAt.multiset_prod","start":[695,0],"end":[703,78],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Mul.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g✝ : E → F\nf'✝ f₀' f₁' g'✝ e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_6\n𝔸 : Type u_7\n𝔸' : Type u_8\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedCommRing 𝔸'\ninst✝² : NormedAlgebra 𝕜 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu✝ : Finset ι\nf : ι → E → 𝔸\nf' : ι → E →L[𝕜] 𝔸\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nu : Multiset ι\nx : E\nh : ∀ i ∈ u, HasStrictFDerivAt (fun x => g i x) (g' i) x\n⊢ HasStrictFDerivAt (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod)\n    (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • g' i) u).sum x","state_after":"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g✝ : E → F\nf'✝ f₀' f₁' g'✝ e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_6\n𝔸 : Type u_7\n𝔸' : Type u_8\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedCommRing 𝔸'\ninst✝² : NormedAlgebra 𝕜 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu✝ : Finset ι\nf : ι → E → 𝔸\nf' : ι → E →L[𝕜] 𝔸\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nu : Multiset ι\nx : E\nh : ∀ i ∈ u, HasStrictFDerivAt (fun x => g i x) (g' i) x\n⊢ HasStrictFDerivAt (fun x => (Multiset.map ((fun x_1 => g x_1 x) ∘ Subtype.val) u.attach).prod)\n    (Multiset.map\n        ((fun x_1 =>\n            (Multiset.map (fun x_2 => g x_2 x) ((Multiset.map Subtype.val u.attach).erase x_1)).prod • g' x_1) ∘\n          Subtype.val)\n        u.attach).sum\n    x","tactic":"simp only [← Multiset.attach_map_val u, Multiset.map_map]","premises":[{"full_name":"Multiset.attach_map_val","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1354,8],"def_end_pos":[1354,22]},{"full_name":"Multiset.map_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1129,8],"def_end_pos":[1129,15]}]}]}
{"url":"Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean","commit":"","full_name":"intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_lip","start":[67,0],"end":[86,37],"file_path":"Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF_int : IntervalIntegrable (F x₀) μ a b\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → LipschitzOnWith (Real.nnabs (bound t)) (fun x => F x t) (ball x₀ ε)\nbound_integrable : IntervalIntegrable bound μ a b\nh_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasDerivAt (fun x => F x t) (F' t) x₀\n⊢ IntervalIntegrable F' μ a b ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀","state_after":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF_int : IntervalIntegrable (F x₀) μ a b\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nbound_integrable : IntervalIntegrable bound μ a b\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\n⊢ IntervalIntegrable F' μ a b ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀","tactic":"rw [← ae_restrict_iff' measurableSet_uIoc] at h_lipsch h_diff","premises":[{"full_name":"MeasureTheory.ae_restrict_iff'","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[547,8],"def_end_pos":[547,24]},{"full_name":"measurableSet_uIoc","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[421,8],"def_end_pos":[421,26]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF_int : IntervalIntegrable (F x₀) μ a b\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nbound_integrable : IntervalIntegrable bound μ a b\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\n⊢ IntervalIntegrable F' μ a b ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀","state_after":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\n⊢ IntegrableOn F' (Ι a b) μ ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀","tactic":"simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢","premises":[{"full_name":"intervalIntegrable_iff","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[80,8],"def_end_pos":[80,30]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\n⊢ IntegrableOn F' (Ι a b) μ ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀","state_after":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\n⊢ IntegrableOn F' (Ι a b) μ ∧\n    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)\n      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀","tactic":"simp only [intervalIntegral_eq_integral_uIoc]","premises":[{"full_name":"intervalIntegral.intervalIntegral_eq_integral_uIoc","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[443,8],"def_end_pos":[443,41]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\n⊢ IntegrableOn F' (Ι a b) μ ∧\n    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)\n      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀","state_after":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\nthis :\n  Integrable F' (μ.restrict (Ι a b)) ∧\n    HasDerivAt (fun x => ∫ (a : ℝ) in Ι a b, F x a ∂μ) (∫ (a : ℝ) in Ι a b, F' a ∂μ) x₀\n⊢ IntegrableOn F' (Ι a b) μ ∧\n    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)\n      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀","tactic":"have := hasDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas h_lipsch\n    bound_integrable h_diff","premises":[{"full_name":"hasDerivAt_integral_of_dominated_loc_of_lip","def_path":"Mathlib/Analysis/Calculus/ParametricIntegral.lean","def_pos":[253,8],"def_end_pos":[253,51]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\nthis :\n  Integrable F' (μ.restrict (Ι a b)) ∧\n    HasDerivAt (fun x => ∫ (a : ℝ) in Ι a b, F x a ∂μ) (∫ (a : ℝ) in Ι a b, F' a ∂μ) x₀\n⊢ IntegrableOn F' (Ι a b) μ ∧\n    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)\n      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀","state_after":"no goals","tactic":"exact ⟨this.1, this.2.const_smul _⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"HasDerivAt.const_smul","def_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","def_pos":[149,15],"def_end_pos":[149,36]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.coe_πs","start":[870,0],"end":[872,5],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nf : LocallyConstant ↑(π C fun x => ord I x < o) ℤ\n⊢ ⇑((πs C o) f) = ⇑f ∘ ProjRestrict C fun x => ord I x < o","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Algebra/Operations.lean","commit":"","full_name":"Submodule.mem_span_mul_finite_of_mem_mul","start":[324,0],"end":[327,90],"file_path":"Mathlib/Algebra/Algebra/Operations.lean","tactics":[{"state_before":"ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P✝ Q✝ : Submodule R A\nm n : A\nP Q : Submodule R A\nx : A\nhx : x ∈ P * Q\n⊢ x ∈ span R (↑P * ↑Q)","state_after":"no goals","tactic":"rwa [← Submodule.span_eq P, ← Submodule.span_eq Q, Submodule.span_mul_span] at hx","premises":[{"full_name":"Submodule.span_eq","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[85,8],"def_end_pos":[85,15]},{"full_name":"Submodule.span_mul_span","def_path":"Mathlib/Algebra/Algebra/Operations.lean","def_pos":[173,8],"def_end_pos":[173,21]}]}]}
{"url":"Mathlib/Dynamics/Minimal.lean","commit":"","full_name":"isMinimal_iff_closed_smul_invariant","start":[96,0],"end":[104,40],"file_path":"Mathlib/Dynamics/Minimal.lean","tactics":[{"state_before":"M : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\n⊢ IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ","state_after":"case mp\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\n⊢ IsMinimal M α → ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\n\ncase mpr\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\n⊢ (∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ) → IsMinimal M α","tactic":"constructor","premises":[]},{"state_before":"case mpr\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\n⊢ (∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ) → IsMinimal M α","state_after":"case mpr.refine_1\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ IsClosed (closure (orbit M x✝))\n\ncase mpr.refine_2\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ ∀ (c : M), c • closure (orbit M x✝) ⊆ closure (orbit M x✝)\n\ncase mpr.refine_3\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ ¬closure (orbit M x✝) = ∅","tactic":"refine fun H ↦ ⟨fun _ ↦ dense_iff_closure_eq.2 <| (H _ ?_ ?_).resolve_left ?_⟩","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]},{"full_name":"dense_iff_closure_eq","def_path":"Mathlib/Topology/Basic.lean","def_pos":[502,8],"def_end_pos":[502,28]}]},{"state_before":"case mpr.refine_1\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ IsClosed (closure (orbit M x✝))\n\ncase mpr.refine_2\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ ∀ (c : M), c • closure (orbit M x✝) ⊆ closure (orbit M x✝)\n\ncase mpr.refine_3\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ ¬closure (orbit M x✝) = ∅","state_after":"no goals","tactic":"exacts [isClosed_closure, fun _ ↦ smul_closure_orbit_subset _ _,\n    (orbit_nonempty _).closure.ne_empty]","premises":[{"full_name":"MulAction.orbit_nonempty","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[65,8],"def_end_pos":[65,22]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]},{"full_name":"smul_closure_orbit_subset","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[160,8],"def_end_pos":[160,33]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Bernstein.lean","commit":"","full_name":"bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt","start":[134,0],"end":[149,66],"file_path":"Mathlib/RingTheory/Polynomial/Bernstein.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nn ν k : ℕ\n⊢ k < ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n ν)) = 0","state_after":"case zero\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\n⊢ k < 0 → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n 0)) = 0\n\ncase succ\nR : Type u_1\ninst✝ : CommRing R\nn k ν : ℕ\n⊢ k < ν + 1 → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0","tactic":"cases' ν with ν","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","commit":"","full_name":"CategoryTheory.IsPullback.isoIsPullback_inv_snd","start":[296,0],"end":[299,52],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nP' : C\nfst' : P' ⟶ X\nsnd' : P' ⟶ Y\nh : IsPullback fst snd f g\nh' : IsPullback fst' snd' f g\n⊢ (isoIsPullback X Y h h').inv ≫ snd = snd'","state_after":"no goals","tactic":"simp only [Iso.inv_comp_eq, isoIsPullback_hom_snd]","premises":[{"full_name":"CategoryTheory.IsPullback.isoIsPullback_hom_snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","def_pos":[287,8],"def_end_pos":[287,29]},{"full_name":"CategoryTheory.Iso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[178,8],"def_end_pos":[178,19]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Associated.lean","commit":"","full_name":"Prime.exists_mem_multiset_map_dvd","start":[36,0],"end":[39,32],"file_path":"Mathlib/Algebra/BigOperators/Associated.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CommMonoidWithZero α\np : α\nhp : Prime p\ns : Multiset β\nf : β → α\nh : p ∣ (Multiset.map f s).prod\n⊢ ∃ a ∈ s, p ∣ f a","state_after":"no goals","tactic":"simpa only [exists_prop, Multiset.mem_map, exists_exists_and_eq_and] using\n    hp.exists_mem_multiset_dvd h","premises":[{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]},{"full_name":"Prime.exists_mem_multiset_dvd","def_path":"Mathlib/Algebra/BigOperators/Associated.lean","def_pos":[27,8],"def_end_pos":[27,31]},{"full_name":"exists_exists_and_eq_and","def_path":"Mathlib/Logic/Basic.lean","def_pos":[560,16],"def_end_pos":[560,40]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]}]}
{"url":"Mathlib/Topology/Algebra/UniformGroup.lean","commit":"","full_name":"UniformInducing.uniformAddGroup","start":[199,0],"end":[206,89],"file_path":"Mathlib/Topology/Algebra/UniformGroup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝⁹ : UniformSpace α\ninst✝⁸ : Group α\ninst✝⁷ : UniformGroup α\ninst✝⁶ : Group β\nγ : Type u_3\ninst✝⁵ : Group γ\ninst✝⁴ : UniformSpace γ\ninst✝³ : UniformGroup γ\ninst✝² : UniformSpace β\nF : Type u_4\ninst✝¹ : FunLike F β γ\ninst✝ : MonoidHomClass F β γ\nf : F\nhf : UniformInducing ⇑f\n⊢ UniformContinuous fun p => p.1 / p.2","state_after":"α : Type u_1\nβ : Type u_2\ninst✝⁹ : UniformSpace α\ninst✝⁸ : Group α\ninst✝⁷ : UniformGroup α\ninst✝⁶ : Group β\nγ : Type u_3\ninst✝⁵ : Group γ\ninst✝⁴ : UniformSpace γ\ninst✝³ : UniformGroup γ\ninst✝² : UniformSpace β\nF : Type u_4\ninst✝¹ : FunLike F β γ\ninst✝ : MonoidHomClass F β γ\nf : F\nhf : UniformInducing ⇑f\n⊢ UniformContinuous fun x => f x.1 / f x.2","tactic":"simp_rw [hf.uniformContinuous_iff, Function.comp_def, map_div]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"UniformInducing.uniformContinuous_iff","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[90,8],"def_end_pos":[90,45]},{"full_name":"map_div","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[415,8],"def_end_pos":[415,15]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝⁹ : UniformSpace α\ninst✝⁸ : Group α\ninst✝⁷ : UniformGroup α\ninst✝⁶ : Group β\nγ : Type u_3\ninst✝⁵ : Group γ\ninst✝⁴ : UniformSpace γ\ninst✝³ : UniformGroup γ\ninst✝² : UniformSpace β\nF : Type u_4\ninst✝¹ : FunLike F β γ\ninst✝ : MonoidHomClass F β γ\nf : F\nhf : UniformInducing ⇑f\n⊢ UniformContinuous fun x => f x.1 / f x.2","state_after":"no goals","tactic":"exact uniformContinuous_div.comp (hf.uniformContinuous.prod_map hf.uniformContinuous)","premises":[{"full_name":"UniformContinuous.comp","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[980,15],"def_end_pos":[980,37]},{"full_name":"UniformContinuous.prod_map","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[1435,8],"def_end_pos":[1435,34]},{"full_name":"UniformInducing.uniformContinuous","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[87,8],"def_end_pos":[87,41]},{"full_name":"uniformContinuous_div","def_path":"Mathlib/Topology/Algebra/UniformGroup.lean","def_pos":[70,8],"def_end_pos":[70,29]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","commit":"","full_name":"SimpleGraph.Subgraph.coe_deleteEdges_eq","start":[984,0],"end":[987,6],"file_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","tactics":[{"state_before":"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : G.Subgraph\ns✝ s : Set (Sym2 V)\n⊢ (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map Subtype.val ⁻¹' s)","state_after":"case Adj.h.mk.h.mk.a\nι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : G.Subgraph\ns✝ s : Set (Sym2 V)\nv : V\nhv : v ∈ (G'.deleteEdges s).verts\nw : V\nhw : w ∈ (G'.deleteEdges s).verts\n⊢ (G'.deleteEdges s).coe.Adj ⟨v, hv⟩ ⟨w, hw⟩ ↔ (G'.coe.deleteEdges (Sym2.map Subtype.val ⁻¹' s)).Adj ⟨v, hv⟩ ⟨w, hw⟩","tactic":"ext ⟨v, hv⟩ ⟨w, hw⟩","premises":[]},{"state_before":"case Adj.h.mk.h.mk.a\nι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : G.Subgraph\ns✝ s : Set (Sym2 V)\nv : V\nhv : v ∈ (G'.deleteEdges s).verts\nw : V\nhw : w ∈ (G'.deleteEdges s).verts\n⊢ (G'.deleteEdges s).coe.Adj ⟨v, hv⟩ ⟨w, hw⟩ ↔ (G'.coe.deleteEdges (Sym2.map Subtype.val ⁻¹' s)).Adj ⟨v, hv⟩ ⟨w, hw⟩","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Adhesive.lean","commit":"","full_name":"CategoryTheory.IsPushout.IsVanKampen.flip","start":[57,0],"end":[61,49],"file_path":"Mathlib/CategoryTheory/Adhesive.lean","tactics":[{"state_before":"J : Type v'\ninst✝¹ : Category.{u', v'} J\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' : H.IsVanKampen\n⊢ ⋯.IsVanKampen","state_after":"J : Type v'\ninst✝¹ : Category.{u', v'} J\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' : H.IsVanKampen\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ Y\nαY : Y' ⟶ X\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX g\nhg : IsPullback g' αW αY f\nhh : CommSq h' αX αZ i\nhi : CommSq i' αY αZ h\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ IsPullback h' αX αZ i ∧ IsPullback i' αY αZ h","tactic":"introv W' hf hg hh hi w","premises":[]},{"state_before":"J : Type v'\ninst✝¹ : Category.{u', v'} J\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' : H.IsVanKampen\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ Y\nαY : Y' ⟶ X\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX g\nhg : IsPullback g' αW αY f\nhh : CommSq h' αX αZ i\nhi : CommSq i' αY αZ h\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ IsPullback h' αX αZ i ∧ IsPullback i' αY αZ h","state_after":"no goals","tactic":"simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using\n    H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip","premises":[{"full_name":"CategoryTheory.CommSq.flip","def_path":"Mathlib/CategoryTheory/CommSq.lean","def_pos":[52,8],"def_end_pos":[52,12]},{"full_name":"CategoryTheory.IsPullback.flip_iff","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","def_pos":[552,8],"def_end_pos":[552,16]},{"full_name":"CategoryTheory.IsPushout.flip_iff","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean","def_pos":[854,8],"def_end_pos":[854,16]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
{"url":"Mathlib/RingTheory/Algebraic.lean","commit":"","full_name":"Algebra.isAlgebraic_iff_isIntegral","start":[217,0],"end":[220,55],"file_path":"Mathlib/RingTheory/Algebraic.lean","tactics":[{"state_before":"K : Type u\nA : Type v\ninst✝² : Field K\ninst✝¹ : Ring A\ninst✝ : Algebra K A\n⊢ Algebra.IsAlgebraic K A ↔ Algebra.IsIntegral K A","state_after":"no goals","tactic":"rw [Algebra.isAlgebraic_def, Algebra.isIntegral_def,\n      forall_congr' fun _ ↦ isAlgebraic_iff_isIntegral]","premises":[{"full_name":"Algebra.isAlgebraic_def","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[59,6],"def_end_pos":[59,29]},{"full_name":"Algebra.isIntegral_def","def_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Defs.lean","def_pos":[37,6],"def_end_pos":[37,28]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"forall_congr'","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[207,8],"def_end_pos":[207,21]},{"full_name":"isAlgebraic_iff_isIntegral","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[211,8],"def_end_pos":[211,34]}]}]}
{"url":"Mathlib/Algebra/Group/Action/Defs.lean","commit":"","full_name":"Commute.smul_left_iff","start":[484,0],"end":[485,65],"file_path":"Mathlib/Algebra/Group/Action/Defs.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁷ : Monoid M\ninst✝⁶ : MulAction M α\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ng✝ : G\na✝ b✝ : α\ninst✝³ : Mul H\ninst✝² : MulAction G H\ninst✝¹ : SMulCommClass G H H\ninst✝ : IsScalarTower G H H\ng : G\na b : H\n⊢ Commute (g • a) b ↔ Commute a b","state_after":"no goals","tactic":"rw [Commute.symm_iff, Commute.smul_right_iff, Commute.symm_iff]","premises":[{"full_name":"Commute.smul_right_iff","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[481,14],"def_end_pos":[481,36]},{"full_name":"Commute.symm_iff","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[70,18],"def_end_pos":[70,26]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/FieldTheory/Finite/Basic.lean","commit":"","full_name":"FiniteField.cast_card_eq_zero","start":[250,0],"end":[251,6],"file_path":"Mathlib/FieldTheory/Finite/Basic.lean","tactics":[{"state_before":"K : Type u_1\nR : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\n⊢ ↑q = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/QuaternionBasis.lean","commit":"","full_name":"QuaternionAlgebra.Basis.self_k","start":[58,0],"end":[67,28],"file_path":"Mathlib/Algebra/QuaternionBasis.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\n⊢ { re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 } = c₁ • 1","state_after":"no goals","tactic":"ext <;> simp","premises":[]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\n⊢ { re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 } = c₂ • 1","state_after":"no goals","tactic":"ext <;> simp","premises":[]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\n⊢ { re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 } =\n    { re := 0, imI := 0, imJ := 0, imK := 1 }","state_after":"no goals","tactic":"ext <;> simp","premises":[]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\n⊢ { re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 } =\n    -{ re := 0, imI := 0, imJ := 0, imK := 1 }","state_after":"no goals","tactic":"ext <;> simp","premises":[]}]}
{"url":"Mathlib/Data/Nat/Pairing.lean","commit":"","full_name":"Nat.left_le_pair","start":[97,0],"end":[97,89],"file_path":"Mathlib/Data/Nat/Pairing.lean","tactics":[{"state_before":"a b : ℕ\n⊢ a ≤ pair a b","state_after":"no goals","tactic":"simpa using unpair_left_le (pair a b)","premises":[{"full_name":"Nat.pair","def_path":"Mathlib/Data/Nat/Pairing.lean","def_pos":[33,4],"def_end_pos":[33,8]},{"full_name":"Nat.unpair_left_le","def_path":"Mathlib/Data/Nat/Pairing.lean","def_pos":[93,8],"def_end_pos":[93,22]}]}]}
{"url":"Mathlib/GroupTheory/Torsion.lean","commit":"","full_name":"AddMonoid.isTorsionFree_iff_torsion_eq_bot","start":[317,0],"end":[321,43],"file_path":"Mathlib/GroupTheory/Torsion.lean","tactics":[{"state_before":"G✝ : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G✝\nG : Type u_3\ninst✝ : CommGroup G\n⊢ IsTorsionFree G ↔ CommGroup.torsion G = ⊥","state_after":"G✝ : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G✝\nG : Type u_3\ninst✝ : CommGroup G\n⊢ (∀ (g : G), g ≠ 1 → ¬IsOfFinOrder g) ↔ ∀ ⦃x : G⦄, x ∈ CommGroup.torsion G → x ∈ ⊥","tactic":"rw [IsTorsionFree, eq_bot_iff, SetLike.le_def]","premises":[{"full_name":"Monoid.IsTorsionFree","def_path":"Mathlib/GroupTheory/Torsion.lean","def_pos":[302,4],"def_end_pos":[302,17]},{"full_name":"SetLike.le_def","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[191,8],"def_end_pos":[191,14]},{"full_name":"eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[282,8],"def_end_pos":[282,18]}]},{"state_before":"G✝ : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G✝\nG : Type u_3\ninst✝ : CommGroup G\n⊢ (∀ (g : G), g ≠ 1 → ¬IsOfFinOrder g) ↔ ∀ ⦃x : G⦄, x ∈ CommGroup.torsion G → x ∈ ⊥","state_after":"no goals","tactic":"simp [not_imp_not, CommGroup.mem_torsion]","premises":[{"full_name":"CommGroup.mem_torsion","def_path":"Mathlib/GroupTheory/Torsion.lean","def_pos":[273,8],"def_end_pos":[273,19]},{"full_name":"not_imp_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[290,8],"def_end_pos":[290,19]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Bernstein.lean","commit":"","full_name":"bernsteinPolynomial.flip","start":[71,0],"end":[73,64],"file_path":"Mathlib/RingTheory/Polynomial/Bernstein.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ν ≤ n\n⊢ (bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν)","state_after":"no goals","tactic":"simp [bernsteinPolynomial, h, tsub_tsub_assoc, mul_right_comm]","premises":[{"full_name":"bernsteinPolynomial","def_path":"Mathlib/RingTheory/Polynomial/Bernstein.lean","def_pos":[49,4],"def_end_pos":[49,23]},{"full_name":"mul_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[156,8],"def_end_pos":[156,22]},{"full_name":"tsub_tsub_assoc","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[185,8],"def_end_pos":[185,23]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","commit":"","full_name":"Real.negMulLog_eq_neg","start":[85,0],"end":[85,85],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","tactics":[{"state_before":"⊢ negMulLog = fun x => -(x * log x)","state_after":"no goals","tactic":"simp [negMulLog_def]","premises":[{"full_name":"Real.negMulLog_def","def_path":"Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean","def_pos":[83,6],"def_end_pos":[83,19]}]}]}
{"url":"Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean","commit":"","full_name":"Module.End.IsSemisimple.genEigenspace_eq_eigenspace","start":[49,0],"end":[53,100],"file_path":"Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nhf : f.IsSemisimple\nμ : R\nk : ℕ\nhk : 0 < k\n⊢ (f.genEigenspace μ) k = f.eigenspace μ","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nhf : f.IsSemisimple\nμ : R\nk : ℕ\nhk : 0 < k\nm : M\nhm : m ∈ (f.genEigenspace μ) k\n⊢ f m = μ • m","tactic":"refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (eigenspace_le_genEigenspace hk)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Module.End.eigenspace_le_genEigenspace","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[232,8],"def_end_pos":[232,35]},{"full_name":"Module.End.mem_eigenspace_iff","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[97,8],"def_end_pos":[97,26]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nhf : f.IsSemisimple\nμ : R\nk : ℕ\nhk : 0 < k\nm : M\nhm : m ∈ (f.genEigenspace μ) k\n⊢ f m = μ • m","state_after":"no goals","tactic":"exact apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub hm rfl hf (by simp)","premises":[{"full_name":"Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub","def_path":"Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean","def_pos":[28,6],"def_end_pos":[28,78]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","commit":"","full_name":"AlgebraicGeometry.LocallyRingedSpace.iso_hom_val_base_inv_val_base","start":[305,0],"end":[309,6],"file_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","tactics":[{"state_before":"X✝ X Y : LocallyRingedSpace\ne : X ≅ Y\n⊢ e.hom.val.base ≫ e.inv.val.base = 𝟙 ↑X.toPresheafedSpace","state_after":"X✝ X Y : LocallyRingedSpace\ne : X ≅ Y\n⊢ (e.hom ≫ e.inv).val.base = 𝟙 ↑X.toPresheafedSpace","tactic":"rw [← SheafedSpace.comp_base, ← LocallyRingedSpace.comp_val]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.comp_val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"AlgebraicGeometry.SheafedSpace.comp_base","def_path":"Mathlib/Geometry/RingedSpace/SheafedSpace.lean","def_pos":[130,8],"def_end_pos":[130,17]}]},{"state_before":"X✝ X Y : LocallyRingedSpace\ne : X ≅ Y\n⊢ (e.hom ≫ e.inv).val.base = 𝟙 ↑X.toPresheafedSpace","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Equiv.lean","commit":"","full_name":"MvPolynomial.pUnitAlgEquiv_symm_apply","start":[59,0],"end":[84,30],"file_path":"Mathlib/Algebra/MvPolynomial/Equiv.lean","tactics":[{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\n⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X)","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.1732 + 1} R := eval₂RingHom C (X PUnit.unit)\n⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X)","tactic":"let f : R[X] →+* MvPolynomial PUnit R := Polynomial.eval₂RingHom MvPolynomial.C (X PUnit.unit)","premises":[{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"MvPolynomial.C","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[166,4],"def_end_pos":[166,5]},{"full_name":"MvPolynomial.X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[178,4],"def_end_pos":[178,5]},{"full_name":"PUnit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[122,10],"def_end_pos":[122,15]},{"full_name":"PUnit.unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[124,4],"def_end_pos":[124,8]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.eval₂RingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[235,4],"def_end_pos":[235,16]},{"full_name":"RingHom","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[297,10],"def_end_pos":[297,17]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.1732 + 1} R := eval₂RingHom C (X PUnit.unit)\n⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X)","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.1732 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X)","tactic":"let g : MvPolynomial PUnit R →+* R[X] := eval₂Hom Polynomial.C fun _ => Polynomial.X","premises":[{"full_name":"MvPolynomial","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[84,4],"def_end_pos":[84,16]},{"full_name":"MvPolynomial.eval₂Hom","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[961,4],"def_end_pos":[961,12]},{"full_name":"PUnit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[122,10],"def_end_pos":[122,15]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"RingHom","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[297,10],"def_end_pos":[297,17]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.1732 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ LeftInverse (Polynomial.eval₂ C (X PUnit.unit)) (eval₂ Polynomial.C fun x => Polynomial.X)","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.2149 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ ∀ (p : MvPolynomial PUnit.{?u.2149 + 1} R), (f.comp g) p = p","tactic":"show ∀ p, f.comp g p = p","premises":[{"full_name":"RingHom.comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[563,4],"def_end_pos":[563,8]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.2149 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ ∀ (p : MvPolynomial PUnit.{?u.2149 + 1} R), (f.comp g) p = p","state_after":"case hC\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.2149 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ (f.comp g).comp C = C\n\ncase hX\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.2149 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ ∀ (n : PUnit.{?u.2149 + 1}), (f.comp g) (X n) = X n","tactic":"apply is_id","premises":[{"full_name":"MvPolynomial.is_id","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[425,8],"def_end_pos":[425,13]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np : R[X]\na : R\n⊢ eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C a)) = Polynomial.C a","state_after":"no goals","tactic":"rw [Polynomial.eval₂_C, MvPolynomial.eval₂_C]","premises":[{"full_name":"MvPolynomial.eval₂_C","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[915,8],"def_end_pos":[915,15]},{"full_name":"Polynomial.eval₂_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[61,8],"def_end_pos":[61,15]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ p q : R[X]\nhp : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) p) = p\nhq : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) q) = q\n⊢ eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (p + q)) = p + q","state_after":"no goals","tactic":"rw [Polynomial.eval₂_add, MvPolynomial.eval₂_add, hp, hq]","premises":[{"full_name":"MvPolynomial.eval₂_add","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[907,8],"def_end_pos":[907,17]},{"full_name":"Polynomial.eval₂_add","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[77,8],"def_end_pos":[77,17]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ : R[X]\np : ℕ\nn : R\nx✝ :\n  eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ p)) =\n    Polynomial.C n * Polynomial.X ^ p\n⊢ eval₂ Polynomial.C (fun x => Polynomial.X)\n      (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ (p + 1))) =\n    Polynomial.C n * Polynomial.X ^ (p + 1)","state_after":"no goals","tactic":"rw [Polynomial.eval₂_mul, Polynomial.eval₂_pow, Polynomial.eval₂_X, Polynomial.eval₂_C,\n        eval₂_mul, eval₂_C, eval₂_pow, eval₂_X]","premises":[{"full_name":"MvPolynomial.eval₂_C","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[915,8],"def_end_pos":[915,15]},{"full_name":"MvPolynomial.eval₂_X","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[923,8],"def_end_pos":[923,15]},{"full_name":"MvPolynomial.eval₂_mul","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[947,8],"def_end_pos":[947,17]},{"full_name":"MvPolynomial.eval₂_pow","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[954,8],"def_end_pos":[954,17]},{"full_name":"Polynomial.eval₂_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[61,8],"def_end_pos":[61,15]},{"full_name":"Polynomial.eval₂_X","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[64,8],"def_end_pos":[64,15]},{"full_name":"Polynomial.eval₂_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[223,8],"def_end_pos":[223,17]},{"full_name":"Polynomial.eval₂_pow","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[244,8],"def_end_pos":[244,17]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Pairwise.lean","commit":"","full_name":"List.pairwise_pair","start":[106,0],"end":[106,70],"file_path":".lake/packages/batteries/Batteries/Data/List/Pairwise.lean","tactics":[{"state_before":"α : Type u_1\nR : α → α → Prop\na b : α\n⊢ Pairwise R [a, b] ↔ R a b","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Finsupp/Basic.lean","commit":"","full_name":"Finsupp.mapDomain_comapDomain","start":[691,0],"end":[695,5],"file_path":"Mathlib/Data/Finsupp/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : AddCommMonoid M\nf : α → β\nhf : Injective f\nl : β →₀ M\nhl : ↑l.support ⊆ Set.range f\n⊢ mapDomain f (comapDomain f l ⋯) = l","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : AddCommMonoid M\nf : α → β\nhf : Injective f\nl : β →₀ M\nhl : ↑l.support ⊆ Set.range f\n⊢ mapDomain f (comapDomain f l ⋯) =\n    mapDomain (⇑{ toFun := f, inj' := hf }) (comapDomain (⇑{ toFun := f, inj' := hf }) l ⋯)","tactic":"conv_rhs => rw [← embDomain_comapDomain (f := ⟨f, hf⟩) hl (M := M), embDomain_eq_mapDomain]","premises":[{"full_name":"Finsupp.embDomain_comapDomain","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[634,6],"def_end_pos":[634,27]},{"full_name":"Finsupp.embDomain_eq_mapDomain","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[522,8],"def_end_pos":[522,30]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : AddCommMonoid M\nf : α → β\nhf : Injective f\nl : β →₀ M\nhl : ↑l.support ⊆ Set.range f\n⊢ mapDomain f (comapDomain f l ⋯) =\n    mapDomain (⇑{ toFun := f, inj' := hf }) (comapDomain (⇑{ toFun := f, inj' := hf }) l ⋯)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","commit":"","full_name":"Finset.ruzsa_triangle_inequality_add_add_sub","start":[66,0],"end":[71,51],"file_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A * C).card * B.card ≤ (A * B).card * (B / C).card","state_after":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A / C⁻¹).card * B.card ≤ (A * B).card * (B * C⁻¹).card","tactic":"rw [← div_inv_eq_mul, div_eq_mul_inv B]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"div_inv_eq_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[501,8],"def_end_pos":[501,22]}]},{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A / C⁻¹).card * B.card ≤ (A * B).card * (B * C⁻¹).card","state_after":"no goals","tactic":"exact ruzsa_triangle_inequality_div_mul_mul _ _ _","premises":[{"full_name":"Finset.ruzsa_triangle_inequality_div_mul_mul","def_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","def_pos":[54,8],"def_end_pos":[54,45]}]}]}
{"url":"Mathlib/Data/Sym/Basic.lean","commit":"","full_name":"Sym.exists_cons_of_mem","start":[276,0],"end":[284,9],"file_path":"Mathlib/Data/Sym/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nn n' m : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh : a ∈ s\n⊢ ∃ t, s = a ::ₛ t","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\n⊢ ∃ t, s = a ::ₛ t","tactic":"obtain ⟨m, h⟩ := Multiset.exists_cons_of_mem h","premises":[{"full_name":"Multiset.exists_cons_of_mem","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[232,8],"def_end_pos":[232,26]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\n⊢ ∃ t, s = a ::ₛ t","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\nthis : Multiset.card m = n\n⊢ ∃ t, s = a ::ₛ t","tactic":"have : Multiset.card m = n := by\n    apply_fun Multiset.card at h\n    rw [s.2, Multiset.card_cons, add_left_inj] at h\n    exact h.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Multiset.card","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[659,4],"def_end_pos":[659,8]},{"full_name":"Multiset.card_cons","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[683,8],"def_end_pos":[683,17]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"add_left_inj","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[80,2],"def_end_pos":[80,13]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\nthis : Multiset.card m = n\n⊢ ∃ t, s = a ::ₛ t","state_after":"case h\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\nthis : Multiset.card m = n\n⊢ s = a ::ₛ ⟨m, this⟩","tactic":"use ⟨m, this⟩","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\nthis : Multiset.card m = n\n⊢ s = a ::ₛ ⟨m, this⟩","state_after":"case h.a\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\nthis : Multiset.card m = n\n⊢ ↑s = ↑(a ::ₛ ⟨m, this⟩)","tactic":"apply Subtype.ext","premises":[{"full_name":"Subtype.ext","def_path":"Mathlib/Data/Subtype.lean","def_pos":[59,18],"def_end_pos":[59,21]}]},{"state_before":"case h.a\nα : Type u_1\nβ : Type u_2\nn n' m✝ : ℕ\ns✝ : Sym α n\na✝ b : α\ns : Sym α (n + 1)\na : α\nh✝ : a ∈ s\nm : Multiset α\nh : ↑s = a ::ₘ m\nthis : Multiset.card m = n\n⊢ ↑s = ↑(a ::ₛ ⟨m, this⟩)","state_after":"no goals","tactic":"exact h","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","commit":"","full_name":"abs_circleMap_zero","start":[106,0],"end":[107,95],"file_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","tactics":[{"state_before":"E : Type u_1\ninst✝ : NormedAddCommGroup E\nR θ : ℝ\n⊢ Complex.abs (circleMap 0 R θ) = |R|","state_after":"no goals","tactic":"simp [circleMap]","premises":[{"full_name":"circleMap","def_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","def_pos":[85,4],"def_end_pos":[85,13]}]}]}
{"url":"Mathlib/Data/Set/Basic.lean","commit":"","full_name":"Set.ne_univ_iff_exists_not_mem","start":[574,0],"end":[575,41],"file_path":"Mathlib/Data/Set/Basic.lean","tactics":[{"state_before":"α✝ : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α✝\ns✝ s₁ s₂ t t₁ t₂ u : Set α✝\nα : Type u_1\ns : Set α\n⊢ s ≠ univ ↔ ∃ a, a ∉ s","state_after":"no goals","tactic":"rw [← not_forall, ← eq_univ_iff_forall]","premises":[{"full_name":"Classical.not_forall","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[137,20],"def_end_pos":[137,30]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.eq_univ_iff_forall","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[558,8],"def_end_pos":[558,26]}]}]}
{"url":"Mathlib/Analysis/Normed/Field/Basic.lean","commit":"","full_name":"DilationEquiv.mulLeft_symm_apply","start":[747,0],"end":[753,53],"file_path":"Mathlib/Analysis/Normed/Field/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝ : NormedDivisionRing α\na✝ a : α\nha : a ≠ 0\nx y : α\n⊢ edist ((Equiv.mulLeft₀ a ha).toFun x) ((Equiv.mulLeft₀ a ha).toFun y) = ↑‖a‖₊ * edist x y","state_after":"no goals","tactic":"simp [edist_nndist, nndist_eq_nnnorm, ← mul_sub]","premises":[{"full_name":"edist_nndist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[275,8],"def_end_pos":[275,20]}]}]}
{"url":"Mathlib/Data/Part.lean","commit":"","full_name":"Part.bind_some_eq_map","start":[439,0],"end":[440,26],"file_path":"Mathlib/Data/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nx : Part α\n⊢ ∀ (a : β), a ∈ x.bind (some ∘ f) ↔ a ∈ map f x","state_after":"no goals","tactic":"simp [eq_comm]","premises":[{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","commit":"","full_name":"seminormFromBounded_one_le","start":[220,0],"end":[230,63],"file_path":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\n⊢ seminormFromBounded' f 1 ≤ 1","state_after":"case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nf_ne_zero : f ≠ 0\n⊢ seminormFromBounded' f 1 ≤ 1\n\ncase neg\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nf_ne_zero : ¬f ≠ 0\n⊢ seminormFromBounded' f 1 ≤ 1","tactic":"by_cases f_ne_zero : f ≠ 0","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","commit":"","full_name":"Matrix.det_one_sub_mul_comm","start":[406,0],"end":[408,95],"file_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁶ : Fintype l\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\n⊢ (1 - A * B).det = (1 - B * A).det","state_after":"no goals","tactic":"rw [sub_eq_add_neg, ← Matrix.neg_mul, det_one_add_mul_comm, Matrix.mul_neg, ← sub_eq_add_neg]","premises":[{"full_name":"Matrix.det_one_add_mul_comm","def_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","def_pos":[395,8],"def_end_pos":[395,28]},{"full_name":"Matrix.mul_neg","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1068,18],"def_end_pos":[1068,25]},{"full_name":"Matrix.neg_mul","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1063,18],"def_end_pos":[1063,25]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/Analysis/Convex/Segment.lean","commit":"","full_name":"Convex.mem_Ioc","start":[514,0],"end":[529,80],"file_path":"Mathlib/Analysis/Convex/Segment.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x < y\n⊢ z ∈ Ioc x y ↔ ∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z","state_after":"case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x < y\nhz : z ∈ Ioc x y\n⊢ ∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z\n\ncase refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\nπ : ι → Type u_6\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x < y\n⊢ (∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z) → z ∈ Ioc x y","tactic":"refine ⟨fun hz => ?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Combinatorics/Additive/Energy.lean","commit":"","full_name":"Finset.mulEnergy_eq_card_filter","start":[114,0],"end":[116,69],"file_path":"Mathlib/Combinatorics/Additive/Energy.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns✝ s₁ s₂ t✝ t₁ t₂ s t : Finset α\n⊢ ∀ (i : (α × α) × α × α),\n    i ∈ filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) ((s ×ˢ s) ×ˢ t ×ˢ t) ↔\n      (Equiv.prodProdProdComm α α α α) i ∈\n        filter\n          (fun x =>\n            match x with\n            | ((a, b), c, d) => a * b = c * d)\n          ((s ×ˢ t) ×ˢ s ×ˢ t)","state_after":"no goals","tactic":"simp [and_and_and_comm]","premises":[{"full_name":"and_and_and_comm","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[65,8],"def_end_pos":[65,24]}]}]}
{"url":"Mathlib/Algebra/Homology/HomologicalComplex.lean","commit":"","full_name":"HomologicalComplex.xPrevIsoSelf_comp_dTo","start":[460,0],"end":[462,55],"file_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\nj : ι\nh : ¬c.Rel (c.prev j) j\n⊢ (C.xPrevIsoSelf h).inv ≫ C.dTo j = 0","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","commit":"","full_name":"UniformOnFun.hasBasis_uniformity_of_basis_aux₁","start":[622,0],"end":[627,64],"file_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\np : ι → Prop\ns : ι → Set (β × β)\nhb : (𝓤 β).HasBasis p s\nS : Set α\n⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis p fun i => UniformOnFun.gen 𝔖 S (s i)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\np : ι → Prop\ns : ι → Set (β × β)\nhb : (𝓤 β).HasBasis p s\nS : Set α\n⊢ (comap (Prod.map S.restrict S.restrict) (𝓤 (↑S → β))).HasBasis p fun i =>\n    Prod.map (S.restrict ∘ ⇑UniformFun.toFun) (S.restrict ∘ ⇑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β (s i)","tactic":"simp_rw [UniformOnFun.gen_eq_preimage_restrict, uniformity_comap]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"UniformOnFun.gen_eq_preimage_restrict","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[568,18],"def_end_pos":[568,42]},{"full_name":"uniformity_comap","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[1115,8],"def_end_pos":[1115,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\np : ι → Prop\ns : ι → Set (β × β)\nhb : (𝓤 β).HasBasis p s\nS : Set α\n⊢ (comap (Prod.map S.restrict S.restrict) (𝓤 (↑S → β))).HasBasis p fun i =>\n    Prod.map (S.restrict ∘ ⇑UniformFun.toFun) (S.restrict ∘ ⇑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β (s i)","state_after":"no goals","tactic":"exact (UniformFun.hasBasis_uniformity_of_basis S β hb).comap _","premises":[{"full_name":"Filter.HasBasis.comap","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[697,8],"def_end_pos":[697,22]},{"full_name":"UniformFun.hasBasis_uniformity_of_basis","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[303,18],"def_end_pos":[303,46]}]}]}
{"url":"Mathlib/Data/Matrix/Basic.lean","commit":"","full_name":"Matrix.dotProduct_add","start":[713,0],"end":[715,52],"file_path":"Mathlib/Data/Matrix/Basic.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx y : n → α\n⊢ u ⬝ᵥ (v + w) = u ⬝ᵥ v + u ⬝ᵥ w","state_after":"no goals","tactic":"simp [dotProduct, mul_add, Finset.sum_add_distrib]","premises":[{"full_name":"Finset.sum_add_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[768,2],"def_end_pos":[768,13]},{"full_name":"Matrix.dotProduct","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[662,4],"def_end_pos":[662,14]}]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"writtenInExtChartAt_extChartAt_symm","start":[1288,0],"end":[1290,28],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx : M\ny : E\nh : y ∈ (extChartAt I x).target\n⊢ writtenInExtChartAt 𝓘(𝕜, E) I (↑(extChartAt I x) x) (↑(extChartAt I x).symm) y = y","state_after":"no goals","tactic":"simp_all only [mfld_simps]","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Coeff.lean","commit":"","full_name":"Polynomial.coeff_mul_X_pow","start":[223,0],"end":[234,54],"file_path":"Mathlib/Algebra/Polynomial/Coeff.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ (p * X ^ n).coeff (d + n) = p.coeff d","state_after":"case h₀\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ ∀ b ∈ antidiagonal (d + n), b ≠ (d, n) → p.coeff b.1 * (X ^ n).coeff b.2 = 0\n\ncase h₁\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ (d, n) ∉ antidiagonal (d + n) → p.coeff (d, n).1 * (X ^ n).coeff (d, n).2 = 0","tactic":"rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one]","premises":[{"full_name":"Finset.sum_eq_single","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[866,2],"def_end_pos":[866,13]},{"full_name":"Polynomial.coeff_X_pow","def_path":"Mathlib/Algebra/Polynomial/Coeff.lean","def_pos":[185,8],"def_end_pos":[185,19]},{"full_name":"Polynomial.coeff_mul","def_path":"Mathlib/Algebra/Polynomial/Coeff.lean","def_pos":[113,8],"def_end_pos":[113,17]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.GoodProducts.maxToGood_injective","start":[1661,0],"end":[1668,70],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\n⊢ Function.Injective (MaxToGood C hC hsC ho h₁)","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : MaxToGood C hC hsC ho h₁ m = MaxToGood C hC hsC ho h₁ n\n⊢ m = n","tactic":"intro m n h","premises":[]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : MaxToGood C hC hsC ho h₁ m = MaxToGood C hC hsC ho h₁ n\n⊢ m = n","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : MaxToGood C hC hsC ho h₁ m = MaxToGood C hC hsC ho h₁ n\n⊢ ↑↑m = ↑↑n","tactic":"apply Subtype.ext ∘ Subtype.ext","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Subtype.ext","def_path":"Mathlib/Data/Subtype.lean","def_pos":[59,18],"def_end_pos":[59,21]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : MaxToGood C hC hsC ho h₁ m = MaxToGood C hC hsC ho h₁ n\n⊢ ↑↑m = ↑↑n","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : ↑(MaxToGood C hC hsC ho h₁ m) = ↑(MaxToGood C hC hsC ho h₁ n)\n⊢ ↑↑m = ↑↑n","tactic":"rw [Subtype.ext_iff] at h","premises":[{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : ↑(MaxToGood C hC hsC ho h₁ m) = ↑(MaxToGood C hC hsC ho h₁ n)\n⊢ ↑↑m = ↑↑n","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : (↑m).Tail = (↑n).Tail\n⊢ ↑↑m = ↑↑n","tactic":"dsimp [MaxToGood] at h","premises":[{"full_name":"Profinite.NobelingProof.GoodProducts.MaxToGood","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[1656,4],"def_end_pos":[1656,13]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))\nm n : ↑(MaxProducts C ho)\nh : (↑m).Tail = (↑n).Tail\n⊢ ↑↑m = ↑↑n","state_after":"no goals","tactic":"rw [max_eq_o_cons_tail C hsC ho m, max_eq_o_cons_tail C hsC ho n, h]","premises":[{"full_name":"Profinite.NobelingProof.GoodProducts.max_eq_o_cons_tail","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[1499,8],"def_end_pos":[1499,39]}]}]}
{"url":"Mathlib/FieldTheory/PolynomialGaloisGroup.lean","commit":"","full_name":"Polynomial.Gal.restrictDvd_def","start":[233,0],"end":[242,13],"file_path":"Mathlib/FieldTheory/PolynomialGaloisGroup.lean","tactics":[{"state_before":"F : Type u_1\ninst✝³ : Field F\np q : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Decidable (q = 0)\nhpq : p ∣ q\n⊢ restrictDvd hpq = if hq : q = 0 then 1 else restrict p q.SplittingField","state_after":"F : Type u_1\ninst✝³ : Field F\np q : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Decidable (q = 0)\nhpq : p ∣ q\n⊢ (if hq : q = 0 then 1 else restrict p q.SplittingField) = if hq : q = 0 then 1 else restrict p q.SplittingField","tactic":"unfold restrictDvd","premises":[{"full_name":"Polynomial.Gal.restrictDvd","def_path":"Mathlib/FieldTheory/PolynomialGaloisGroup.lean","def_pos":[226,4],"def_end_pos":[226,15]}]},{"state_before":"F : Type u_1\ninst✝³ : Field F\np q : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Decidable (q = 0)\nhpq : p ∣ q\n⊢ (if hq : q = 0 then 1 else restrict p q.SplittingField) = if hq : q = 0 then 1 else restrict p q.SplittingField","state_after":"no goals","tactic":"convert rfl","premises":[{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Logic/Equiv/List.lean","commit":"","full_name":"Denumerable.list_ofNat_zero","start":[234,0],"end":[235,93],"file_path":"Mathlib/Logic/Equiv/List.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\n⊢ ofNat (List α) 0 = []","state_after":"no goals","tactic":"rw [← @encode_list_nil α, ofNat_encode]","premises":[{"full_name":"Denumerable.ofNat_encode","def_path":"Mathlib/Logic/Denumerable.lean","def_pos":[64,8],"def_end_pos":[64,20]},{"full_name":"Encodable.encode_list_nil","def_path":"Mathlib/Logic/Equiv/List.lean","def_pos":[54,8],"def_end_pos":[54,23]}]}]}
{"url":"Mathlib/Data/ENNReal/Real.lean","commit":"","full_name":"ENNReal.toReal_max","start":[129,0],"end":[133,68],"file_path":"Mathlib/Data/ENNReal/Real.lean","tactics":[{"state_before":"a b c d : ℝ≥0∞\nr p q : ℝ≥0\nhr : a ≠ ⊤\nhp : b ≠ ⊤\nh : a ≤ b\n⊢ (max a b).toReal = max a.toReal b.toReal","state_after":"no goals","tactic":"simp only [h, (ENNReal.toReal_le_toReal hr hp).2 h, max_eq_right]","premises":[{"full_name":"ENNReal.toReal_le_toReal","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[68,8],"def_end_pos":[68,24]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"max_eq_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[121,8],"def_end_pos":[121,20]}]},{"state_before":"a b c d : ℝ≥0∞\nr p q : ℝ≥0\nhr : a ≠ ⊤\nhp : b ≠ ⊤\nh : b ≤ a\n⊢ (max a b).toReal = max a.toReal b.toReal","state_after":"no goals","tactic":"simp only [h, (ENNReal.toReal_le_toReal hp hr).2 h, max_eq_left]","premises":[{"full_name":"ENNReal.toReal_le_toReal","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[68,8],"def_end_pos":[68,24]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"max_eq_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[118,8],"def_end_pos":[118,19]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","commit":"","full_name":"Polynomial.Chebyshev.map_U","start":[234,0],"end":[244,10],"file_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nn : ℤ\n⊢ map f (U R n) = U S n","state_after":"no goals","tactic":"induction n using Polynomial.Chebyshev.induct with\n  | zero => simp\n  | one => simp\n  | add_two n ih1 ih2 =>\n    simp_rw [U_add_two, Polynomial.map_sub, Polynomial.map_mul, Polynomial.map_ofNat, map_X, ih1,\n      ih2]\n  | neg_add_one n ih1 ih2 =>\n    simp_rw [U_sub_one, Polynomial.map_sub, Polynomial.map_mul, Polynomial.map_ofNat, map_X, ih1,\n      ih2]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Polynomial.Chebyshev.U_add_two","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[139,8],"def_end_pos":[139,17]},{"full_name":"Polynomial.Chebyshev.U_sub_one","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[149,8],"def_end_pos":[149,17]},{"full_name":"Polynomial.Chebyshev.induct","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[71,18],"def_end_pos":[71,24]},{"full_name":"Polynomial.map_X","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[605,8],"def_end_pos":[605,13]},{"full_name":"Polynomial.map_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[626,18],"def_end_pos":[626,25]},{"full_name":"Polynomial.map_ofNat","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[662,18],"def_end_pos":[662,27]},{"full_name":"Polynomial.map_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1085,18],"def_end_pos":[1085,25]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Pointwise.lean","commit":"","full_name":"neg_thickening","start":[78,0],"end":[81,5],"file_path":"Mathlib/Analysis/Normed/Group/Pointwise.lean","tactics":[{"state_before":"E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ (thickening δ s)⁻¹ = thickening δ s⁻¹","state_after":"E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ {x | EMetric.infEdist x s < ENNReal.ofReal δ}⁻¹ = {x | EMetric.infEdist x⁻¹ s < ENNReal.ofReal δ}","tactic":"simp_rw [thickening, ← infEdist_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Metric.thickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[49,4],"def_end_pos":[49,14]},{"full_name":"infEdist_inv","def_path":"Mathlib/Analysis/Normed/Group/Pointwise.lean","def_pos":[64,8],"def_end_pos":[64,20]}]},{"state_before":"E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ {x | EMetric.infEdist x s < ENNReal.ofReal δ}⁻¹ = {x | EMetric.infEdist x⁻¹ s < ENNReal.ofReal δ}","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","commit":"","full_name":"MeasureTheory.integral_indicator","start":[167,0],"end":[180,35],"file_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ","state_after":"case pos\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\nhfi : IntegrableOn f s μ\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ\n\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\nhfi : ¬IntegrableOn f s μ\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ","tactic":"by_cases hfi : IntegrableOn f s μ","premises":[{"full_name":"MeasureTheory.IntegrableOn","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[80,4],"def_end_pos":[80,16]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\nhfi : IntegrableOn f s μ\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ\n\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\nhfi : ¬IntegrableOn f s μ\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ","state_after":"case neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\nhfi : ¬IntegrableOn f s μ\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ\n\ncase pos\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X → E\ns t : Set X\nμ ν : Measure X\nl l' : Filter X\nhs : MeasurableSet s\nhfi : IntegrableOn f s μ\n⊢ ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ","tactic":"swap","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","commit":"","full_name":"Matrix.det_one_add_mul_comm","start":[393,0],"end":[400,87],"file_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁶ : Fintype l\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\n⊢ (1 + A * B).det = (fromBlocks 1 (-A) B 1).det","state_after":"no goals","tactic":"rw [det_fromBlocks_one₂₂, Matrix.neg_mul, sub_neg_eq_add]","premises":[{"full_name":"Matrix.det_fromBlocks_one₂₂","def_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","def_pos":[388,8],"def_end_pos":[388,28]},{"full_name":"Matrix.neg_mul","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1063,18],"def_end_pos":[1063,25]},{"full_name":"sub_neg_eq_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[500,2],"def_end_pos":[500,13]}]},{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁶ : Fintype l\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix m n α\nB : Matrix n m α\n⊢ (fromBlocks 1 (-A) B 1).det = (1 + B * A).det","state_after":"no goals","tactic":"rw [det_fromBlocks_one₁₁, Matrix.mul_neg, sub_neg_eq_add]","premises":[{"full_name":"Matrix.det_fromBlocks_one₁₁","def_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","def_pos":[371,8],"def_end_pos":[371,28]},{"full_name":"Matrix.mul_neg","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1068,18],"def_end_pos":[1068,25]},{"full_name":"sub_neg_eq_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[500,2],"def_end_pos":[500,13]}]}]}
{"url":"Mathlib/Topology/VectorBundle/Basic.lean","commit":"","full_name":"Trivialization.symmL_apply","start":[385,0],"end":[395,83],"file_path":"Mathlib/Topology/VectorBundle/Basic.lean","tactics":[{"state_before":"R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpace F E)\ninst✝² : (x : B) → TopologicalSpace (E x)\ninst✝¹ : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst✝ : Trivialization.IsLinear R e\nb : B\n⊢ Continuous { toFun := e.symm b, map_add' := ⋯, map_smul' := ⋯ }.toFun","state_after":"case pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpace F E)\ninst✝² : (x : B) → TopologicalSpace (E x)\ninst✝¹ : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst✝ : Trivialization.IsLinear R e\nb : B\nhb : b ∈ e.baseSet\n⊢ Continuous { toFun := e.symm b, map_add' := ⋯, map_smul' := ⋯ }.toFun\n\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpace F E)\ninst✝² : (x : B) → TopologicalSpace (E x)\ninst✝¹ : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst✝ : Trivialization.IsLinear R e\nb : B\nhb : b ∉ e.baseSet\n⊢ Continuous { toFun := e.symm b, map_add' := ⋯, map_smul' := ⋯ }.toFun","tactic":"by_cases hb : b ∈ e.baseSet","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Trivialization.baseSet","def_path":"Mathlib/Topology/FiberBundle/Trivialization.lean","def_pos":[266,2],"def_end_pos":[266,9]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Order/CauSeq/Basic.lean","commit":"","full_name":"rat_inv_continuous_lemma","start":[69,0],"end":[80,8],"file_path":"Mathlib/Algebra/Order/CauSeq/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\n⊢ ∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\n⊢ abv (a⁻¹ - b⁻¹) < ε","tactic":"refine ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\n⊢ abv (a⁻¹ - b⁻¹) < ε","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\n⊢ abv (a⁻¹ - b⁻¹) < ε","tactic":"have a0 := K0.trans_le ha","premises":[]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\n⊢ abv (a⁻¹ - b⁻¹) < ε","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ abv (a⁻¹ - b⁻¹) < ε","tactic":"have b0 := K0.trans_le hb","premises":[]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ abv (a⁻¹ - b⁻¹) < ε","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ (abv a)⁻¹ * abv (a - b) * (abv b)⁻¹ < ε","tactic":"rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,\n    abv_inv abv, abv_sub abv]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsAbsoluteValue.abv_inv","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[410,8],"def_end_pos":[410,15]},{"full_name":"IsAbsoluteValue.abv_mul","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[291,6],"def_end_pos":[291,13]},{"full_name":"IsAbsoluteValue.abv_pos","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[312,8],"def_end_pos":[312,15]},{"full_name":"IsAbsoluteValue.abv_sub","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[368,8],"def_end_pos":[368,15]},{"full_name":"inv_sub_inv'","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[147,8],"def_end_pos":[147,20]}]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ (abv a)⁻¹ * abv (a - b) * (abv b)⁻¹ < ε","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ abv a * ((abv a)⁻¹ * abv (a - b) * (abv b)⁻¹) * abv b < abv a * ε * abv b","tactic":"refine lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right ?_ b0.le) a0.le","premises":[{"full_name":"lt_of_mul_lt_mul_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[205,8],"def_end_pos":[205,29]},{"full_name":"lt_of_mul_lt_mul_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[208,8],"def_end_pos":[208,30]}]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ abv a * ((abv a)⁻¹ * abv (a - b) * (abv b)⁻¹) * abv b < abv a * ε * abv b","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ abv (a - b) < abv a * ε * abv b","tactic":"rw [mul_assoc, inv_mul_cancel_right₀ b0.ne', ← mul_assoc, mul_inv_cancel a0.ne', one_mul]","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"inv_mul_cancel_right₀","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[249,8],"def_end_pos":[249,29]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_inv_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[195,14],"def_end_pos":[195,28]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ abv (a - b) < abv a * ε * abv b","state_after":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ K * ε * K ≤ abv a * ε * abv b","tactic":"refine h.trans_le ?_","premises":[]},{"state_before":"α : Type u_1\nβ✝ : Type u_2\ninst✝⁴ : LinearOrderedField α\ninst✝³ : Ring β✝\nabv✝ : β✝ → α\ninst✝² : IsAbsoluteValue abv✝\nβ : Type u_3\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K : α\nε0 : 0 < ε\nK0 : 0 < K\na b : β\nha : K ≤ abv a\nhb : K ≤ abv b\nh : abv (a - b) < K * ε * K\na0 : 0 < abv a\nb0 : 0 < abv b\n⊢ K * ε * K ≤ abv a * ε * abv b","state_after":"no goals","tactic":"gcongr","premises":[]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","commit":"","full_name":"Filter.EventuallyEq.hasFDerivAtFilter_iff","start":[807,0],"end":[810,76],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh₀ : f₀ =ᶠ[L] f₁\nhx : f₀ x = f₁ x\nh₁ : ∀ (x : E), f₀' x = f₁' x\n⊢ HasFDerivAtFilter f₀ f₀' x L ↔ HasFDerivAtFilter f₁ f₁' x L","state_after":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh₀ : f₀ =ᶠ[L] f₁\nhx : f₀ x = f₁ x\nh₁ : ∀ (x : E), f₀' x = f₁' x\n⊢ ((fun x' => f₀ x' - f₀ x - f₀' (x' - x)) =o[L] fun x' => x' - x) ↔\n    (fun x' => f₁ x' - f₁ x - f₁' (x' - x)) =o[L] fun x' => x' - x","tactic":"simp only [hasFDerivAtFilter_iff_isLittleO]","premises":[{"full_name":"hasFDerivAtFilter_iff_isLittleO","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[136,9],"def_end_pos":[136,40]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh₀ : f₀ =ᶠ[L] f₁\nhx : f₀ x = f₁ x\nh₁ : ∀ (x : E), f₀' x = f₁' x\n⊢ ((fun x' => f₀ x' - f₀ x - f₀' (x' - x)) =o[L] fun x' => x' - x) ↔\n    (fun x' => f₁ x' - f₁ x - f₁' (x' - x)) =o[L] fun x' => x' - x","state_after":"no goals","tactic":"exact isLittleO_congr (h₀.mono fun y hy => by simp only [hy, h₁, hx]) .rfl","premises":[{"full_name":"Asymptotics.isLittleO_congr","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[328,8],"def_end_pos":[328,23]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.EventuallyEq.rfl","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1303,18],"def_end_pos":[1303,34]}]}]}
{"url":"Mathlib/NumberTheory/SmoothNumbers.lean","commit":"","full_name":"Nat.smoothNumbers_succ","start":[347,0],"end":[350,94],"file_path":"Mathlib/NumberTheory/SmoothNumbers.lean","tactics":[{"state_before":"N : ℕ\nhN : ¬Prime N\n⊢ N.succ.smoothNumbers = N.smoothNumbers","state_after":"no goals","tactic":"simp only [smoothNumbers_eq_factoredNumbers, Finset.range_succ, factoredNumbers_insert _ hN]","premises":[{"full_name":"Finset.range_succ","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2465,8],"def_end_pos":[2465,18]},{"full_name":"Nat.factoredNumbers_insert","def_path":"Mathlib/NumberTheory/SmoothNumbers.lean","def_pos":[159,6],"def_end_pos":[159,28]},{"full_name":"Nat.smoothNumbers_eq_factoredNumbers","def_path":"Mathlib/NumberTheory/SmoothNumbers.lean","def_pos":[277,6],"def_end_pos":[277,38]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.get?_modifyNth_ne","start":[361,0],"end":[364,10],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nf : α → α\nm n : Nat\nl : List α\nh : m ≠ n\n⊢ (modifyNth f m l).get? n = l.get? n","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Preadditive/Mat.lean","commit":"","full_name":"CategoryTheory.Functor.mapMatId_inv_app","start":[247,0],"end":[254,31],"file_path":"Mathlib/CategoryTheory/Preadditive/Mat.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nM : Mat_ C\n⊢ (𝟭 C).mapMat_.obj M = (𝟭 (Mat_ C)).obj M","state_after":"case mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\n⊢ (𝟭 C).mapMat_.obj (Mat_.mk ι✝ X✝) = (𝟭 (Mat_ C)).obj (Mat_.mk ι✝ X✝)","tactic":"cases M","premises":[]},{"state_before":"case mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\n⊢ (𝟭 C).mapMat_.obj (Mat_.mk ι✝ X✝) = (𝟭 (Mat_ C)).obj (Mat_.mk ι✝ X✝)","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nM N : Mat_ C\nf : M ⟶ N\n⊢ (𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) N).hom = ((fun M => eqToIso ⋯) M).hom ≫ (𝟭 (Mat_ C)).map f","state_after":"case H\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nM N : Mat_ C\nf : M ⟶ N\ni✝ : ((𝟭 C).mapMat_.obj M).ι\nj✝ : ((𝟭 (Mat_ C)).obj N).ι\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) N).hom) i✝ j✝ = (((fun M => eqToIso ⋯) M).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝","tactic":"ext","premises":[]},{"state_before":"case H\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nM N : Mat_ C\nf : M ⟶ N\ni✝ : ((𝟭 C).mapMat_.obj M).ι\nj✝ : ((𝟭 (Mat_ C)).obj N).ι\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) N).hom) i✝ j✝ = (((fun M => eqToIso ⋯) M).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝","state_after":"case H.mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nN : Mat_ C\nj✝ : ((𝟭 (Mat_ C)).obj N).ι\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\nf : Mat_.mk ι✝ X✝ ⟶ N\ni✝ : ((𝟭 C).mapMat_.obj (Mat_.mk ι✝ X✝)).ι\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) N).hom) i✝ j✝ =\n    (((fun M => eqToIso ⋯) (Mat_.mk ι✝ X✝)).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝","tactic":"cases M","premises":[]},{"state_before":"case H.mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nN : Mat_ C\nj✝ : ((𝟭 (Mat_ C)).obj N).ι\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\nf : Mat_.mk ι✝ X✝ ⟶ N\ni✝ : ((𝟭 C).mapMat_.obj (Mat_.mk ι✝ X✝)).ι\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) N).hom) i✝ j✝ =\n    (((fun M => eqToIso ⋯) (Mat_.mk ι✝ X✝)).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝","state_after":"case H.mk.mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nι✝¹ : Type\nfintype✝¹ : Fintype ι✝¹\nX✝¹ : ι✝¹ → C\ni✝ : ((𝟭 C).mapMat_.obj (Mat_.mk ι✝¹ X✝¹)).ι\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\nj✝ : ((𝟭 (Mat_ C)).obj (Mat_.mk ι✝ X✝)).ι\nf : Mat_.mk ι✝¹ X✝¹ ⟶ Mat_.mk ι✝ X✝\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) (Mat_.mk ι✝ X✝)).hom) i✝ j✝ =\n    (((fun M => eqToIso ⋯) (Mat_.mk ι✝¹ X✝¹)).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝","tactic":"cases N","premises":[]},{"state_before":"case H.mk.mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nι✝¹ : Type\nfintype✝¹ : Fintype ι✝¹\nX✝¹ : ι✝¹ → C\ni✝ : ((𝟭 C).mapMat_.obj (Mat_.mk ι✝¹ X✝¹)).ι\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\nj✝ : ((𝟭 (Mat_ C)).obj (Mat_.mk ι✝ X✝)).ι\nf : Mat_.mk ι✝¹ X✝¹ ⟶ Mat_.mk ι✝ X✝\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) (Mat_.mk ι✝ X✝)).hom) i✝ j✝ =\n    (((fun M => eqToIso ⋯) (Mat_.mk ι✝¹ X✝¹)).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝","state_after":"no goals","tactic":"simp [comp_dite, dite_comp]","premises":[{"full_name":"CategoryTheory.comp_dite","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[228,8],"def_end_pos":[228,17]},{"full_name":"CategoryTheory.dite_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[232,8],"def_end_pos":[232,17]}]}]}
{"url":"Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean","commit":"","full_name":"TensorProduct.sum_tmul_eq_zero_of_vanishesTrivially","start":[83,0],"end":[94,86],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhmn : VanishesTrivially R m n\n⊢ ∑ i : ι, m i ⊗ₜ[R] n i = 0","state_after":"case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ i : ι, m i ⊗ₜ[R] n i = 0","tactic":"obtain ⟨κ, _, a, y, h₁, h₂⟩ := hmn","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ i : ι, m i ⊗ₜ[R] n i = 0","state_after":"case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ x : ι, ∑ x_1 : κ, a x x_1 • m x ⊗ₜ[R] y x_1 = 0","tactic":"simp_rw [h₁, tmul_sum, tmul_smul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"TensorProduct.tmul_smul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[351,8],"def_end_pos":[351,17]},{"full_name":"TensorProduct.tmul_sum","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[439,8],"def_end_pos":[439,16]}]},{"state_before":"case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ x : ι, ∑ x_1 : κ, a x x_1 • m x ⊗ₜ[R] y x_1 = 0","state_after":"case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ y_1 : κ, ∑ x : ι, a x y_1 • m x ⊗ₜ[R] y y_1 = 0","tactic":"rw [Finset.sum_comm]","premises":[{"full_name":"Finset.sum_comm","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[816,2],"def_end_pos":[816,13]}]},{"state_before":"case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ y_1 : κ, ∑ x : ι, a x y_1 • m x ⊗ₜ[R] y y_1 = 0","state_after":"no goals","tactic":"simp_rw [← tmul_smul, ← smul_tmul, ← sum_tmul, h₂, zero_tmul, Finset.sum_const_zero]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.sum_const_zero","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[362,2],"def_end_pos":[362,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"TensorProduct.smul_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[242,8],"def_end_pos":[242,17]},{"full_name":"TensorProduct.sum_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[432,8],"def_end_pos":[432,16]},{"full_name":"TensorProduct.tmul_smul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[351,8],"def_end_pos":[351,17]},{"full_name":"TensorProduct.zero_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[183,8],"def_end_pos":[183,17]}]}]}
{"url":"Mathlib/Topology/Order/Hom/Esakia.lean","commit":"","full_name":"EsakiaHom.cancel_left","start":[312,0],"end":[315,80],"file_path":"Mathlib/Topology/Order/Hom/Esakia.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : Preorder α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Preorder β\ninst✝³ : TopologicalSpace γ\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : Preorder δ\ng : EsakiaHom β γ\nf₁ f₂ : EsakiaHom α β\nhg : Injective ⇑g\nh : g.comp f₁ = g.comp f₂\na : α\n⊢ g (f₁ a) = g (f₂ a)","state_after":"no goals","tactic":"rw [← comp_apply, h, comp_apply]","premises":[{"full_name":"EsakiaHom.comp_apply","def_path":"Mathlib/Topology/Order/Hom/Esakia.lean","def_pos":[293,8],"def_end_pos":[293,18]}]}]}
{"url":"Mathlib/Data/ZMod/Basic.lean","commit":"","full_name":"ZMod.intCast_cast_sub","start":[578,0],"end":[579,87],"file_path":"Mathlib/Data/ZMod/Basic.lean","tactics":[{"state_before":"m n : ℕ\nx y : ZMod n\n⊢ (x - y).cast = (x.cast - y.cast) % ↑n","state_after":"no goals","tactic":"rw [← ZMod.coe_intCast, Int.cast_sub, ZMod.intCast_zmod_cast, ZMod.intCast_zmod_cast]","premises":[{"full_name":"Int.cast_sub","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[107,8],"def_end_pos":[107,16]},{"full_name":"ZMod.coe_intCast","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[564,8],"def_end_pos":[564,19]},{"full_name":"ZMod.intCast_zmod_cast","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[211,8],"def_end_pos":[211,25]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Infsep.lean","commit":"","full_name":"Set.einfsep_pos_of_finite","start":[243,0],"end":[249,47],"file_path":"Mathlib/Topology/MetricSpace/Infsep.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : EMetricSpace α\nx y z : α\ns t : Set α\nC : ℝ≥0∞\nsC : Set ℝ≥0∞\ninst✝ : Finite ↑s\n⊢ 0 < s.einfsep","state_after":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : EMetricSpace α\nx y z : α\ns t : Set α\nC : ℝ≥0∞\nsC : Set ℝ≥0∞\ninst✝ : Finite ↑s\nval✝ : Fintype ↑s\n⊢ 0 < s.einfsep","tactic":"cases nonempty_fintype s","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : EMetricSpace α\nx y z : α\ns t : Set α\nC : ℝ≥0∞\nsC : Set ℝ≥0∞\ninst✝ : Finite ↑s\nval✝ : Fintype ↑s\n⊢ 0 < s.einfsep","state_after":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : EMetricSpace α\nx y z : α\ns t : Set α\nC : ℝ≥0∞\nsC : Set ℝ≥0∞\ninst✝ : Finite ↑s\nval✝ : Fintype ↑s\nhs : s.Nontrivial\n⊢ 0 < s.einfsep\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : EMetricSpace α\nx y z : α\ns t : Set α\nC : ℝ≥0∞\nsC : Set ℝ≥0∞\ninst✝ : Finite ↑s\nval✝ : Fintype ↑s\nhs : ¬s.Nontrivial\n⊢ 0 < s.einfsep","tactic":"by_cases hs : s.Nontrivial","premises":[{"full_name":"Set.Nontrivial","def_path":"Mathlib/Data/Set/Subsingleton.lean","def_pos":[122,14],"def_end_pos":[122,24]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Finsupp/AList.lean","commit":"","full_name":"AList.empty_lookupFinsupp","start":[90,0],"end":[94,8],"file_path":"Mathlib/Data/Finsupp/AList.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\ninst✝ : Zero M\n⊢ ∅.lookupFinsupp = 0","state_after":"no goals","tactic":"classical\n    ext\n    simp","premises":[]}]}
{"url":"Mathlib/Analysis/RCLike/Lemmas.lean","commit":"","full_name":"RCLike.reCLM_norm","start":[64,0],"end":[68,6],"file_path":"Mathlib/Analysis/RCLike/Lemmas.lean","tactics":[{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ ‖reCLM‖ = 1","state_after":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ 1 ≤ ‖reLm.mkContinuous 1 ⋯‖","tactic":"apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)","premises":[{"full_name":"LinearMap.mkContinuous_norm_le","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean","def_pos":[450,8],"def_end_pos":[450,28]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]},{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ 1 ≤ ‖reLm.mkContinuous 1 ⋯‖","state_after":"case h.e'_3\nK : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ 1 = ‖reCLM 1‖ / ‖1‖","tactic":"convert ContinuousLinearMap.ratio_le_opNorm (reCLM : K →L[ℝ] ℝ) (1 : K)","premises":[{"full_name":"ContinuousLinearMap","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[224,10],"def_end_pos":[224,29]},{"full_name":"ContinuousLinearMap.ratio_le_opNorm","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean","def_pos":[229,8],"def_end_pos":[229,23]},{"full_name":"RCLike.reCLM","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[865,18],"def_end_pos":[865,23]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"case h.e'_3\nK : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ 1 = ‖reCLM 1‖ / ‖1‖","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Order/GroupWithZero/Canonical.lean","commit":"","full_name":"pow_pos_iff","start":[95,0],"end":[95,98],"file_path":"Mathlib/Algebra/Order/GroupWithZero/Canonical.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : LinearOrderedCommMonoidWithZero α\na b c d x y z : α\nn : ℕ\ninst✝ : NoZeroDivisors α\nhn : n ≠ 0\n⊢ 0 < a ^ n ↔ 0 < a","state_after":"no goals","tactic":"simp_rw [zero_lt_iff, pow_ne_zero_iff hn]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"pow_ne_zero_iff","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[169,6],"def_end_pos":[169,21]},{"full_name":"zero_lt_iff","def_path":"Mathlib/Algebra/Order/GroupWithZero/Canonical.lean","def_pos":[81,8],"def_end_pos":[81,19]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/Basic.lean","commit":"","full_name":"CategoryTheory.Abelian.epi_fst_of_isLimit","start":[674,0],"end":[677,87],"file_path":"Mathlib/CategoryTheory/Abelian/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\ns : PullbackCone f g\nhs : IsLimit s\n⊢ Epi s.fst","state_after":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\ns : PullbackCone f g\nhs : IsLimit s\nthis : Epi ((limit.cone (cospan f g)).π.app WalkingCospan.left)\n⊢ Epi s.fst","tactic":"haveI : Epi (NatTrans.app (limit.cone (cospan f g)).π WalkingCospan.left) :=\n    Abelian.epi_pullback_of_epi_g f g","premises":[{"full_name":"CategoryTheory.Abelian.epi_pullback_of_epi_g","def_path":"Mathlib/CategoryTheory/Abelian/Basic.lean","def_pos":[638,9],"def_end_pos":[638,30]},{"full_name":"CategoryTheory.Epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[241,6],"def_end_pos":[241,9]},{"full_name":"CategoryTheory.Limits.Cone.π","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[119,2],"def_end_pos":[119,3]},{"full_name":"CategoryTheory.Limits.WalkingCospan.left","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean","def_pos":[38,7],"def_end_pos":[38,25]},{"full_name":"CategoryTheory.Limits.cospan","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean","def_pos":[165,4],"def_end_pos":[165,10]},{"full_name":"CategoryTheory.Limits.limit.cone","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[131,4],"def_end_pos":[131,14]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\ns : PullbackCone f g\nhs : IsLimit s\nthis : Epi ((limit.cone (cospan f g)).π.app WalkingCospan.left)\n⊢ Epi s.fst","state_after":"no goals","tactic":"apply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _)","premises":[{"full_name":"CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso_hom_comp","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[137,8],"def_end_pos":[137,39]},{"full_name":"CategoryTheory.Limits.limit.isLimit","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[156,4],"def_end_pos":[156,17]},{"full_name":"CategoryTheory.epi_of_epi_fac","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[310,8],"def_end_pos":[310,22]}]}]}
{"url":"Mathlib/Data/Finset/Pointwise.lean","commit":"","full_name":"Finset.inv_smul_finset_distrib","start":[1699,0],"end":[1701,32],"file_path":"Mathlib/Data/Finset/Pointwise.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : DecidableEq β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ns✝ t : Finset β\na✝ : α\nb : β\ninst✝ : DecidableEq α\na : α\ns : Finset α\n⊢ (a • s)⁻¹ = op a⁻¹ • s⁻¹","state_after":"case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : DecidableEq β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ns✝ t : Finset β\na✝¹ : α\nb : β\ninst✝ : DecidableEq α\na : α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ (a • s)⁻¹ ↔ a✝ ∈ op a⁻¹ • s⁻¹","tactic":"ext","premises":[]},{"state_before":"case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : DecidableEq β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ns✝ t : Finset β\na✝¹ : α\nb : β\ninst✝ : DecidableEq α\na : α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ (a • s)⁻¹ ↔ a✝ ∈ op a⁻¹ • s⁻¹","state_after":"no goals","tactic":"simp [← inv_smul_mem_iff]","premises":[{"full_name":"Finset.inv_smul_mem_iff","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[1622,8],"def_end_pos":[1622,24]}]}]}
{"url":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","commit":"","full_name":"MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive","start":[266,0],"end":[278,41],"file_path":"Mathlib/MeasureTheory/Decomposition/Jordan.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u","tactic":"rw [restrict_le_restrict_iff] at hsu hsv","premises":[{"full_name":"MeasureTheory.VectorMeasure.restrict_le_restrict_iff","def_path":"Mathlib/MeasureTheory/Measure/VectorMeasure.lean","def_pos":[744,8],"def_end_pos":[744,32]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u","state_after":"case left\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0\n\ncase right\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u","tactic":"on_goal 1 =>\n    have a := hsu (hu.diff hv) diff_subset\n    have b := hsv (hv.diff hu) diff_subset\n    erw [of_union (Set.disjoint_of_subset_left diff_subset disjoint_sdiff_self_right)\n        (hu.diff hv) (hv.diff hu)] at hs\n    rw [zero_apply] at a b\n    constructor","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MeasurableSet.diff","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[176,18],"def_end_pos":[176,36]},{"full_name":"MeasureTheory.VectorMeasure.of_union","def_path":"Mathlib/MeasureTheory/Measure/VectorMeasure.lean","def_pos":[164,8],"def_end_pos":[164,16]},{"full_name":"MeasureTheory.VectorMeasure.zero_apply","def_path":"Mathlib/MeasureTheory/Measure/VectorMeasure.lean","def_pos":[256,8],"def_end_pos":[256,18]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]},{"full_name":"Set.disjoint_of_subset_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1236,6],"def_end_pos":[1236,29]},{"full_name":"disjoint_sdiff_self_right","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[198,8],"def_end_pos":[198,33]}]},{"state_before":"case left\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0\n\ncase right\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u","state_after":"no goals","tactic":"all_goals first | linarith | assumption","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean","commit":"","full_name":"CategoryTheory.Meq.congr_apply","start":[55,0],"end":[59,5],"file_path":"Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category.{max v u, w} D\ninst✝ : ConcreteCategory D\nX : C\nP : Cᵒᵖ ⥤ D\nS : J.Cover X\nx : Meq P S\nY : C\nf g : Y ⟶ X\nh : f = g\nhf : S.sieve.arrows f\n⊢ S.sieve.arrows g","state_after":"no goals","tactic":"simpa only [← h] using hf","premises":[]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category.{max v u, w} D\ninst✝ : ConcreteCategory D\nX : C\nP : Cᵒᵖ ⥤ D\nS : J.Cover X\nx : Meq P S\nY : C\nf g : Y ⟶ X\nh : f = g\nhf : S.sieve.arrows f\n⊢ ↑x { Y := Y, f := f, hf := hf } = ↑x { Y := Y, f := g, hf := ⋯ }","state_after":"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category.{max v u, w} D\ninst✝ : ConcreteCategory D\nX : C\nP : Cᵒᵖ ⥤ D\nS : J.Cover X\nx : Meq P S\nY : C\nf : Y ⟶ X\nhf : S.sieve.arrows f\n⊢ ↑x { Y := Y, f := f, hf := hf } = ↑x { Y := Y, f := f, hf := ⋯ }","tactic":"subst h","premises":[]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category.{max v u, w} D\ninst✝ : ConcreteCategory D\nX : C\nP : Cᵒᵖ ⥤ D\nS : J.Cover X\nx : Meq P S\nY : C\nf : Y ⟶ X\nhf : S.sieve.arrows f\n⊢ ↑x { Y := Y, f := f, hf := hf } = ↑x { Y := Y, f := f, hf := ⋯ }","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Abelian/Basic.lean","commit":"","full_name":"CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation_m","start":[343,0],"end":[350,36],"file_path":"Mathlib/CategoryTheory/Abelian/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\n⊢ Mono (Abelian.factorThruCoimage f)","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","commit":"","full_name":"WeierstrassCurve.Φ_three","start":[382,0],"end":[384,86],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","tactics":[{"state_before":"R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW : WeierstrassCurve R\n⊢ W.Φ 3 = X * W.Ψ₃ ^ 2 - W.preΨ₄ * W.Ψ₂Sq","state_after":"no goals","tactic":"erw [Φ_ofNat, preΨ'_three, mul_one, preΨ'_four, preΨ'_two, mul_one, if_pos even_two]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"WeierstrassCurve.preΨ'_four","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","def_pos":[179,6],"def_end_pos":[179,16]},{"full_name":"WeierstrassCurve.preΨ'_three","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","def_pos":[175,6],"def_end_pos":[175,17]},{"full_name":"WeierstrassCurve.preΨ'_two","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","def_pos":[171,6],"def_end_pos":[171,15]},{"full_name":"WeierstrassCurve.Φ_ofNat","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","def_pos":[361,6],"def_end_pos":[361,13]},{"full_name":"even_two","def_path":"Mathlib/Algebra/Ring/Parity.lean","def_pos":[77,14],"def_end_pos":[77,22]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]}]}
{"url":"Mathlib/GroupTheory/Coxeter/Length.lean","commit":"","full_name":"CoxeterSystem.length_mul_ge_length_sub_length'","start":[111,0],"end":[113,78],"file_path":"Mathlib/GroupTheory/Coxeter/Length.lean","tactics":[{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₁ w₂ : W\n⊢ cs.length w₂ - cs.length w₁ ≤ cs.length (w₁ * w₂)","state_after":"no goals","tactic":"simpa [Nat.sub_le_of_le_add, add_comm] using cs.length_mul_le w₁⁻¹ (w₁ * w₂)","premises":[{"full_name":"CoxeterSystem.length_mul_le","def_path":"Mathlib/GroupTheory/Coxeter/Length.lean","def_pos":[100,8],"def_end_pos":[100,21]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Nat.sub_le_of_le_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[963,8],"def_end_pos":[963,24]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
{"url":"Mathlib/Data/Finset/Image.lean","commit":"","full_name":"Finset.coe_image","start":[353,0],"end":[355,75],"file_path":"Mathlib/Data/Finset/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\n⊢ ∀ (x : β), x ∈ ↑(image f s) ↔ x ∈ f '' ↑s","state_after":"no goals","tactic":"simp only [mem_coe, mem_image, Set.mem_image, implies_true]","premises":[{"full_name":"Finset.mem_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[195,8],"def_end_pos":[195,15]},{"full_name":"Finset.mem_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[303,8],"def_end_pos":[303,17]},{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"implies_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[132,16],"def_end_pos":[132,28]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"add_sub_one_nsmul","start":[194,0],"end":[196,67],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Monoid M\na✝ b c : M\nm n : ℕ\nhn : n ≠ 0\na : M\n⊢ a ^ (n - 1) * a = a ^ n","state_after":"no goals","tactic":"rw [← pow_succ, Nat.sub_add_cancel $ Nat.one_le_iff_ne_zero.2 hn]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.one_le_iff_ne_zero","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[132,6],"def_end_pos":[132,24]},{"full_name":"Nat.sub_add_cancel","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[893,26],"def_end_pos":[893,40]},{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]}]}]}
{"url":"Mathlib/Algebra/Lie/InvariantForm.lean","commit":"","full_name":"LieAlgebra.InvariantForm.restrict_nondegenerate","start":[137,0],"end":[140,75],"file_path":"Mathlib/Algebra/Lie/InvariantForm.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : Field K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : LieRingModule L M\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I\nI : LieIdeal K L\nhI : IsAtom I\n⊢ (Φ.restrict (lieIdealSubalgebra K L I).toSubmodule).Nondegenerate","state_after":"K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : Field K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : LieRingModule L M\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I\nI : LieIdeal K L\nhI : IsAtom I\n⊢ IsCompl (lieIdealSubalgebra K L I).toSubmodule (Φ.orthogonal (lieIdealSubalgebra K L I).toSubmodule)","tactic":"rw [LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal hΦ_refl]","premises":[{"full_name":"LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal","def_path":"Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean","def_pos":[360,8],"def_end_pos":[360,53]}]},{"state_before":"K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : Field K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : LieRingModule L M\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I\nI : LieIdeal K L\nhI : IsAtom I\n⊢ IsCompl (lieIdealSubalgebra K L I).toSubmodule (Φ.orthogonal (lieIdealSubalgebra K L I).toSubmodule)","state_after":"no goals","tactic":"exact orthogonal_isCompl_coe_submodule Φ hΦ_nondeg hΦ_inv hΦ_refl hL I hI","premises":[{"full_name":"LieAlgebra.InvariantForm.orthogonal_isCompl_coe_submodule","def_path":"Mathlib/Algebra/Lie/InvariantForm.lean","def_pos":[125,6],"def_end_pos":[125,38]}]}]}
{"url":"Mathlib/LinearAlgebra/SesquilinearForm.lean","commit":"","full_name":"LinearMap.IsAlt.neg","start":[264,0],"end":[269,10],"file_path":"Mathlib/LinearAlgebra/SesquilinearForm.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\n⊢ -(B x) y = (B y) x","state_after":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : (B (y + x)) (y + x) = 0\n⊢ -(B x) y = (B y) x","tactic":"have H1 : B (y + x) (y + x) = 0 := self_eq_zero H (y + x)","premises":[{"full_name":"LinearMap.IsAlt.self_eq_zero","def_path":"Mathlib/LinearAlgebra/SesquilinearForm.lean","def_pos":[243,8],"def_end_pos":[243,26]}]},{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : (B (y + x)) (y + x) = 0\n⊢ -(B x) y = (B y) x","state_after":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : (B x) y + (B y) x = 0\n⊢ -(B x) y = (B y) x","tactic":"simp? [map_add, self_eq_zero H] at H1 says\n    simp only [map_add, add_apply, self_eq_zero H, zero_add, add_zero] at H1","premises":[{"full_name":"LinearMap.IsAlt.self_eq_zero","def_path":"Mathlib/LinearAlgebra/SesquilinearForm.lean","def_pos":[243,8],"def_end_pos":[243,26]},{"full_name":"LinearMap.add_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[819,8],"def_end_pos":[819,17]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : (B x) y + (B y) x = 0\n⊢ -(B x) y = (B y) x","state_after":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : -(B x) y = (B y) x\n⊢ -(B x) y = (B y) x","tactic":"rw [add_eq_zero_iff_neg_eq] at H1","premises":[{"full_name":"add_eq_zero_iff_neg_eq","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[641,2],"def_end_pos":[641,13]}]},{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nMₗ₁ : Type u_9\nMₗ₁' : Type u_10\nMₗ₂ : Type u_11\nMₗ₂' : Type u_12\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\nn : Type u_19\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : -(B x) y = (B y) x\n⊢ -(B x) y = (B y) x","state_after":"no goals","tactic":"exact H1","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/ExactSequence.lean","commit":"","full_name":"CategoryTheory.ComposableArrows.exact₀","start":[159,0],"end":[162,41],"file_path":"Mathlib/Algebra/Homology/ExactSequence.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS✝ : ComposableArrows C n\nS : ComposableArrows C 0\ni : ℕ\nhi : autoParam (i + 2 ≤ 0) _auto✝\n⊢ (S.sc ⋯ i ⋯).Exact","state_after":"no goals","tactic":"simp [autoParam] at hi","premises":[{"full_name":"autoParam","def_path":".lake/packages/lean4/src/lean/Init/Tactics.lean","def_pos":[1614,7],"def_end_pos":[1614,16]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Cofinality.lean","commit":"","full_name":"Cardinal.nfpBFamily_lt_ord_of_isRegular","start":[1045,0],"end":[1049,66],"file_path":"Mathlib/SetTheory/Cardinal/Cofinality.lean","tactics":[{"state_before":"α : Type u_1\nr : α → α → Prop\no : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u} → Ordinal.{u}\nc : Cardinal.{u}\nhc : c.IsRegular\nho : o.card < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), ∀ b < c.ord, f i hi b < c.ord\na : Ordinal.{u}\n⊢ lift.{u, u} o.card < c","state_after":"no goals","tactic":"rwa [lift_id]","premises":[{"full_name":"Cardinal.lift_id","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[203,8],"def_end_pos":[203,15]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Preadditive.lean","commit":"","full_name":"CategoryTheory.leftDistributor_assoc","start":[179,0],"end":[190,6],"file_path":"Mathlib/CategoryTheory/Monoidal/Preadditive.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\n⊢ ((asIso (𝟙 X) ⊗ leftDistributor Y f) ≪≫ leftDistributor X fun j => Y ⊗ f j) =\n    (α_ X Y (⨁ f)).symm ≪≫ leftDistributor (X ⊗ Y) f ≪≫ biproduct.mapIso fun j => α_ X Y (f j)","state_after":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ ((asIso (𝟙 X) ⊗ leftDistributor Y f) ≪≫ leftDistributor X fun j => Y ⊗ f j).hom ≫\n      biproduct.π (fun j => X ⊗ Y ⊗ f j) j✝ =\n    ((α_ X Y (⨁ f)).symm ≪≫ leftDistributor (X ⊗ Y) f ≪≫ biproduct.mapIso fun j => α_ X Y (f j)).hom ≫\n      biproduct.π (fun j => X ⊗ Y ⊗ f j) j✝","tactic":"ext","premises":[]},{"state_before":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ ((asIso (𝟙 X) ⊗ leftDistributor Y f) ≪≫ leftDistributor X fun j => Y ⊗ f j).hom ≫\n      biproduct.π (fun j => X ⊗ Y ⊗ f j) j✝ =\n    ((α_ X Y (⨁ f)).symm ≪≫ leftDistributor (X ⊗ Y) f ≪≫ biproduct.mapIso fun j => α_ X Y (f j)).hom ≫\n      biproduct.π (fun j => X ⊗ Y ⊗ f j) j✝","state_after":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then\n      ∑ x : J,\n        (𝟙 X ⊗ Y ◁ biproduct.π f x) ≫ (𝟙 X ⊗ biproduct.ι (fun j => Y ⊗ f j) x) ≫ X ◁ biproduct.π (fun j => Y ⊗ f j) j✝\n    else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (X ⊗ Y) ◁ biproduct.π f j✝ ≫ (α_ X Y (f j✝)).hom else 0","tactic":"simp only [Category.comp_id, Category.assoc, eqToHom_refl, Iso.trans_hom, Iso.symm_hom,\n    asIso_hom, comp_zero, comp_dite, Preadditive.sum_comp, Preadditive.comp_sum, tensor_sum,\n    id_tensor_comp, tensorIso_hom, leftDistributor_hom, biproduct.mapIso_hom, biproduct.ι_map,\n    biproduct.ι_π, Finset.sum_dite_irrel, Finset.sum_dite_eq', Finset.sum_const_zero]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Iso.symm_hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"CategoryTheory.Iso.trans_hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[128,9],"def_end_pos":[128,14]},{"full_name":"CategoryTheory.Limits.biproduct.mapIso_hom","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[590,2],"def_end_pos":[590,7]},{"full_name":"CategoryTheory.Limits.biproduct.ι_map","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[569,8],"def_end_pos":[569,23]},{"full_name":"CategoryTheory.Limits.biproduct.ι_π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[450,8],"def_end_pos":[450,21]},{"full_name":"CategoryTheory.Limits.comp_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[61,8],"def_end_pos":[61,17]},{"full_name":"CategoryTheory.MonoidalCategory.id_tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[749,8],"def_end_pos":[749,22]},{"full_name":"CategoryTheory.Preadditive.comp_sum","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[162,8],"def_end_pos":[162,16]},{"full_name":"CategoryTheory.Preadditive.sum_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]},{"full_name":"CategoryTheory.asIso_hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[285,8],"def_end_pos":[285,17]},{"full_name":"CategoryTheory.comp_dite","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[228,8],"def_end_pos":[228,17]},{"full_name":"CategoryTheory.eqToHom_refl","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[44,8],"def_end_pos":[44,20]},{"full_name":"CategoryTheory.leftDistributor_hom","def_path":"Mathlib/CategoryTheory/Monoidal/Preadditive.lean","def_pos":[140,8],"def_end_pos":[140,27]},{"full_name":"CategoryTheory.tensorIso_hom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[394,2],"def_end_pos":[394,7]},{"full_name":"CategoryTheory.tensor_sum","def_path":"Mathlib/CategoryTheory/Monoidal/Preadditive.lean","def_pos":[105,8],"def_end_pos":[105,18]},{"full_name":"Finset.sum_const_zero","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[362,2],"def_end_pos":[362,13]},{"full_name":"Finset.sum_dite_eq'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1116,2],"def_end_pos":[1116,13]},{"full_name":"Finset.sum_dite_irrel","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1158,2],"def_end_pos":[1158,13]}]},{"state_before":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then\n      ∑ x : J,\n        (𝟙 X ⊗ Y ◁ biproduct.π f x) ≫ (𝟙 X ⊗ biproduct.ι (fun j => Y ⊗ f j) x) ≫ X ◁ biproduct.π (fun j => Y ⊗ f j) j✝\n    else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (X ⊗ Y) ◁ biproduct.π f j✝ ≫ (α_ X Y (f j✝)).hom else 0","state_after":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then\n      ∑ x : J,\n        (𝟙 X ⊗ 𝟙 Y ⊗ biproduct.π f x) ≫\n          (𝟙 X ⊗ biproduct.ι (fun j => Y ⊗ f j) x) ≫ (𝟙 X ⊗ biproduct.π (fun j => Y ⊗ f j) j✝)\n    else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (𝟙 (X ⊗ Y) ⊗ biproduct.π f j✝) ≫ (α_ X Y (f j✝)).hom else 0","tactic":"simp_rw [← id_tensorHom]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CategoryTheory.MonoidalCategory.id_tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[222,8],"def_end_pos":[222,20]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then\n      ∑ x : J,\n        (𝟙 X ⊗ 𝟙 Y ⊗ biproduct.π f x) ≫\n          (𝟙 X ⊗ biproduct.ι (fun j => Y ⊗ f j) x) ≫ (𝟙 X ⊗ biproduct.π (fun j => Y ⊗ f j) j✝)\n    else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (𝟙 (X ⊗ Y) ⊗ biproduct.π f j✝) ≫ (α_ X Y (f j✝)).hom else 0","state_after":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then ∑ x : J, 𝟙 X ⊗ (𝟙 Y ⊗ biproduct.π f x) ≫ if h : x = j✝ then eqToHom ⋯ else 0 else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (𝟙 (X ⊗ Y) ⊗ biproduct.π f j✝) ≫ (α_ X Y (f j✝)).hom else 0","tactic":"simp only [← id_tensor_comp, biproduct.ι_π]","premises":[{"full_name":"CategoryTheory.Limits.biproduct.ι_π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[450,8],"def_end_pos":[450,21]},{"full_name":"CategoryTheory.MonoidalCategory.id_tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[749,8],"def_end_pos":[749,22]}]},{"state_before":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then ∑ x : J, 𝟙 X ⊗ (𝟙 Y ⊗ biproduct.π f x) ≫ if h : x = j✝ then eqToHom ⋯ else 0 else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (𝟙 (X ⊗ Y) ⊗ biproduct.π f j✝) ≫ (α_ X Y (f j✝)).hom else 0","state_after":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then\n      ∑ x : J,\n        if h : x = j✝ then (𝟙 X ⊗ 𝟙 Y ⊗ biproduct.π f x) ≫ (𝟙 X ⊗ eqToHom ⋯)\n        else (𝟙 X ⊗ 𝟙 Y ⊗ biproduct.π f x) ≫ (𝟙 X ⊗ 0)\n    else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (𝟙 (X ⊗ Y) ⊗ biproduct.π f j✝) ≫ (α_ X Y (f j✝)).hom else 0","tactic":"simp only [id_tensor_comp, tensor_dite, comp_dite]","premises":[{"full_name":"CategoryTheory.MonoidalCategory.id_tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[749,8],"def_end_pos":[749,22]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_dite","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[431,8],"def_end_pos":[431,19]},{"full_name":"CategoryTheory.comp_dite","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[228,8],"def_end_pos":[228,17]}]},{"state_before":"case w.w\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nf : J → C\nj✝ : J\n⊢ (if j✝ ∈ Finset.univ then\n      ∑ x : J,\n        if h : x = j✝ then (𝟙 X ⊗ 𝟙 Y ⊗ biproduct.π f x) ≫ (𝟙 X ⊗ eqToHom ⋯)\n        else (𝟙 X ⊗ 𝟙 Y ⊗ biproduct.π f x) ≫ (𝟙 X ⊗ 0)\n    else 0) =\n    if j✝ ∈ Finset.univ then (α_ X Y (⨁ f)).inv ≫ (𝟙 (X ⊗ Y) ⊗ biproduct.π f j✝) ≫ (α_ X Y (f j✝)).hom else 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/GroupTheory/SpecificGroups/Alternating.lean","commit":"","full_name":"alternatingGroup.nontrivial_of_three_le_card","start":[204,0],"end":[209,46],"file_path":"Mathlib/GroupTheory/SpecificGroups/Alternating.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\n⊢ Nontrivial ↥(alternatingGroup α)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ Nontrivial ↥(alternatingGroup α)","tactic":"haveI := Fintype.one_lt_card_iff_nontrivial.1 (lt_trans (by decide) h3)","premises":[{"full_name":"Fintype.one_lt_card_iff_nontrivial","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[517,8],"def_end_pos":[517,34]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"lt_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[74,8],"def_end_pos":[74,16]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ Nontrivial ↥(alternatingGroup α)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 1 < card ↥(alternatingGroup α)","tactic":"rw [← Fintype.one_lt_card_iff_nontrivial]","premises":[{"full_name":"Fintype.one_lt_card_iff_nontrivial","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[517,8],"def_end_pos":[517,34]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 1 < card ↥(alternatingGroup α)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 2 * 1 < 2 * card ↥(alternatingGroup α)","tactic":"refine lt_of_mul_lt_mul_left ?_ (le_of_lt Nat.prime_two.pos)","premises":[{"full_name":"Nat.Prime.pos","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[51,8],"def_end_pos":[51,17]},{"full_name":"Nat.prime_two","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[143,8],"def_end_pos":[143,17]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"lt_of_mul_lt_mul_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[205,8],"def_end_pos":[205,29]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 2 * 1 < 2 * card ↥(alternatingGroup α)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ (2 * 1).succ ≤ (card α).factorial","tactic":"rw [two_mul_card_alternatingGroup, card_perm, ← Nat.succ_le_iff]","premises":[{"full_name":"Fintype.card_perm","def_path":"Mathlib/Data/Fintype/Perm.lean","def_pos":[150,8],"def_end_pos":[150,25]},{"full_name":"Nat.succ_le_iff","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[109,6],"def_end_pos":[109,17]},{"full_name":"two_mul_card_alternatingGroup","def_path":"Mathlib/GroupTheory/SpecificGroups/Alternating.lean","def_pos":[88,8],"def_end_pos":[88,37]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ (2 * 1).succ ≤ (card α).factorial","state_after":"no goals","tactic":"exact le_trans h3 (card α).self_le_factorial","premises":[{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Nat.self_le_factorial","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[119,8],"def_end_pos":[119,25]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]}]}
{"url":"Mathlib/Data/Real/EReal.lean","commit":"","full_name":"EReal.induction₂","start":[154,0],"end":[179,19],"file_path":"Mathlib/Data/Real/EReal.lean","tactics":[{"state_before":"P : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\n⊢ P ⊥ ↑y","state_after":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : y < 0\n⊢ P ⊥ ↑y\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ⊥ ↑0\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : 0 < y\n⊢ P ⊥ ↑y","tactic":"rcases lt_trichotomy y 0 with (hy | rfl | hy)","premises":[{"full_name":"lt_trichotomy","def_path":"Mathlib/Order/Defs.lean","def_pos":[266,8],"def_end_pos":[266,21]}]},{"state_before":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : y < 0\n⊢ P ⊥ ↑y\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ⊥ ↑0\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : 0 < y\n⊢ P ⊥ ↑y","state_after":"no goals","tactic":"exacts [bot_neg y hy, bot_zero, bot_pos y hy]","premises":[]},{"state_before":"P : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\n⊢ P ↑x ⊥","state_after":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : x < 0\n⊢ P ↑x ⊥\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ↑0 ⊥\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : 0 < x\n⊢ P ↑x ⊥","tactic":"rcases lt_trichotomy x 0 with (hx | rfl | hx)","premises":[{"full_name":"lt_trichotomy","def_path":"Mathlib/Order/Defs.lean","def_pos":[266,8],"def_end_pos":[266,21]}]},{"state_before":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : x < 0\n⊢ P ↑x ⊥\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ↑0 ⊥\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : 0 < x\n⊢ P ↑x ⊥","state_after":"no goals","tactic":"exacts [neg_bot x hx, zero_bot, pos_bot x hx]","premises":[]},{"state_before":"P : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\n⊢ P ↑x ⊤","state_after":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : x < 0\n⊢ P ↑x ⊤\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ↑0 ⊤\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : 0 < x\n⊢ P ↑x ⊤","tactic":"rcases lt_trichotomy x 0 with (hx | rfl | hx)","premises":[{"full_name":"lt_trichotomy","def_path":"Mathlib/Order/Defs.lean","def_pos":[266,8],"def_end_pos":[266,21]}]},{"state_before":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : x < 0\n⊢ P ↑x ⊤\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ↑0 ⊤\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\nx : ℝ\nhx : 0 < x\n⊢ P ↑x ⊤","state_after":"no goals","tactic":"exacts [neg_top x hx, zero_top, pos_top x hx]","premises":[]},{"state_before":"P : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\n⊢ P ⊤ ↑y","state_after":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : y < 0\n⊢ P ⊤ ↑y\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ⊤ ↑0\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : 0 < y\n⊢ P ⊤ ↑y","tactic":"rcases lt_trichotomy y 0 with (hy | rfl | hy)","premises":[{"full_name":"lt_trichotomy","def_path":"Mathlib/Order/Defs.lean","def_pos":[266,8],"def_end_pos":[266,21]}]},{"state_before":"case inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : y < 0\n⊢ P ⊤ ↑y\n\ncase inr.inl\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\n⊢ P ⊤ ↑0\n\ncase inr.inr\nP : EReal → EReal → Prop\ntop_top : P ⊤ ⊤\ntop_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x\ntop_zero : P ⊤ 0\ntop_neg : ∀ x < 0, P ⊤ ↑x\ntop_bot : P ⊤ ⊥\npos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤\npos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥\nzero_top : P 0 ⊤\ncoe_coe : ∀ (x y : ℝ), P ↑x ↑y\nzero_bot : P 0 ⊥\nneg_top : ∀ x < 0, P ↑x ⊤\nneg_bot : ∀ x < 0, P ↑x ⊥\nbot_top : P ⊥ ⊤\nbot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x\nbot_zero : P ⊥ 0\nbot_neg : ∀ x < 0, P ⊥ ↑x\nbot_bot : P ⊥ ⊥\ny : ℝ\nhy : 0 < y\n⊢ P ⊤ ↑y","state_after":"no goals","tactic":"exacts [top_neg y hy, top_zero, top_pos y hy]","premises":[]}]}
{"url":"Mathlib/Data/Real/Irrational.lean","commit":"","full_name":"Irrational.of_int_mul","start":[342,0],"end":[343,39],"file_path":"Mathlib/Data/Real/Irrational.lean","tactics":[{"state_before":"q : ℚ\nx y : ℝ\nm : ℤ\nh : Irrational (↑m * x)\n⊢ Irrational (↑↑m * x)","state_after":"no goals","tactic":"rwa [cast_intCast]","premises":[{"full_name":"Rat.cast_intCast","def_path":"Mathlib/Data/Rat/Cast/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]}]}]}
{"url":"Mathlib/Analysis/PSeries.lean","commit":"","full_name":"NNReal.summable_rpow_inv","start":[354,0],"end":[357,30],"file_path":"Mathlib/Analysis/PSeries.lean","tactics":[{"state_before":"p : ℝ\n⊢ (Summable fun n => (↑n ^ p)⁻¹) ↔ 1 < p","state_after":"no goals","tactic":"simp [← NNReal.summable_coe]","premises":[{"full_name":"NNReal.summable_coe","def_path":"Mathlib/Topology/Instances/NNReal.lean","def_pos":[179,8],"def_end_pos":[179,20]}]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Basic.lean","commit":"","full_name":"Nat.tendsto_div_const_atTop","start":[656,0],"end":[657,48],"file_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nR : Type u_4\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\nn : ℕ\nhn : n ≠ 0\n⊢ Tendsto (fun x => x / n) atTop atTop","state_after":"no goals","tactic":"rw [Tendsto, map_div_atTop_eq_nat n hn.bot_lt]","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.map_div_atTop_eq_nat","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1541,8],"def_end_pos":[1541,28]},{"full_name":"Ne.bot_lt","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[301,8],"def_end_pos":[301,17]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","commit":"","full_name":"Sum.getRight?_eq_some_iff","start":[68,0],"end":[69,65],"file_path":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","tactics":[{"state_before":"β✝ : Type u_1\nb : β✝\nα✝ : Type u_2\nx : α✝ ⊕ β✝\n⊢ x.getRight? = some b ↔ x = inr b","state_after":"no goals","tactic":"cases x <;> simp only [getRight?, Option.some.injEq, inr.injEq]","premises":[{"full_name":"Sum.getRight?","def_path":".lake/packages/lean4/src/lean/Init/Data/Sum.lean","def_pos":[20,4],"def_end_pos":[20,13]}]}]}
{"url":"Mathlib/LinearAlgebra/LinearIndependent.lean","commit":"","full_name":"LinearIndependent.fin_cons'","start":[345,0],"end":[356,41],"file_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","tactics":[{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : LinearIndependent R v\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\n⊢ LinearIndependent R (Fin.cons x v)","state_after":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\n⊢ ∀ (g : Fin (m + 1) → R), ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0 → ∀ (i : Fin (m + 1)), g i = 0","tactic":"rw [Fintype.linearIndependent_iff] at hli ⊢","premises":[{"full_name":"Fintype.linearIndependent_iff","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[168,8],"def_end_pos":[168,37]}]},{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\n⊢ ∀ (g : Fin (m + 1) → R), ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0 → ∀ (i : Fin (m + 1)), g i = 0","state_after":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\ntotal_eq : ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0\nj : Fin (m + 1)\n⊢ g j = 0","tactic":"rintro g total_eq j","premises":[]},{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\ntotal_eq : ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0\nj : Fin (m + 1)\n⊢ g j = 0","state_after":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g x.succ • v x = 0\n⊢ g j = 0","tactic":"simp_rw [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ] at total_eq","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Fin.cons_succ","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"Fin.cons_zero","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Fin.sum_univ_succ","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[66,2],"def_end_pos":[66,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g x.succ • v x = 0\n⊢ g j = 0","state_after":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g x.succ • v x = 0\nthis : g 0 = 0\n⊢ g j = 0","tactic":"have : g 0 = 0 := by\n    refine x_ortho (g 0) ⟨∑ i : Fin m, g i.succ • v i, ?_⟩ total_eq\n    exact sum_mem fun i _ => smul_mem _ _ (subset_span ⟨i, rfl⟩)","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Fin.succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Basic.lean","def_pos":[34,4],"def_end_pos":[34,8]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Submodule.smul_mem","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[194,8],"def_end_pos":[194,16]},{"full_name":"Submodule.subset_span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[72,8],"def_end_pos":[72,19]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"sum_mem","def_path":"Mathlib/Algebra/Group/Submonoid/Membership.lean","def_pos":[87,2],"def_end_pos":[87,13]}]},{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g x.succ • v x = 0\nthis : g 0 = 0\n⊢ g j = 0","state_after":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : ∑ x : Fin m, g x.succ • v x = 0\nthis : g 0 = 0\n⊢ g j = 0","tactic":"rw [this, zero_smul, zero_add] at total_eq","premises":[{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]},{"state_before":"ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : ↥(span R (range v))), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : ∑ x : Fin m, g x.succ • v x = 0\nthis : g 0 = 0\n⊢ g j = 0","state_after":"no goals","tactic":"exact Fin.cases this (hli _ total_eq) j","premises":[{"full_name":"Fin.cases","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[654,20],"def_end_pos":[654,25]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/Independent.lean","commit":"","full_name":"Affine.Simplex.face_centroid_eq_centroid","start":[881,0],"end":[888,6],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Independent.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : fs.card = m + 1\n⊢ centroid k univ (s.face h).points = centroid k fs s.points","state_after":"case h.e'_3.h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : fs.card = m + 1\n⊢ fs = map (fs.orderEmbOfFin h).toEmbedding univ","tactic":"convert (Finset.univ.centroid_map k (fs.orderEmbOfFin h).toEmbedding s.points).symm","premises":[{"full_name":"Affine.Simplex.points","def_path":"Mathlib/LinearAlgebra/AffineSpace/Independent.lean","def_pos":[763,2],"def_end_pos":[763,8]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset.centroid_map","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[790,8],"def_end_pos":[790,20]},{"full_name":"Finset.orderEmbOfFin","def_path":"Mathlib/Data/Finset/Sort.lean","def_pos":[140,4],"def_end_pos":[140,17]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]}]},{"state_before":"case h.e'_3.h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : fs.card = m + 1\n⊢ fs = map (fs.orderEmbOfFin h).toEmbedding univ","state_after":"case h.e'_3.h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : fs.card = m + 1\n⊢ ↑fs = Set.range ⇑(fs.orderEmbOfFin h).toEmbedding","tactic":"rw [← Finset.coe_inj, Finset.coe_map, Finset.coe_univ, Set.image_univ]","premises":[{"full_name":"Finset.coe_inj","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[224,8],"def_end_pos":[224,15]},{"full_name":"Finset.coe_map","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[100,8],"def_end_pos":[100,15]},{"full_name":"Finset.coe_univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[84,8],"def_end_pos":[84,16]},{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]}]},{"state_before":"case h.e'_3.h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : fs.card = m + 1\n⊢ ↑fs = Set.range ⇑(fs.orderEmbOfFin h).toEmbedding","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/RCLike/Basic.lean","commit":"","full_name":"RCLike.re_ofReal_mul","start":[213,0],"end":[215,62],"file_path":"Mathlib/Analysis/RCLike/Basic.lean","tactics":[{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nr : ℝ\nz : K\n⊢ re (↑r * z) = r * re z","state_after":"no goals","tactic":"simp only [mul_re, ofReal_im, zero_mul, ofReal_re, sub_zero]","premises":[{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"RCLike.mul_re","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[120,8],"def_end_pos":[120,14]},{"full_name":"RCLike.ofReal_im","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[116,8],"def_end_pos":[116,17]},{"full_name":"RCLike.ofReal_re","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[112,8],"def_end_pos":[112,17]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"mul_div_left_comm","start":[548,0],"end":[549,64],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a * (b / c) = b * (a / c)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.tendsto_atTop_diagonal","start":[1349,0],"end":[1351,37],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeSup α\n⊢ Tendsto (fun a => (a, a)) atTop atTop","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeSup α\n⊢ Tendsto (fun a => (a, a)) atTop (atTop ×ˢ atTop)","tactic":"rw [← prod_atTop_atTop_eq]","premises":[{"full_name":"Filter.prod_atTop_atTop_eq","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1317,8],"def_end_pos":[1317,27]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeSup α\n⊢ Tendsto (fun a => (a, a)) atTop (atTop ×ˢ atTop)","state_after":"no goals","tactic":"exact tendsto_id.prod_mk tendsto_id","premises":[{"full_name":"Filter.Tendsto.prod_mk","def_path":"Mathlib/Order/Filter/Prod.lean","def_pos":[144,8],"def_end_pos":[144,23]},{"full_name":"Filter.tendsto_id","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2649,8],"def_end_pos":[2649,18]}]}]}
{"url":"Mathlib/Topology/EMetricSpace/Basic.lean","commit":"","full_name":"EMetric.diam_pi_le_of_le","start":[883,0],"end":[887,62],"file_path":"Mathlib/Topology/EMetricSpace/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝² : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t : Set α\nπ : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (π b)\ns : (b : β) → Set (π b)\nc : ℝ≥0∞\nh : ∀ (b : β), diam (s b) ≤ c\n⊢ diam (univ.pi s) ≤ c","state_after":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝² : PseudoEMetricSpace α\nx✝ y✝ z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t : Set α\nπ : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (π b)\ns : (b : β) → Set (π b)\nc : ℝ≥0∞\nh : ∀ (b : β), diam (s b) ≤ c\nx : (i : β) → π i\nhx : x ∈ univ.pi s\ny : (i : β) → π i\nhy : y ∈ univ.pi s\n⊢ ∀ (b : β), edist (x b) (y b) ≤ c","tactic":"refine diam_le fun x hx y hy => edist_pi_le_iff.mpr ?_","premises":[{"full_name":"EMetric.diam_le","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[806,8],"def_end_pos":[806,15]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"edist_pi_le_iff","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[448,8],"def_end_pos":[448,23]}]},{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝² : PseudoEMetricSpace α\nx✝ y✝ z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t : Set α\nπ : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (π b)\ns : (b : β) → Set (π b)\nc : ℝ≥0∞\nh : ∀ (b : β), diam (s b) ≤ c\nx : (i : β) → π i\nhx : x ∈ univ.pi s\ny : (i : β) → π i\nhy : y ∈ univ.pi s\n⊢ ∀ (b : β), edist (x b) (y b) ≤ c","state_after":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝² : PseudoEMetricSpace α\nx✝ y✝ z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t : Set α\nπ : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (π b)\ns : (b : β) → Set (π b)\nc : ℝ≥0∞\nh : ∀ (b : β), diam (s b) ≤ c\nx : (i : β) → π i\nhx : ∀ (i : β), x i ∈ s i\ny : (i : β) → π i\nhy : ∀ (i : β), y i ∈ s i\n⊢ ∀ (b : β), edist (x b) (y b) ≤ c","tactic":"rw [mem_univ_pi] at hx hy","premises":[{"full_name":"Set.mem_univ_pi","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[224,8],"def_end_pos":[224,19]}]},{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\ninst✝² : PseudoEMetricSpace α\nx✝ y✝ z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t : Set α\nπ : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (π b)\ns : (b : β) → Set (π b)\nc : ℝ≥0∞\nh : ∀ (b : β), diam (s b) ≤ c\nx : (i : β) → π i\nhx : ∀ (i : β), x i ∈ s i\ny : (i : β) → π i\nhy : ∀ (i : β), y i ∈ s i\n⊢ ∀ (b : β), edist (x b) (y b) ≤ c","state_after":"no goals","tactic":"exact fun b => diam_le_iff.1 (h b) (x b) (hx b) (y b) (hy b)","premises":[{"full_name":"EMetric.diam_le_iff","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[790,8],"def_end_pos":[790,19]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","commit":"","full_name":"MeasureTheory.Measure.mkMetric_mono_smul","start":[444,0],"end":[449,51],"file_path":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","tactics":[{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝³ : EMetricSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ⊤\nh0 : c ≠ 0\nhle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂\ns : Set X\n⊢ (mkMetric m₁) s ≤ (c • mkMetric m₂) s","state_after":"ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝³ : EMetricSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ⊤\nh0 : c ≠ 0\nhle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂\ns : Set X\n⊢ (OuterMeasure.mkMetric m₁) s ≤ (c • ⇑(OuterMeasure.mkMetric m₂)) s","tactic":"rw [← OuterMeasure.coe_mkMetric, coe_smul, ← OuterMeasure.coe_mkMetric]","premises":[{"full_name":"MeasureTheory.Measure.coe_smul","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[807,8],"def_end_pos":[807,16]},{"full_name":"MeasureTheory.OuterMeasure.coe_mkMetric","def_path":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","def_pos":[436,8],"def_end_pos":[436,33]}]},{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝³ : EMetricSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ⊤\nh0 : c ≠ 0\nhle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂\ns : Set X\n⊢ (OuterMeasure.mkMetric m₁) s ≤ (c • ⇑(OuterMeasure.mkMetric m₂)) s","state_after":"no goals","tactic":"exact OuterMeasure.mkMetric_mono_smul hc h0 hle s","premises":[{"full_name":"MeasureTheory.OuterMeasure.mkMetric_mono_smul","def_path":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","def_pos":[318,8],"def_end_pos":[318,26]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Ordinal.lean","commit":"","full_name":"Cardinal.aleph_le_beth","start":[402,0],"end":[410,62],"file_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","tactics":[{"state_before":"o : Ordinal.{u_1}\n⊢ aleph o ≤ beth o","state_after":"no goals","tactic":"induction o using limitRecOn with\n  | H₁ => simp\n  | H₂ o h =>\n    rw [aleph_succ, beth_succ, succ_le_iff]\n    exact (cantor _).trans_le (power_le_power_left two_ne_zero h)\n  | H₃ o ho IH =>\n    rw [aleph_limit ho, beth_limit ho]\n    exact ciSup_mono (bddAbove_of_small _) fun x => IH x.1 x.2","premises":[{"full_name":"Cardinal.aleph_limit","def_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","def_pos":[244,8],"def_end_pos":[244,19]},{"full_name":"Cardinal.aleph_succ","def_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","def_pos":[238,8],"def_end_pos":[238,18]},{"full_name":"Cardinal.bddAbove_of_small","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[837,8],"def_end_pos":[837,25]},{"full_name":"Cardinal.beth_limit","def_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","def_pos":[372,8],"def_end_pos":[372,18]},{"full_name":"Cardinal.beth_succ","def_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","def_pos":[369,8],"def_end_pos":[369,17]},{"full_name":"Cardinal.cantor","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[618,8],"def_end_pos":[618,14]},{"full_name":"Cardinal.power_le_power_left","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[606,8],"def_end_pos":[606,27]},{"full_name":"Order.succ_le_iff","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[323,8],"def_end_pos":[323,19]},{"full_name":"Ordinal.limitRecOn","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[276,4],"def_end_pos":[276,14]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]},{"full_name":"ciSup_mono","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Basic.lean","def_pos":[714,8],"def_end_pos":[714,18]},{"full_name":"two_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[54,6],"def_end_pos":[54,17]}]}]}
{"url":"Mathlib/CategoryTheory/Category/PartialFun.lean","commit":"","full_name":"partialFunEquivPointed_functor_obj_X","start":[119,0],"end":[154,62],"file_path":"Mathlib/CategoryTheory/Category/PartialFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nX : PartialFun\na : (partialFunToPointed ⋙ pointedToPartialFun).obj X\n⊢ (fun a => ⟨some a, ⋯⟩) ((fun a => Option.get ↑a ⋯) a) = a","state_after":"no goals","tactic":"simp only [some_get, Subtype.coe_eta]","premises":[{"full_name":"Option.some_get","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean","def_pos":[33,16],"def_end_pos":[33,24]},{"full_name":"Subtype.coe_eta","def_path":"Mathlib/Data/Subtype.lean","def_pos":[83,8],"def_end_pos":[83,15]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ b ∈\n      ((𝟭 PartialFun).map f ≫\n          ((fun X =>\n                PartialFun.Iso.mk\n                  { toFun := fun a => ⟨some a, ⋯⟩, invFun := fun a => Option.get ↑a ⋯, left_inv := ⋯, right_inv := ⋯ })\n              Y✝).hom)\n        a ↔\n    b ∈\n      (((fun X =>\n                PartialFun.Iso.mk\n                  { toFun := fun a => ⟨some a, ⋯⟩, invFun := fun a => Option.get ↑a ⋯, left_inv := ⋯, right_inv := ⋯ })\n              X✝).hom ≫\n          (partialFunToPointed ⋙ pointedToPartialFun).map f)\n        a","state_after":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ (b ∈ (f a).bind fun x => Part.some ⟨some x, ⋯⟩) ↔\n    b ∈\n      (Part.some ⟨some a, ⋯⟩).bind\n        (PFun.toSubtype (fun x => ¬x = none) (Option.elim' none fun a => (f a).toOption) ∘ Subtype.val)","tactic":"dsimp [PartialFun.Iso.mk, CategoryStruct.comp, pointedToPartialFun]","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"PartialFun.Iso.mk","def_path":"Mathlib/CategoryTheory/Category/PartialFun.lean","def_pos":[64,4],"def_end_pos":[64,10]},{"full_name":"pointedToPartialFun","def_path":"Mathlib/CategoryTheory/Category/PartialFun.lean","def_pos":[88,4],"def_end_pos":[88,23]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ (b ∈ (f a).bind fun x => Part.some ⟨some x, ⋯⟩) ↔\n    b ∈\n      (Part.some ⟨some a, ⋯⟩).bind\n        (PFun.toSubtype (fun x => ¬x = none) (Option.elim' none fun a => (f a).toOption) ∘ Subtype.val)","state_after":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ (b ∈ (f a).bind fun x => Part.some ⟨some x, ⋯⟩) ↔\n    b ∈ (PFun.toSubtype (fun x => ¬x = none) (Option.elim' none fun a => (f a).toOption) ∘ Subtype.val) ⟨some a, ⋯⟩","tactic":"rw [Part.bind_some]","premises":[{"full_name":"Part.bind_some","def_path":"Mathlib/Data/Part.lean","def_pos":[433,8],"def_end_pos":[433,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ (b ∈ (f a).bind fun x => Part.some ⟨some x, ⋯⟩) ↔\n    b ∈ (PFun.toSubtype (fun x => ¬x = none) (Option.elim' none fun a => (f a).toOption) ∘ Subtype.val) ⟨some a, ⋯⟩","state_after":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ (∃ a_1 ∈ f a, b ∈ Part.some ⟨some a_1, ⋯⟩) ↔ ↑b = Option.elim' none (fun a => (f a).toOption) ↑⟨some a, ⋯⟩","tactic":"refine (Part.mem_bind_iff.trans ?_).trans PFun.mem_toSubtype_iff.symm","premises":[{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"PFun.mem_toSubtype_iff","def_path":"Mathlib/Data/PFun.lean","def_pos":[469,8],"def_end_pos":[469,25]},{"full_name":"Part.mem_bind_iff","def_path":"Mathlib/Data/Part.lean","def_pos":[413,8],"def_end_pos":[413,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝\n⊢ (∃ a_1 ∈ f a, b ∈ Part.some ⟨some a_1, ⋯⟩) ↔ ↑b = Option.elim' none (fun a => (f a).toOption) ↑⟨some a, ⋯⟩","state_after":"case mk.none\nα : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nhb : none ≠ (partialFunToPointed.obj Y✝).point\n⊢ (∃ a_1 ∈ f a, ⟨none, hb⟩ ∈ Part.some ⟨some a_1, ⋯⟩) ↔\n    ↑⟨none, hb⟩ = Option.elim' none (fun a => (f a).toOption) ↑⟨some a, ⋯⟩\n\ncase mk.some\nα : Type u_1\nβ : Type u_2\nX✝ Y✝ : PartialFun\nf : X✝ ⟶ Y✝\na : (𝟭 PartialFun).obj X✝\nb : Y✝\nhb : some b ≠ (partialFunToPointed.obj Y✝).point\n⊢ (∃ a_1 ∈ f a, ⟨some b, hb⟩ ∈ Part.some ⟨some a_1, ⋯⟩) ↔\n    ↑⟨some b, hb⟩ = Option.elim' none (fun a => (f a).toOption) ↑⟨some a, ⋯⟩","tactic":"obtain ⟨b | b, hb⟩ := b","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nX : Pointed\n⊢ ((pointedToPartialFun ⋙ partialFunToPointed).obj X).X ≃ ((𝟭 Pointed).obj X).X","state_after":"no goals","tactic":"classical exact Equiv.optionSubtypeNe X.point","premises":[{"full_name":"Equiv.optionSubtypeNe","def_path":"Mathlib/Logic/Equiv/Option.lean","def_pos":[242,4],"def_end_pos":[242,19]},{"full_name":"Pointed.point","def_path":"Mathlib/CategoryTheory/Category/Pointed.lean","def_pos":[32,2],"def_end_pos":[32,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nX : Pointed\n⊢ (Equiv.optionSubtypeNe X.point) ((pointedToPartialFun ⋙ partialFunToPointed).obj X).point = ((𝟭 Pointed).obj X).point","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nX Y : Pointed\nf : X ⟶ Y\na : ((pointedToPartialFun ⋙ partialFunToPointed).obj X).X\n⊢ ((pointedToPartialFun ⋙ partialFunToPointed).map f ≫\n          ((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) Y).hom).toFun\n      a =\n    (((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) X).hom ≫ (𝟭 Pointed).map f).toFun a","state_after":"case none\nα : Type u_1\nβ : Type u_2\nX Y : Pointed\nf : X ⟶ Y\n⊢ ((pointedToPartialFun ⋙ partialFunToPointed).map f ≫\n          ((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) Y).hom).toFun\n      none =\n    (((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) X).hom ≫ (𝟭 Pointed).map f).toFun none\n\ncase some.mk\nα : Type u_1\nβ : Type u_2\nX Y : Pointed\nf : X ⟶ Y\na : X.X\nha : a ≠ X.point\n⊢ ((pointedToPartialFun ⋙ partialFunToPointed).map f ≫\n          ((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) Y).hom).toFun\n      (some ⟨a, ha⟩) =\n    (((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) X).hom ≫ (𝟭 Pointed).map f).toFun (some ⟨a, ha⟩)","tactic":"obtain _ | ⟨a, ha⟩ := a","premises":[]},{"state_before":"case some.mk\nα : Type u_1\nβ : Type u_2\nX Y : Pointed\nf : X ⟶ Y\na : X.X\nha : a ≠ X.point\n⊢ ((pointedToPartialFun ⋙ partialFunToPointed).map f ≫\n          ((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) Y).hom).toFun\n      (some ⟨a, ha⟩) =\n    (((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) X).hom ≫ (𝟭 Pointed).map f).toFun (some ⟨a, ha⟩)","state_after":"no goals","tactic":"simp_all [Option.casesOn'_eq_elim, Part.elim_toOption]","premises":[{"full_name":"Option.casesOn'_eq_elim","def_path":"Mathlib/Data/Option/Basic.lean","def_pos":[309,6],"def_end_pos":[309,22]},{"full_name":"Part.elim_toOption","def_path":"Mathlib/Data/Part.lean","def_pos":[262,8],"def_end_pos":[262,21]}]}]}
{"url":"Mathlib/Computability/Language.lean","commit":"","full_name":"Language.map_id","start":[151,0],"end":[152,63],"file_path":"Mathlib/Computability/Language.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ (map id) l = l","state_after":"no goals","tactic":"simp [map]","premises":[{"full_name":"Language.map","def_path":"Mathlib/Computability/Language.lean","def_pos":[144,4],"def_end_pos":[144,7]}]}]}
{"url":"Mathlib/Probability/Moments.lean","commit":"","full_name":"ProbabilityTheory.aestronglyMeasurable_exp_mul_add","start":[218,0],"end":[223,48],"file_path":"Mathlib/Probability/Moments.lean","tactics":[{"state_before":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_int_X : AEStronglyMeasurable (fun ω => rexp (t * X ω)) μ\nh_int_Y : AEStronglyMeasurable (fun ω => rexp (t * Y ω)) μ\n⊢ AEStronglyMeasurable (fun ω => rexp (t * (X + Y) ω)) μ","state_after":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_int_X : AEStronglyMeasurable (fun ω => rexp (t * X ω)) μ\nh_int_Y : AEStronglyMeasurable (fun ω => rexp (t * Y ω)) μ\n⊢ AEStronglyMeasurable (fun ω => rexp (t * X ω) * rexp (t * Y ω)) μ","tactic":"simp_rw [Pi.add_apply, mul_add, exp_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"Real.exp_add","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[687,15],"def_end_pos":[687,22]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_int_X : AEStronglyMeasurable (fun ω => rexp (t * X ω)) μ\nh_int_Y : AEStronglyMeasurable (fun ω => rexp (t * Y ω)) μ\n⊢ AEStronglyMeasurable (fun ω => rexp (t * X ω) * rexp (t * Y ω)) μ","state_after":"no goals","tactic":"exact AEStronglyMeasurable.mul h_int_X h_int_Y","premises":[{"full_name":"MeasureTheory.AEStronglyMeasurable.mul","def_path":"Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean","def_pos":[1197,18],"def_end_pos":[1197,21]}]}]}
{"url":"Mathlib/Algebra/Homology/Additive.lean","commit":"","full_name":"HomologicalComplex.singleMapHomologicalComplex_hom_app_self","start":[268,0],"end":[272,84],"file_path":"Mathlib/Algebra/Homology/Additive.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u\ninst✝¹¹ : Category.{v, u} V\ninst✝¹⁰ : Preadditive V\nW : Type u_2\ninst✝⁹ : Category.{?u.135432, u_2} W\ninst✝⁸ : Preadditive W\nW₁ : Type u_3\nW₂ : Type u_4\ninst✝⁷ : Category.{u_6, u_3} W₁\ninst✝⁶ : Category.{u_5, u_4} W₂\ninst✝⁵ : HasZeroMorphisms W₁\ninst✝⁴ : HasZeroMorphisms W₂\nc✝ : ComplexShape ι\nC D E : HomologicalComplex V c✝\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\ninst✝³ : HasZeroObject W₁\ninst✝² : HasZeroObject W₂\nF : W₁ ⥤ W₂\ninst✝¹ : F.PreservesZeroMorphisms\nc : ComplexShape ι\ninst✝ : DecidableEq ι\nj : ι\nX : W₁\n⊢ ((singleMapHomologicalComplex F c j).hom.app X).f j =\n    F.map (singleObjXSelf c j X).hom ≫ (singleObjXSelf c j (F.obj X)).inv","state_after":"no goals","tactic":"simp [singleMapHomologicalComplex, singleObjXSelf, singleObjXIsoOfEq, eqToHom_map]","premises":[{"full_name":"CategoryTheory.eqToHom_map","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[268,8],"def_end_pos":[268,19]},{"full_name":"HomologicalComplex.singleMapHomologicalComplex","def_path":"Mathlib/Algebra/Homology/Additive.lean","def_pos":[241,18],"def_end_pos":[241,45]},{"full_name":"HomologicalComplex.singleObjXIsoOfEq","def_path":"Mathlib/Algebra/Homology/Single.lean","def_pos":[73,18],"def_end_pos":[73,35]},{"full_name":"HomologicalComplex.singleObjXSelf","def_path":"Mathlib/Algebra/Homology/Single.lean","def_pos":[78,18],"def_end_pos":[78,32]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","commit":"","full_name":"ProbabilityTheory.IsRatCondKernelCDFAux.iInf_rat_gt_eq","start":[391,0],"end":[399,63],"file_path":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\n⊢ ∀ᵐ (t : β) ∂ν a, ∀ (q : ℚ), ⨅ r, f (a, t) ↑r = f (a, t) q","state_after":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\n⊢ ∀ (i : ℚ), ∀ᵐ (a_1 : β) ∂ν a, ⨅ r, f (a, a_1) ↑r = f (a, a_1) i","tactic":"rw [ae_all_iff]","premises":[{"full_name":"MeasureTheory.ae_all_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[93,8],"def_end_pos":[93,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\n⊢ ∀ (i : ℚ), ∀ᵐ (a_1 : β) ∂ν a, ⨅ r, f (a, a_1) ↑r = f (a, a_1) i","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\n⊢ ∀ (s : Set β), MeasurableSet s → (ν a) s < ⊤ → IntegrableOn (fun a_3 => ⨅ r, f (a, a_3) ↑r) s (ν a)\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\n⊢ ∀ (s : Set β), MeasurableSet s → (ν a) s < ⊤ → IntegrableOn (fun a_3 => f (a, a_3) q) s (ν a)\n\ncase refine_3\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\n⊢ ∀ (s : Set β), MeasurableSet s → (ν a) s < ⊤ → ∫ (x : β) in s, ⨅ r, f (a, x) ↑r ∂ν a = ∫ (x : β) in s, f (a, x) q ∂ν a","tactic":"refine fun q ↦ ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (μ := ν a) ?_ ?_ ?_","premises":[{"full_name":"MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite","def_path":"Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean","def_pos":[438,8],"def_end_pos":[438,53]}]}]}
{"url":"Mathlib/Algebra/Group/Semiconj/Defs.lean","commit":"","full_name":"AddSemiconjBy.conj_iff","start":[124,0],"end":[130,74],"file_path":"Mathlib/Algebra/Group/Semiconj/Defs.lean","tactics":[{"state_before":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ SemiconjBy (b * a * b⁻¹) (b * x * b⁻¹) (b * y * b⁻¹) ↔ SemiconjBy a x y","state_after":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * a * b⁻¹ * (b * x * b⁻¹) = b * y * b⁻¹ * (b * a * b⁻¹) ↔ a * x = y * a","tactic":"unfold SemiconjBy","premises":[{"full_name":"SemiconjBy","def_path":"Mathlib/Algebra/Group/Semiconj/Defs.lean","def_pos":[38,4],"def_end_pos":[38,14]}]},{"state_before":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * a * b⁻¹ * (b * x * b⁻¹) = b * y * b⁻¹ * (b * a * b⁻¹) ↔ a * x = y * a","state_after":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * a * x * b⁻¹ = b * y * a * b⁻¹ ↔ a * x = y * a","tactic":"simp only [← mul_assoc, inv_mul_cancel_right]","premises":[{"full_name":"inv_mul_cancel_right","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1071,8],"def_end_pos":[1071,28]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * a * x * b⁻¹ = b * y * a * b⁻¹ ↔ a * x = y * a","state_after":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * (a * (x * b⁻¹)) = b * (y * (a * b⁻¹)) ↔ a * x = y * a","tactic":"repeat rw [mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * (a * (x * b⁻¹)) = b * (y * (a * b⁻¹)) ↔ a * x = y * a","state_after":"no goals","tactic":"rw [mul_left_cancel_iff, ← mul_assoc, ← mul_assoc, mul_right_cancel_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_left_cancel_iff","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[219,8],"def_end_pos":[219,27]},{"full_name":"mul_right_cancel_iff","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[233,8],"def_end_pos":[233,28]}]}]}
{"url":"Mathlib/RingTheory/Smooth/Basic.lean","commit":"","full_name":"Algebra.FormallySmooth.comp","start":[175,0],"end":[183,67],"file_path":"Mathlib/RingTheory/Smooth/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝⁸ : CommSemiring R\nA : Type u\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\nB : Type u\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FormallySmooth R A\ninst✝ : FormallySmooth A B\n⊢ FormallySmooth R B","state_after":"case comp_surjective\nR : Type u\ninst✝⁸ : CommSemiring R\nA : Type u\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\nB : Type u\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FormallySmooth R A\ninst✝ : FormallySmooth A B\n⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),\n    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp","tactic":"constructor","premises":[]},{"state_before":"case comp_surjective\nR : Type u\ninst✝⁸ : CommSemiring R\nA : Type u\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\nB : Type u\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FormallySmooth R A\ninst✝ : FormallySmooth A B\n⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),\n    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp","state_after":"case comp_surjective\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","tactic":"intro C _ _ I hI f","premises":[]},{"state_before":"case comp_surjective\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","state_after":"case comp_surjective.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","tactic":"obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))","premises":[{"full_name":"AlgHom.comp","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[274,4],"def_end_pos":[274,8]},{"full_name":"Algebra.FormallySmooth.comp_surjective","def_path":"Mathlib/RingTheory/Smooth/Basic.lean","def_pos":[50,2],"def_end_pos":[50,17]},{"full_name":"IsScalarTower.toAlgHom","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[133,4],"def_end_pos":[133,12]}]},{"state_before":"case comp_surjective.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","state_after":"case comp_surjective.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","tactic":"letI := f'.toRingHom.toAlgebra","premises":[{"full_name":"RingHom.toAlgebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[198,4],"def_end_pos":[198,21]}]},{"state_before":"case comp_surjective.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","state_after":"case comp_surjective.intro.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\nf'' : B →ₐ[A] C\ne' :\n  (Ideal.Quotient.mkₐ A I).comp f'' =\n    let __src := f.toRingHom;\n    { toRingHom := __src, commutes' := ⋯ }\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","tactic":"obtain ⟨f'', e'⟩ :=\n    FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }","premises":[{"full_name":"AlgHom.congr_fun","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[183,18],"def_end_pos":[183,27]},{"full_name":"Algebra.FormallySmooth.comp_surjective","def_path":"Mathlib/RingTheory/Smooth/Basic.lean","def_pos":[50,2],"def_end_pos":[50,17]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]},{"state_before":"case comp_surjective.intro.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\nf'' : B →ₐ[A] C\ne' :\n  (Ideal.Quotient.mkₐ A I).comp f'' =\n    let __src := f.toRingHom;\n    { toRingHom := __src, commutes' := ⋯ }\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","state_after":"case comp_surjective.intro.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\nf'' : B →ₐ[A] C\ne' :\n  AlgHom.restrictScalars R ((Ideal.Quotient.mkₐ A I).comp f'') =\n    AlgHom.restrictScalars R\n      (let __src := f.toRingHom;\n      { toRingHom := __src, commutes' := ⋯ })\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","tactic":"apply_fun AlgHom.restrictScalars R at e'","premises":[{"full_name":"AlgHom.restrictScalars","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[187,4],"def_end_pos":[187,19]}]},{"state_before":"case comp_surjective.intro.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\nf'' : B →ₐ[A] C\ne' :\n  AlgHom.restrictScalars R ((Ideal.Quotient.mkₐ A I).comp f'') =\n    AlgHom.restrictScalars R\n      (let __src := f.toRingHom;\n      { toRingHom := __src, commutes' := ⋯ })\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","state_after":"no goals","tactic":"exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩","premises":[{"full_name":"AlgHom.ext","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[190,8],"def_end_pos":[190,11]},{"full_name":"AlgHom.restrictScalars","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[187,4],"def_end_pos":[187,19]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","commit":"","full_name":"CategoryTheory.Limits.Cofork.ofπ_pt","start":[330,0],"end":[337,68],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf✝ g : X ⟶ Y\nP : C\nπ : Y ⟶ P\nw : f✝ ≫ π = g ≫ π\ni j : WalkingParallelPair\nf : i ⟶ j\n⊢ (parallelPair f✝ g).map f ≫ (fun X_1 => WalkingParallelPair.casesOn X_1 (f✝ ≫ π) π) j =\n    (fun X_1 => WalkingParallelPair.casesOn X_1 (f✝ ≫ π) π) i ≫ ((Functor.const WalkingParallelPair).obj P).map f","state_after":"no goals","tactic":"cases f <;> dsimp <;> simp [w]","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","commit":"","full_name":"NNReal.orderIsoRpow_apply","start":[334,0],"end":[341,62],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","tactics":[{"state_before":"w x✝ y✝ z y : ℝ\nhy : 0 < y\nx : ℝ≥0\n⊢ (fun x => x ^ y) ((fun x => x ^ (1 / y)) x) = x","state_after":"w x✝ y✝ z y : ℝ\nhy : 0 < y\nx : ℝ≥0\n⊢ (x ^ (1 / y)) ^ y = x","tactic":"dsimp","premises":[]},{"state_before":"w x✝ y✝ z y : ℝ\nhy : 0 < y\nx : ℝ≥0\n⊢ (x ^ (1 / y)) ^ y = x","state_after":"no goals","tactic":"rw [← rpow_mul, one_div_mul_cancel hy.ne.symm, rpow_one]","premises":[{"full_name":"NNReal.rpow_mul","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[81,8],"def_end_pos":[81,16]},{"full_name":"NNReal.rpow_one","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"one_div_mul_cancel","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[293,6],"def_end_pos":[293,24]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean","commit":"","full_name":"CategoryTheory.isSheaf_coherent","start":[27,0],"end":[39,22],"file_path":"Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Precoherent C\nP : Cᵒᵖ ⥤ Type w\n⊢ Presieve.IsSheaf (coherentTopology C) P ↔\n    ∀ (B : C) (α : Type) [inst : Finite α] (X : α → C) (π : (a : α) → X a ⟶ B),\n      EffectiveEpiFamily X π → Presieve.IsSheafFor P (Presieve.ofArrows X π)","state_after":"case mp\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Precoherent C\nP : Cᵒᵖ ⥤ Type w\n⊢ Presieve.IsSheaf (coherentTopology C) P →\n    ∀ (B : C) (α : Type) [inst : Finite α] (X : α → C) (π : (a : α) → X a ⟶ B),\n      EffectiveEpiFamily X π → Presieve.IsSheafFor P (Presieve.ofArrows X π)\n\ncase mpr\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Precoherent C\nP : Cᵒᵖ ⥤ Type w\n⊢ (∀ (B : C) (α : Type) [inst : Finite α] (X : α → C) (π : (a : α) → X a ⟶ B),\n      EffectiveEpiFamily X π → Presieve.IsSheafFor P (Presieve.ofArrows X π)) →\n    Presieve.IsSheaf (coherentTopology C) P","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Group/Pointwise.lean","commit":"","full_name":"sub_ball","start":[182,0],"end":[183,82],"file_path":"Mathlib/Analysis/Normed/Group/Pointwise.lean","tactics":[{"state_before":"E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ s / ball x δ = x⁻¹ • thickening δ s","state_after":"no goals","tactic":"simp [div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean","commit":"","full_name":"NNReal.eventually_pow_one_div_le","start":[390,0],"end":[397,88],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean","tactics":[{"state_before":"x y : ℝ≥0\nhy : 1 < y\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y","state_after":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < (y - 1 + 1) ^ m\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y","tactic":"obtain ⟨m, hm⟩ := add_one_pow_unbounded_of_pos x (tsub_pos_of_lt hy)","premises":[{"full_name":"add_one_pow_unbounded_of_pos","def_path":"Mathlib/Algebra/Order/Archimedean.lean","def_pos":[126,8],"def_end_pos":[126,36]},{"full_name":"tsub_pos_of_lt","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[297,8],"def_end_pos":[297,22]}]},{"state_before":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < (y - 1 + 1) ^ m\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y","state_after":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y","tactic":"rw [tsub_add_cancel_of_le hy.le] at hm","premises":[{"full_name":"tsub_add_cancel_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[28,8],"def_end_pos":[28,29]}]},{"state_before":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y","state_after":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (1 / ↑n) ≤ y","tactic":"refine eventually_atTop.2 ⟨m + 1, fun n hn => ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.eventually_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[155,8],"def_end_pos":[155,24]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (1 / ↑n) ≤ y","state_after":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (↑n)⁻¹ ≤ y","tactic":"simp only [one_div]","premises":[{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]}]},{"state_before":"case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (↑n)⁻¹ ≤ y","state_after":"no goals","tactic":"simpa only [NNReal.rpow_inv_le_iff (Nat.cast_pos.2 <| m.succ_pos.trans_le hn),\n    NNReal.rpow_natCast] using hm.le.trans (pow_le_pow_right hy.le (m.le_succ.trans hn))","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"NNReal.rpow_inv_le_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[209,8],"def_end_pos":[209,23]},{"full_name":"NNReal.rpow_natCast","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[113,8],"def_end_pos":[113,20]},{"full_name":"Nat.cast_pos","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[53,8],"def_end_pos":[53,16]},{"full_name":"Nat.le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1684,8],"def_end_pos":[1684,19]},{"full_name":"Nat.succ_pos","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1693,8],"def_end_pos":[1693,20]},{"full_name":"pow_le_pow_right","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[84,8],"def_end_pos":[84,24]}]}]}
{"url":"Mathlib/LinearAlgebra/BilinearForm/Properties.lean","commit":"","full_name":"LinearMap.BilinForm.compLeft_injective","start":[357,0],"end":[362,66],"file_path":"Mathlib/LinearAlgebra/BilinearForm/Properties.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\n⊢ φ = ψ","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw : M₁\n⊢ φ w = ψ w","tactic":"ext w","premises":[]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw : M₁\n⊢ φ w = ψ w","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw : M₁\n⊢ ∀ (n : M₁), (B (φ w - ψ w)) n = 0","tactic":"refine eq_of_sub_eq_zero (b _ ?_)","premises":[{"full_name":"eq_of_sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[380,2],"def_end_pos":[380,13]}]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw : M₁\n⊢ ∀ (n : M₁), (B (φ w - ψ w)) n = 0","state_after":"case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw v : M₁\n⊢ (B (φ w - ψ w)) v = 0","tactic":"intro v","premises":[]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw v : M₁\n⊢ (B (φ w - ψ w)) v = 0","state_after":"no goals","tactic":"rw [sub_left, ← compLeft_apply, ← compLeft_apply, ← h, sub_self]","premises":[{"full_name":"LinearMap.BilinForm.compLeft_apply","def_path":"Mathlib/LinearAlgebra/BilinearForm/Hom.lean","def_pos":[207,8],"def_end_pos":[207,22]},{"full_name":"LinearMap.BilinForm.sub_left","def_path":"Mathlib/LinearAlgebra/BilinearForm/Basic.lean","def_pos":[81,8],"def_end_pos":[81,16]},{"full_name":"sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[695,29],"def_end_pos":[695,37]}]}]}
{"url":"Mathlib/Control/Bitraversable/Lemmas.lean","commit":"","full_name":"Bitraversable.tfst_eq_fst_id","start":[99,0],"end":[102,30],"file_path":"Mathlib/Control/Bitraversable/Lemmas.lean","tactics":[{"state_before":"t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ✝ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : LawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα α' β : Type u\nf : α → α'\nx : t α β\n⊢ tfst (pure ∘ f) x = pure (fst f x)","state_after":"no goals","tactic":"apply bitraverse_eq_bimap_id","premises":[{"full_name":"LawfulBitraversable.bitraverse_eq_bimap_id","def_path":"Mathlib/Control/Bitraversable/Basic.lean","def_pos":[67,2],"def_end_pos":[67,24]}]}]}
{"url":"Mathlib/Data/Set/Pointwise/Interval.lean","commit":"","full_name":"Set.Ici_mul_Ioi_subset'","start":[107,0],"end":[111,38],"file_path":"Mathlib/Data/Set/Pointwise/Interval.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\n⊢ Ici a * Ioi b ⊆ Ioi (a * b)","state_after":"α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\nthis :\n  ∀ (M : Type ?u.12705) (N : Type ?u.12704) (μ : M → N → N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1\n⊢ Ici a * Ioi b ⊆ Ioi (a * b)","tactic":"haveI := covariantClass_le_of_lt","premises":[{"full_name":"covariantClass_le_of_lt","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean","def_pos":[263,8],"def_end_pos":[263,31]}]},{"state_before":"α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\nthis :\n  ∀ (M : Type ?u.12705) (N : Type ?u.12704) (μ : M → N → N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1\n⊢ Ici a * Ioi b ⊆ Ioi (a * b)","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\nthis :\n  ∀ (M : Type ?u.12705) (N : Type ?u.12704) (μ : M → N → N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1\ny : α\nhya : y ∈ Ici a\nz : α\nhzb : z ∈ Ioi b\n⊢ (fun x x_1 => x * x_1) y z ∈ Ioi (a * b)","tactic":"rintro x ⟨y, hya, z, hzb, rfl⟩","premises":[]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\nthis :\n  ∀ (M : Type ?u.12705) (N : Type ?u.12704) (μ : M → N → N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1\ny : α\nhya : y ∈ Ici a\nz : α\nhzb : z ∈ Ioi b\n⊢ (fun x x_1 => x * x_1) y z ∈ Ioi (a * b)","state_after":"no goals","tactic":"exact mul_lt_mul_of_le_of_lt hya hzb","premises":[{"full_name":"mul_lt_mul_of_le_of_lt","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[156,8],"def_end_pos":[156,30]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/CommMon_.lean","commit":"","full_name":"CategoryTheory.LaxBraidedFunctor.mapCommMonFunctor_obj","start":[126,0],"end":[132,42],"file_path":"Mathlib/CategoryTheory/Monoidal/CommMon_.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : BraidedCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\ninst✝ : BraidedCategory D\nX✝ Y✝ : LaxBraidedFunctor C D\nα : X✝ ⟶ Y✝\n⊢ ∀ ⦃X Y : CommMon_ C⦄ (f : X ⟶ Y),\n    X✝.mapCommMon.map f ≫ (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) Y =\n      (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) X ≫ Y✝.mapCommMon.map f","state_after":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : BraidedCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\ninst✝ : BraidedCategory D\nX✝¹ Y✝¹ : LaxBraidedFunctor C D\nα : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : CommMon_ C\nf✝ : X✝ ⟶ Y✝\n⊢ X✝¹.mapCommMon.map f✝ ≫ (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) Y✝ =\n    (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) X✝ ≫ Y✝¹.mapCommMon.map f✝","tactic":"intros","premises":[]},{"state_before":"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : BraidedCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\ninst✝ : BraidedCategory D\nX✝¹ Y✝¹ : LaxBraidedFunctor C D\nα : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : CommMon_ C\nf✝ : X✝ ⟶ Y✝\n⊢ X✝¹.mapCommMon.map f✝ ≫ (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) Y✝ =\n    (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) X✝ ≫ Y✝¹.mapCommMon.map f✝","state_after":"case h\nC : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : BraidedCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\ninst✝ : BraidedCategory D\nX✝¹ Y✝¹ : LaxBraidedFunctor C D\nα : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : CommMon_ C\nf✝ : X✝ ⟶ Y✝\n⊢ (X✝¹.mapCommMon.map f✝ ≫ (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) Y✝).hom =\n    ((fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) X✝ ≫ Y✝¹.mapCommMon.map f✝).hom","tactic":"ext","premises":[]},{"state_before":"case h\nC : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : BraidedCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\ninst✝ : BraidedCategory D\nX✝¹ Y✝¹ : LaxBraidedFunctor C D\nα : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : CommMon_ C\nf✝ : X✝ ⟶ Y✝\n⊢ (X✝¹.mapCommMon.map f✝ ≫ (fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) Y✝).hom =\n    ((fun A => { hom := α.app A.X, one_hom := ⋯, mul_hom := ⋯ }) X✝ ≫ Y✝¹.mapCommMon.map f✝).hom","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","commit":"","full_name":"TendstoLocallyUniformlyOn.differentiableOn","start":[127,0],"end":[146,80],"file_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","tactics":[{"state_before":"E : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\n⊢ DifferentiableOn ℂ f U","state_after":"E : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"rintro x hx","premises":[]},{"state_before":"E : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"obtain ⟨K, ⟨hKx, hK⟩, hKU⟩ := (compact_basis_nhds x).mem_iff.mp (hU.mem_nhds hx)","premises":[{"full_name":"Filter.HasBasis.mem_iff","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[239,8],"def_end_pos":[239,24]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"compact_basis_nhds","def_path":"Mathlib/Topology/Compactness/LocallyCompact.lean","def_pos":[54,8],"def_end_pos":[54,26]}]},{"state_before":"case intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"obtain ⟨δ, _, _, h1⟩ := exists_cthickening_tendstoUniformlyOn hf hF hK hU hKU","premises":[{"full_name":"Complex.exists_cthickening_tendstoUniformlyOn","def_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","def_pos":[121,8],"def_end_pos":[121,45]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"have h2 : interior K ⊆ U := interior_subset.trans hKU","premises":[{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"interior_subset","def_path":"Mathlib/Topology/Basic.lean","def_pos":[222,8],"def_end_pos":[222,23]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"have h3 : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) (interior K) := by\n    filter_upwards [hF] with n h using h.mono h2","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"DifferentiableOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[188,4],"def_end_pos":[188,20]},{"full_name":"DifferentiableOn.mono","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[588,8],"def_end_pos":[588,29]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"have h4 : TendstoLocallyUniformlyOn F f φ (interior K) := hf.mono h2","premises":[{"full_name":"TendstoLocallyUniformlyOn","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[531,4],"def_end_pos":[531,29]},{"full_name":"TendstoLocallyUniformlyOn.mono","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[574,8],"def_end_pos":[574,38]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"have h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K) :=\n    h1.tendstoLocallyUniformlyOn.mono interior_subset","premises":[{"full_name":"Complex.cderiv","def_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","def_pos":[39,18],"def_end_pos":[39,24]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"TendstoLocallyUniformlyOn","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[531,4],"def_end_pos":[531,29]},{"full_name":"TendstoLocallyUniformlyOn.mono","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[574,8],"def_end_pos":[574,38]},{"full_name":"TendstoUniformlyOn.tendstoLocallyUniformlyOn","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[567,18],"def_end_pos":[567,62]},{"full_name":"deriv","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[143,4],"def_end_pos":[143,9]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"interior_subset","def_path":"Mathlib/Topology/Basic.lean","def_pos":[222,8],"def_end_pos":[222,23]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\nh6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"have h6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x := fun x h =>\n    hasDerivAt_of_tendsto_locally_uniformly_on' isOpen_interior h5 h3 (fun _ => h4.tendsto_at) h","premises":[{"full_name":"Complex.cderiv","def_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","def_pos":[39,18],"def_end_pos":[39,24]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"TendstoLocallyUniformlyOn.tendsto_at","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[693,8],"def_end_pos":[693,44]},{"full_name":"hasDerivAt_of_tendsto_locally_uniformly_on'","def_path":"Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean","def_pos":[523,8],"def_end_pos":[523,51]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]},{"full_name":"isOpen_interior","def_path":"Mathlib/Topology/Basic.lean","def_pos":[219,8],"def_end_pos":[219,23]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\nh6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\nh6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x\nh7 : DifferentiableOn ℂ f (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","tactic":"have h7 : DifferentiableOn ℂ f (interior K) := fun x hx =>\n    (h6 x hx).differentiableAt.differentiableWithinAt","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"DifferentiableAt.differentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[576,8],"def_end_pos":[576,47]},{"full_name":"DifferentiableOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[188,4],"def_end_pos":[188,20]},{"full_name":"HasDerivAt.differentiableAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[336,8],"def_end_pos":[336,35]},{"full_name":"interior","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[108,4],"def_end_pos":[108,12]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\nh6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x\nh7 : DifferentiableOn ℂ f (interior K)\n⊢ DifferentiableWithinAt ℂ f U x","state_after":"no goals","tactic":"exact (h7.differentiableAt (interior_mem_nhds.mpr hKx)).differentiableWithinAt","premises":[{"full_name":"DifferentiableAt.differentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[576,8],"def_end_pos":[576,47]},{"full_name":"DifferentiableOn.differentiableAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[505,8],"def_end_pos":[505,41]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"interior_mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1005,8],"def_end_pos":[1005,25]}]}]}
{"url":"Mathlib/MeasureTheory/Function/LpSpace.lean","commit":"","full_name":"MeasureTheory.indicatorConstLp_univ","start":[894,0],"end":[898,48],"file_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) c","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ Memℒp.toLp (Set.univ.indicator fun x => c) ⋯ = Memℒp.toLp (fun x => c) ⋯","tactic":"rw [← Memℒp.toLp_const, indicatorConstLp]","premises":[{"full_name":"MeasureTheory.Memℒp.toLp_const","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[892,6],"def_end_pos":[892,22]},{"full_name":"MeasureTheory.indicatorConstLp","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[747,4],"def_end_pos":[747,20]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ Memℒp.toLp (Set.univ.indicator fun x => c) ⋯ = Memℒp.toLp (fun x => c) ⋯","state_after":"no goals","tactic":"simp only [Set.indicator_univ, Function.const]","premises":[{"full_name":"Function.const","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[66,14],"def_end_pos":[66,28]},{"full_name":"Set.indicator_univ","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[147,2],"def_end_pos":[147,13]}]}]}
{"url":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","commit":"","full_name":"ValuationSubring.image_maximalIdeal","start":[529,0],"end":[536,6],"file_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","tactics":[{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\n⊢ Subtype.val '' ↑(LocalRing.maximalIdeal ↥A) = ↑A.nonunits","state_after":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ a ∈ Subtype.val '' ↑(LocalRing.maximalIdeal ↥A) ↔ a ∈ ↑A.nonunits","tactic":"ext a","premises":[]},{"state_before":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ a ∈ Subtype.val '' ↑(LocalRing.maximalIdeal ↥A) ↔ a ∈ ↑A.nonunits","state_after":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ x ∈ LocalRing.maximalIdeal ↥A, ↑x = a) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A","tactic":"simp only [Set.mem_image, SetLike.mem_coe, mem_nonunits_iff_exists_mem_maximalIdeal]","premises":[{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"SetLike.mem_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[168,8],"def_end_pos":[168,15]},{"full_name":"ValuationSubring.mem_nonunits_iff_exists_mem_maximalIdeal","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[524,8],"def_end_pos":[524,48]}]},{"state_before":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ x ∈ LocalRing.maximalIdeal ↥A, ↑x = a) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A","state_after":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ a_1, ∃ (b : a_1 ∈ ↑A), ⟨a_1, b⟩ ∈ LocalRing.maximalIdeal ↥A ∧ ↑⟨a_1, b⟩ = a) ↔\n    ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A","tactic":"erw [Subtype.exists]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Subtype.exists","def_path":"Mathlib/Data/Subtype.lean","def_pos":[50,18],"def_end_pos":[50,26]}]},{"state_before":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ a_1, ∃ (b : a_1 ∈ ↑A), ⟨a_1, b⟩ ∈ LocalRing.maximalIdeal ↥A ∧ ↑⟨a_1, b⟩ = a) ↔\n    ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A","state_after":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ (x : a ∈ ↑A), ⟨a, ⋯⟩ ∈ LocalRing.maximalIdeal ↥A) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A","tactic":"simp_rw [exists_and_right, exists_eq_right]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"exists_and_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[291,16],"def_end_pos":[291,32]},{"full_name":"exists_eq_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[285,16],"def_end_pos":[285,31]}]},{"state_before":"case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ (x : a ∈ ↑A), ⟨a, ⋯⟩ ∈ LocalRing.maximalIdeal ↥A) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Finsupp.lean","commit":"","full_name":"Finsupp.mem_supported_support","start":[290,0],"end":[291,28],"file_path":"Mathlib/LinearAlgebra/Finsupp.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\np : α →₀ M\n⊢ p ∈ supported M R ↑p.support","state_after":"no goals","tactic":"rw [Finsupp.mem_supported]","premises":[{"full_name":"Finsupp.mem_supported","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[283,8],"def_end_pos":[283,21]}]}]}
{"url":"Mathlib/Data/Complex/Order.lean","commit":"","full_name":"Complex.real_le_real","start":[66,0],"end":[67,87],"file_path":"Mathlib/Data/Complex/Order.lean","tactics":[{"state_before":"x y : ℝ\n⊢ ↑x ≤ ↑y ↔ x ≤ y","state_after":"no goals","tactic":"simp [le_def, ofReal']","premises":[{"full_name":"Complex.le_def","def_path":"Mathlib/Data/Complex/Order.lean","def_pos":[54,8],"def_end_pos":[54,14]},{"full_name":"Complex.ofReal'","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[79,4],"def_end_pos":[79,11]}]}]}
{"url":"Mathlib/Data/Int/GCD.lean","commit":"","full_name":"Int.lcm_dvd","start":[360,0],"end":[363,91],"file_path":"Mathlib/Data/Int/GCD.lean","tactics":[{"state_before":"i j k : ℤ\n⊢ i ∣ k → j ∣ k → ↑(i.lcm j) ∣ k","state_after":"i j k : ℤ\n⊢ i ∣ k → j ∣ k → ↑(i.natAbs.lcm j.natAbs) ∣ k","tactic":"rw [Int.lcm]","premises":[{"full_name":"Int.lcm","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Gcd.lean","def_pos":[41,4],"def_end_pos":[41,7]}]},{"state_before":"i j k : ℤ\n⊢ i ∣ k → j ∣ k → ↑(i.natAbs.lcm j.natAbs) ∣ k","state_after":"i j k : ℤ\nhi : i ∣ k\nhj : j ∣ k\n⊢ ↑(i.natAbs.lcm j.natAbs) ∣ k","tactic":"intro hi hj","premises":[]},{"state_before":"i j k : ℤ\nhi : i ∣ k\nhj : j ∣ k\n⊢ ↑(i.natAbs.lcm j.natAbs) ∣ k","state_after":"no goals","tactic":"exact natCast_dvd.mpr (Nat.lcm_dvd (natAbs_dvd_natAbs.mpr hi) (natAbs_dvd_natAbs.mpr hj))","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.natAbs_dvd_natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[68,16],"def_end_pos":[68,33]},{"full_name":"Int.natCast_dvd","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[458,6],"def_end_pos":[458,17]},{"full_name":"Nat.lcm_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lcm.lean","def_pos":[44,8],"def_end_pos":[44,15]}]}]}
{"url":"Mathlib/NumberTheory/Multiplicity.lean","commit":"","full_name":"Nat.two_pow_sub_pow","start":[337,0],"end":[351,41],"file_path":"Mathlib/NumberTheory/Multiplicity.lean","tactics":[{"state_before":"R : Type u_1\nn✝ x y : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nn : ℕ\nhn : Even n\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n","state_after":"case inl\nR : Type u_1\nn✝ x y : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nn : ℕ\nhn : Even n\nhyx : y ≤ x\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n\ncase inr\nR : Type u_1\nn✝ x y : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nn : ℕ\nhn : Even n\nhyx : x ≤ y\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n","tactic":"obtain hyx | hyx := le_total y x","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]}]}
{"url":"Mathlib/LinearAlgebra/Dimension/Finite.lean","commit":"","full_name":"FiniteDimensional.nontrivial_of_finrank_eq_succ","start":[353,0],"end":[357,58],"file_path":"Mathlib/LinearAlgebra/Dimension/Finite.lean","tactics":[{"state_before":"R : Type u\nM M₁ : Type v\nM' : Type v'\nι : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroSMulDivisors R M\nn : ℕ\nhn : finrank R M = n.succ\n⊢ 0 < finrank ?m.344409 M","state_after":"R : Type u\nM M₁ : Type v\nM' : Type v'\nι : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroSMulDivisors R M\nn : ℕ\nhn : finrank R M = n.succ\n⊢ 0 < n.succ","tactic":"rw [hn]","premises":[]},{"state_before":"R : Type u\nM M₁ : Type v\nM' : Type v'\nι : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroSMulDivisors R M\nn : ℕ\nhn : finrank R M = n.succ\n⊢ 0 < n.succ","state_after":"no goals","tactic":"exact n.succ_pos","premises":[{"full_name":"Nat.succ_pos","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1693,8],"def_end_pos":[1693,20]}]}]}
{"url":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","commit":"","full_name":"CategoryTheory.Subgroupoid.inv_mem_iff","start":[79,0],"end":[84,15],"file_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\nc d : C\nf : c ⟶ d\n⊢ Groupoid.inv f ∈ S.arrows d c ↔ f ∈ S.arrows c d","state_after":"case mp\nC : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\nc d : C\nf : c ⟶ d\n⊢ Groupoid.inv f ∈ S.arrows d c → f ∈ S.arrows c d\n\ncase mpr\nC : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\nc d : C\nf : c ⟶ d\n⊢ f ∈ S.arrows c d → Groupoid.inv f ∈ S.arrows d c","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","commit":"","full_name":"Polynomial.tendsto_atBot_iff_leadingCoeff_nonpos","start":[67,0],"end":[70,45],"file_path":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ Tendsto (fun x => eval x P) atTop atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0","state_after":"no goals","tactic":"simp only [← tendsto_neg_atTop_iff, ← eval_neg, tendsto_atTop_iff_leadingCoeff_nonneg,\n    degree_neg, leadingCoeff_neg, neg_nonneg]","premises":[{"full_name":"Filter.tendsto_neg_atTop_iff","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[770,8],"def_end_pos":[770,29]},{"full_name":"Polynomial.degree_neg","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[452,8],"def_end_pos":[452,18]},{"full_name":"Polynomial.eval_neg","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1115,8],"def_end_pos":[1115,16]},{"full_name":"Polynomial.leadingCoeff_neg","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[476,8],"def_end_pos":[476,24]},{"full_name":"Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg","def_path":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","def_pos":[59,8],"def_end_pos":[59,45]}]}]}
{"url":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","commit":"","full_name":"AlgebraicGeometry.LocallyRingedSpace.comp_ring_hom_ext","start":[236,0],"end":[252,11],"file_path":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","tactics":[{"state_before":"X✝ : LocallyRingedSpace\nr : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\n⊢ X.toΓSpec ≫ Spec.locallyRingedSpaceMap f = β","state_after":"case h\nX✝ : LocallyRingedSpace\nr : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\n⊢ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val = β.val","tactic":"ext1","premises":[]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\n⊢ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val = β.val","state_after":"case h\nX✝ : LocallyRingedSpace\nr : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\n⊢ ∀ (r : ↑R),\n    let U := basicOpen r;\n    (toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) =\n      toOpen (↑R) U ≫ β.val.c.app (op U)","tactic":"refine Spec.basicOpen_hom_ext w ?_","premises":[{"full_name":"AlgebraicGeometry.Spec.basicOpen_hom_ext","def_path":"Mathlib/AlgebraicGeometry/Spec.lean","def_pos":[152,8],"def_end_pos":[152,30]}]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\n⊢ ∀ (r : ↑R),\n    let U := basicOpen r;\n    (toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) =\n      toOpen (↑R) U ≫ β.val.c.app (op U)","state_after":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","tactic":"intro r U","premises":[]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","state_after":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (toOpen (↑R) U ≫\n        (Spec.locallyRingedSpaceMap f).val.c.app (op U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n      X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","tactic":"rw [LocallyRingedSpace.comp_val_c_app]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.comp_val_c_app","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[151,8],"def_end_pos":[151,22]}]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (toOpen (↑R) U ≫\n        (Spec.locallyRingedSpaceMap f).val.c.app (op U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n      X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","state_after":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (CommRingCat.ofHom f ≫\n        toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n      X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","tactic":"erw [toOpen_comp_comap_assoc]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (CommRingCat.ofHom f ≫\n        toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n      X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","state_after":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ CommRingCat.ofHom f ≫\n      (toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n        X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","tactic":"rw [Category.assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ CommRingCat.ofHom f ≫\n      (toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n        X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","state_after":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ CommRingCat.ofHom f ≫ X.presheaf.map ((X.toΓSpecMapBasicOpen (f r)).leTop.op ≫ eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","tactic":"erw [toΓSpecSheafedSpace_app_spec, ← X.presheaf.map_comp]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec","def_path":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","def_pos":[175,19],"def_end_pos":[175,47]},{"full_name":"AlgebraicGeometry.PresheafedSpace.presheaf","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[48,12],"def_end_pos":[48,20]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ CommRingCat.ofHom f ≫ X.presheaf.map ((X.toΓSpecMapBasicOpen (f r)).leTop.op ≫ eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)","state_after":"no goals","tactic":"exact h r","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","commit":"","full_name":"AlgebraicGeometry.ΓSpec.adjunction_unit_app_app_top","start":[425,0],"end":[433,68],"file_path":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","tactics":[{"state_before":"X : Scheme\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","state_after":"X : Scheme\nthis :\n  Scheme.Γ.rightOp.map (adjunction.unit.app X) ≫ adjunction.counit.app (Scheme.Γ.rightOp.obj X) =\n    𝟙 (Scheme.Γ.rightOp.obj X)\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","tactic":"have := ΓSpec.adjunction.left_triangle_components X","premises":[{"full_name":"AlgebraicGeometry.ΓSpec.adjunction","def_path":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","def_pos":[378,4],"def_end_pos":[378,14]},{"full_name":"CategoryTheory.Adjunction.left_triangle_components","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[208,8],"def_end_pos":[208,32]}]},{"state_before":"X : Scheme\nthis :\n  Scheme.Γ.rightOp.map (adjunction.unit.app X) ≫ adjunction.counit.app (Scheme.Γ.rightOp.obj X) =\n    𝟙 (Scheme.Γ.rightOp.obj X)\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","state_after":"X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op ≫ (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op = 𝟙 (op Γ(X, ⊤))\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","tactic":"dsimp at this","premises":[]},{"state_before":"X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op ≫ (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op = 𝟙 (op Γ(X, ⊤))\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","state_after":"X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = 𝟙 (op Γ(X, ⊤)) ≫ inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","tactic":"rw [← IsIso.eq_comp_inv] at this","premises":[{"full_name":"CategoryTheory.IsIso.eq_comp_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[385,8],"def_end_pos":[385,19]}]},{"state_before":"X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = 𝟙 (op Γ(X, ⊤)) ≫ inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","state_after":"X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","tactic":"simp only [adjunction_counit_app, Functor.id_obj, Functor.comp_obj, Functor.rightOp_obj,\n    Scheme.Γ_obj, Category.id_comp] at this","premises":[{"full_name":"AlgebraicGeometry.Scheme.Γ_obj","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[393,8],"def_end_pos":[393,13]},{"full_name":"AlgebraicGeometry.ΓSpec.adjunction_counit_app","def_path":"Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean","def_pos":[403,8],"def_end_pos":[403,29]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.comp_obj","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[99,8],"def_end_pos":[99,11]},{"full_name":"CategoryTheory.Functor.id_obj","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[85,8],"def_end_pos":[85,14]},{"full_name":"CategoryTheory.Functor.rightOp_obj","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[217,2],"def_end_pos":[217,7]}]},{"state_before":"X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom","state_after":"no goals","tactic":"rw [← Quiver.Hom.op_inj.eq_iff, this, ← op_inv, IsIso.Iso.inv_inv]","premises":[{"full_name":"CategoryTheory.IsIso.Iso.inv_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[363,8],"def_end_pos":[363,19]},{"full_name":"CategoryTheory.op_inv","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[141,8],"def_end_pos":[141,14]},{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"Quiver.Hom.op_inj","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[35,8],"def_end_pos":[35,25]}]}]}
{"url":"Mathlib/Topology/Homotopy/Path.lean","commit":"","full_name":"Path.Homotopy.symm₂_toFun","start":[201,0],"end":[215,10],"file_path":"Mathlib/Topology/Homotopy/Path.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\n⊢ ∀ (x : ↑I), { toFun := fun x => F (x.1, σ x.2), continuous_toFun := ⋯ }.toFun (0, x) = p.symm.toContinuousMap x","state_after":"no goals","tactic":"simp [Path.symm]","premises":[{"full_name":"Path.symm","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[157,4],"def_end_pos":[157,8]}]},{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\n⊢ ∀ (x : ↑I), { toFun := fun x => F (x.1, σ x.2), continuous_toFun := ⋯ }.toFun (1, x) = q.symm.toContinuousMap x","state_after":"no goals","tactic":"simp [Path.symm]","premises":[{"full_name":"Path.symm","def_path":"Mathlib/Topology/Connected/PathConnected.lean","def_pos":[157,4],"def_end_pos":[157,8]}]},{"state_before":"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nt x : ↑I\nhx : x ∈ {0, 1}\n⊢ {\n        toFun := fun x =>\n          { toFun := fun x => F (x.1, σ x.2), continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }.toFun\n            (t, x),\n        continuous_toFun := ⋯ }\n      x =\n    p.symm.toContinuousMap x","state_after":"case inl\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nt x : ↑I\nhx : x = 0\n⊢ {\n        toFun := fun x =>\n          { toFun := fun x => F (x.1, σ x.2), continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }.toFun\n            (t, x),\n        continuous_toFun := ⋯ }\n      x =\n    p.symm.toContinuousMap x\n\ncase inr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np q : Path x₀ x₁\nF : p.Homotopy q\nt x : ↑I\nhx : x ∈ {1}\n⊢ {\n        toFun := fun x =>\n          { toFun := fun x => F (x.1, σ x.2), continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }.toFun\n            (t, x),\n        continuous_toFun := ⋯ }\n      x =\n    p.symm.toContinuousMap x","tactic":"cases' hx with hx hx","premises":[]}]}
{"url":"Mathlib/Probability/Kernel/Basic.lean","commit":"","full_name":"ProbabilityTheory.Kernel.setLIntegral_const","start":[475,0],"end":[477,74],"file_path":"Mathlib/Probability/Kernel/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nf : β → ℝ≥0∞\nμ : Measure β\na : α\ns : Set β\n⊢ ∫⁻ (x : β) in s, f x ∂(const α μ) a = ∫⁻ (x : β) in s, f x ∂μ","state_after":"no goals","tactic":"rw [const_apply]","premises":[{"full_name":"ProbabilityTheory.Kernel.const_apply","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[439,8],"def_end_pos":[439,19]}]}]}
{"url":"Mathlib/RingTheory/ClassGroup.lean","commit":"","full_name":"ClassGroup.mk_eq_one_iff","start":[321,0],"end":[335,14],"file_path":"Mathlib/RingTheory/ClassGroup.lean","tactics":[{"state_before":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ mk I = 1 ↔ (↑↑I).IsPrincipal","state_after":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (equiv K) (mk I) = (equiv K) 1 ↔ (↑↑I).IsPrincipal","tactic":"rw [← (ClassGroup.equiv K).injective.eq_iff]","premises":[{"full_name":"ClassGroup.equiv","def_path":"Mathlib/RingTheory/ClassGroup.lean","def_pos":[168,18],"def_end_pos":[168,34]},{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"MulEquiv.injective","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[227,18],"def_end_pos":[227,27]}]},{"state_before":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (equiv K) (mk I) = (equiv K) 1 ↔ (↑↑I).IsPrincipal","state_after":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (∃ x, spanSingleton R⁰ ↑x = ↑I) ↔ (↑↑I).IsPrincipal","tactic":"simp only [equiv_mk, canonicalEquiv_self, RingEquiv.coe_mulEquiv_refl, QuotientGroup.mk'_apply,\n    _root_.map_one, QuotientGroup.eq_one_iff, MonoidHom.mem_range, Units.ext_iff,\n    coe_toPrincipalIdeal, coe_mapEquiv, MulEquiv.refl_apply]","premises":[{"full_name":"ClassGroup.equiv_mk","def_path":"Mathlib/RingTheory/ClassGroup.lean","def_pos":[195,8],"def_end_pos":[195,27]},{"full_name":"FractionalIdeal.canonicalEquiv_self","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[247,8],"def_end_pos":[247,27]},{"full_name":"MonoidHom.mem_range","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1953,8],"def_end_pos":[1953,17]},{"full_name":"MulEquiv.refl_apply","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[328,8],"def_end_pos":[328,18]},{"full_name":"QuotientGroup.eq_one_iff","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[110,8],"def_end_pos":[110,18]},{"full_name":"QuotientGroup.mk'_apply","def_path":"Mathlib/GroupTheory/QuotientGroup.lean","def_pos":[77,8],"def_end_pos":[77,17]},{"full_name":"RingEquiv.coe_mulEquiv_refl","def_path":"Mathlib/Algebra/Ring/Equiv.lean","def_pos":[224,8],"def_end_pos":[224,25]},{"full_name":"Units.coe_mapEquiv","def_path":"Mathlib/Algebra/Group/Units/Equiv.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"Units.ext_iff","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[137,18],"def_end_pos":[137,25]},{"full_name":"coe_toPrincipalIdeal","def_path":"Mathlib/RingTheory/ClassGroup.lean","def_pos":[58,8],"def_end_pos":[58,28]},{"full_name":"map_one","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[190,8],"def_end_pos":[190,15]}]},{"state_before":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (∃ x, spanSingleton R⁰ ↑x = ↑I) ↔ (↑↑I).IsPrincipal","state_after":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (↑↑I).IsPrincipal → ∃ x, spanSingleton R⁰ ↑x = ↑I","tactic":"refine ⟨fun ⟨x, hx⟩ => ⟨⟨x, by rw [← hx, coe_spanSingleton]⟩⟩, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"FractionalIdeal.coe_spanSingleton","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[560,8],"def_end_pos":[560,25]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (↑↑I).IsPrincipal → ∃ x, spanSingleton R⁰ ↑x = ↑I","state_after":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I","tactic":"intro hI","premises":[]},{"state_before":"R : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I","state_after":"case intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I","tactic":"obtain ⟨x, hx⟩ := @Submodule.IsPrincipal.principal _ _ _ _ _ _ hI","premises":[{"full_name":"Submodule.IsPrincipal.principal","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[60,8],"def_end_pos":[60,29]}]},{"state_before":"case intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I","state_after":"case intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I","tactic":"have hx' : (I : FractionalIdeal R⁰ K) = spanSingleton R⁰ x := by\n    apply Subtype.coe_injective\n    simp only [val_eq_coe, hx, coe_spanSingleton]","premises":[{"full_name":"FractionalIdeal","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[77,4],"def_end_pos":[77,19]},{"full_name":"FractionalIdeal.coe_spanSingleton","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[560,8],"def_end_pos":[560,25]},{"full_name":"FractionalIdeal.spanSingleton","def_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","def_pos":[555,16],"def_end_pos":[555,29]},{"full_name":"FractionalIdeal.val_eq_coe","def_path":"Mathlib/RingTheory/FractionalIdeal/Basic.lean","def_pos":[189,8],"def_end_pos":[189,18]},{"full_name":"Subtype.coe_injective","def_path":"Mathlib/Data/Subtype.lean","def_pos":[102,8],"def_end_pos":[102,21]},{"full_name":"nonZeroDivisors","def_path":"Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean","def_pos":[84,4],"def_end_pos":[84,19]}]},{"state_before":"case intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I","state_after":"case intro.refine_1\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ x ≠ 0\n\ncase intro.refine_2\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ spanSingleton R⁰ ↑(Units.mk0 x ?intro.refine_1) = ↑I","tactic":"refine ⟨Units.mk0 x ?_, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Units.mk0","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[161,4],"def_end_pos":[161,7]}]}]}
{"url":"Mathlib/Deprecated/Group.lean","commit":"","full_name":"IsGroupHom.map_inv","start":[246,0],"end":[249,77],"file_path":"Mathlib/Deprecated/Group.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝¹ : Group α\ninst✝ : Group β\nf : α → β\nhf✝ hf : IsGroupHom f\na : α\n⊢ f a⁻¹ * f a = 1","state_after":"no goals","tactic":"rw [← hf.map_mul, inv_mul_self, hf.map_one]","premises":[{"full_name":"IsGroupHom.map_one","def_path":"Mathlib/Deprecated/Group.lean","def_pos":[243,8],"def_end_pos":[243,15]},{"full_name":"IsMulHom.map_mul","def_path":"Mathlib/Deprecated/Group.lean","def_pos":[47,2],"def_end_pos":[47,9]},{"full_name":"inv_mul_self","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1043,8],"def_end_pos":[1043,20]}]}]}
{"url":"Mathlib/CategoryTheory/Adjunction/Basic.lean","commit":"","full_name":"CategoryTheory.Adjunction.comp_unit_app","start":[442,0],"end":[445,24],"file_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nE : Type u₃\nℰ : Category.{v₃, u₃} E\nH : D ⥤ E\nI : E ⥤ D\nadj₁ : F ⊣ G\nadj₂ : H ⊣ I\nX : C\n⊢ (adj₁.comp adj₂).unit.app X = adj₁.unit.app X ≫ G.map (adj₂.unit.app (F.obj X))","state_after":"no goals","tactic":"simp [Adjunction.comp]","premises":[{"full_name":"CategoryTheory.Adjunction.comp","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[436,4],"def_end_pos":[436,8]}]}]}
{"url":"Mathlib/Probability/CondCount.lean","commit":"","full_name":"ProbabilityTheory.condCount_empty_meas","start":[55,0],"end":[56,83],"file_path":"Mathlib/Probability/CondCount.lean","tactics":[{"state_before":"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\n⊢ condCount ∅ = 0","state_after":"no goals","tactic":"simp [condCount]","premises":[{"full_name":"ProbabilityTheory.condCount","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[52,4],"def_end_pos":[52,13]}]}]}
{"url":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","commit":"","full_name":"BoxIntegral.Box.coe_inf","start":[310,0],"end":[320,12],"file_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\nI J : WithBot (Box ι)\n⊢ ↑(I ⊓ J) = ↑I ∩ ↑J","state_after":"case bot\nι : Type u_1\nI J✝ : Box ι\nx y : ι → ℝ\nJ : WithBot (Box ι)\n⊢ ↑(⊥ ⊓ J) = ↑⊥ ∩ ↑J\n\ncase coe\nι : Type u_1\nI J✝ : Box ι\nx y : ι → ℝ\nJ : WithBot (Box ι)\na✝ : Box ι\n⊢ ↑(↑a✝ ⊓ J) = ↑↑a✝ ∩ ↑J","tactic":"induction I","premises":[]},{"state_before":"case coe\nι : Type u_1\nI J✝ : Box ι\nx y : ι → ℝ\nJ : WithBot (Box ι)\na✝ : Box ι\n⊢ ↑(↑a✝ ⊓ J) = ↑↑a✝ ∩ ↑J","state_after":"case coe.bot\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\na✝ : Box ι\n⊢ ↑(↑a✝ ⊓ ⊥) = ↑↑a✝ ∩ ↑⊥\n\ncase coe.coe\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\na✝¹ a✝ : Box ι\n⊢ ↑(↑a✝¹ ⊓ ↑a✝) = ↑↑a✝¹ ∩ ↑↑a✝","tactic":"induction J","premises":[]},{"state_before":"case coe.coe\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\na✝¹ a✝ : Box ι\n⊢ ↑(↑a✝¹ ⊓ ↑a✝) = ↑↑a✝¹ ∩ ↑↑a✝","state_after":"case coe.coe\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\na✝¹ a✝ : Box ι\n⊢ ↑(mk' (a✝¹.lower ⊔ a✝.lower) (a✝¹.upper ⊓ a✝.upper)) = ↑↑a✝¹ ∩ ↑↑a✝","tactic":"change ((mk' _ _ : WithBot (Box ι)) : Set (ι → ℝ)) = _","premises":[{"full_name":"BoxIntegral.Box","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[73,10],"def_end_pos":[73,13]},{"full_name":"BoxIntegral.Box.mk'","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[281,4],"def_end_pos":[281,7]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"WithBot","def_path":"Mathlib/Order/WithBot.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case coe.coe\nι : Type u_1\nI J : Box ι\nx y : ι → ℝ\na✝¹ a✝ : Box ι\n⊢ ↑(mk' (a✝¹.lower ⊔ a✝.lower) (a✝¹.upper ⊓ a✝.upper)) = ↑↑a✝¹ ∩ ↑↑a✝","state_after":"no goals","tactic":"simp only [coe_eq_pi, ← pi_inter_distrib, Ioc_inter_Ioc, Pi.sup_apply, Pi.inf_apply, coe_mk',\n    coe_coe]","premises":[{"full_name":"BoxIntegral.Box.coe_coe","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[254,8],"def_end_pos":[254,15]},{"full_name":"BoxIntegral.Box.coe_eq_pi","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[117,8],"def_end_pos":[117,17]},{"full_name":"BoxIntegral.Box.coe_mk'","def_path":"Mathlib/Analysis/BoxIntegral/Box/Basic.lean","def_pos":[298,8],"def_end_pos":[298,15]},{"full_name":"Pi.inf_apply","def_path":"Mathlib/Order/Lattice.lean","def_pos":[828,8],"def_end_pos":[828,17]},{"full_name":"Pi.sup_apply","def_path":"Mathlib/Order/Lattice.lean","def_pos":[818,8],"def_end_pos":[818,17]},{"full_name":"Set.Ioc_inter_Ioc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[1497,8],"def_end_pos":[1497,21]},{"full_name":"Set.pi_inter_distrib","def_path":"Mathlib/Data/Set/Prod.lean","def_pos":[626,8],"def_end_pos":[626,24]}]}]}
{"url":"Mathlib/Topology/DenseEmbedding.lean","commit":"","full_name":"DenseInducing.mk'","start":[195,0],"end":[199,11],"file_path":"Mathlib/Topology/DenseEmbedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ni✝ : α → β\ndi : DenseInducing i✝\ninst✝¹ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\ninst✝ : TopologicalSpace γ\ni : α → β\nc : Continuous i\ndense : ∀ (x : β), x ∈ closure (range i)\nH : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ (b : α), i b ∈ t → b ∈ s\na : α\n⊢ comap i (𝓝 (i a)) ≤ 𝓝 a","state_after":"no goals","tactic":"simpa [Filter.le_def] using H a","premises":[{"full_name":"Filter.le_def","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[311,8],"def_end_pos":[311,14]}]}]}
{"url":"Mathlib/CategoryTheory/GradedObject.lean","commit":"","full_name":"CategoryTheory.Iso.hom_inv_id_eval","start":[107,0],"end":[111,44],"file_path":"Mathlib/CategoryTheory/GradedObject.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝² : Category.{u_5, u_1} C\ninst✝¹ : Category.{?u.5761, u_2} D\ninst✝ : Category.{?u.5765, u_3} E\nX Y : GradedObject J C\ne : X ≅ Y\nj : J\n⊢ e.hom j ≫ e.inv j = 𝟙 (X j)","state_after":"no goals","tactic":"rw [← GradedObject.categoryOfGradedObjects_comp, e.hom_inv_id,\n    GradedObject.categoryOfGradedObjects_id]","premises":[{"full_name":"CategoryTheory.GradedObject.categoryOfGradedObjects_comp","def_path":"Mathlib/CategoryTheory/GradedObject.lean","def_pos":[60,2],"def_end_pos":[60,8]},{"full_name":"CategoryTheory.GradedObject.categoryOfGradedObjects_id","def_path":"Mathlib/CategoryTheory/GradedObject.lean","def_pos":[60,2],"def_end_pos":[60,8]},{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]}]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.indexOf_cons_ne","start":[816,0],"end":[818,89],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nx✝ : b ≠ a\nh : b ≠ a := x✝\n⊢ indexOf a (b :: l) = (indexOf a l).succ","state_after":"no goals","tactic":"simp only [indexOf, findIdx_cons, Bool.cond_eq_ite, beq_iff_eq, h, ite_false]","premises":[{"full_name":"Bool.cond_eq_ite","def_path":".lake/packages/lean4/src/lean/Init/Data/Bool.lean","def_pos":[446,8],"def_end_pos":[446,19]},{"full_name":"List.findIdx_cons","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[667,8],"def_end_pos":[667,20]},{"full_name":"List.indexOf","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[94,4],"def_end_pos":[94,11]},{"full_name":"beq_iff_eq","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1326,16],"def_end_pos":[1326,26]},{"full_name":"ite_false","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[92,16],"def_end_pos":[92,25]}]}]}
{"url":"Mathlib/Geometry/RingedSpace/Stalks.lean","commit":"","full_name":"AlgebraicGeometry.PresheafedSpace.stalkMap_germ","start":[44,0],"end":[48,71],"file_path":"Mathlib/Geometry/RingedSpace/Stalks.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα : X ⟶ Y\nU : Opens ↑↑Y\nx : ↥((Opens.map α.base).obj U)\n⊢ Y.presheaf.germ ⟨α.base ↑x, ⋯⟩ ≫ Hom.stalkMap α ↑x = α.c.app (op U) ≫ X.presheaf.germ x","state_after":"no goals","tactic":"rw [Hom.stalkMap, stalkFunctor_map_germ_assoc, stalkPushforward_germ]","premises":[{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.stalkMap","def_path":"Mathlib/Geometry/RingedSpace/Stalks.lean","def_pos":[40,4],"def_end_pos":[40,16]},{"full_name":"TopCat.Presheaf.stalkPushforward_germ","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[144,8],"def_end_pos":[144,29]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","commit":"","full_name":"HurwitzZeta.hurwitzOddFEPair_f","start":[302,0],"end":[330,12],"file_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","tactics":[{"state_before":"a : UnitAddCircle\n⊢ 0 < 3 / 2","state_after":"no goals","tactic":"norm_num","premises":[]},{"state_before":"a : UnitAddCircle\nr : ℝ\n⊢ (fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : oddKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r","tactic":"let ⟨v, hv, hv'⟩ := isBigO_atTop_oddKernel a","premises":[{"full_name":"HurwitzZeta.isBigO_atTop_oddKernel","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[268,6],"def_end_pos":[268,28]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : oddKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r","tactic":"rw [← isBigO_norm_left] at hv' ⊢","premises":[{"full_name":"Asymptotics.isBigO_norm_left","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[675,8],"def_end_pos":[675,24]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r","tactic":"simp_rw [Function.comp_def, sub_zero, norm_real]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.norm_real","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[141,8],"def_end_pos":[141,17]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r","state_after":"no goals","tactic":"exact hv'.trans (isLittleO_exp_neg_mul_rpow_atTop hv _).isBigO","premises":[{"full_name":"Asymptotics.IsBigO.trans","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[431,8],"def_end_pos":[431,20]},{"full_name":"Asymptotics.IsLittleO.isBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[193,8],"def_end_pos":[193,24]},{"full_name":"isLittleO_exp_neg_mul_rpow_atTop","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean","def_pos":[304,8],"def_end_pos":[304,40]}]},{"state_before":"a : UnitAddCircle\nr : ℝ\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : sinKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r","tactic":"let ⟨v, hv, hv'⟩ := isBigO_atTop_sinKernel a","premises":[{"full_name":"HurwitzZeta.isBigO_atTop_sinKernel","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[279,6],"def_end_pos":[279,28]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : sinKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r","tactic":"rw [← isBigO_norm_left] at hv' ⊢","premises":[{"full_name":"Asymptotics.isBigO_norm_left","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[675,8],"def_end_pos":[675,24]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r","state_after":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r","tactic":"simp_rw [Function.comp_def, sub_zero, norm_real]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.norm_real","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[141,8],"def_end_pos":[141,17]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]},{"state_before":"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r","state_after":"no goals","tactic":"exact hv'.trans (isLittleO_exp_neg_mul_rpow_atTop hv _).isBigO","premises":[{"full_name":"Asymptotics.IsBigO.trans","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[431,8],"def_end_pos":[431,20]},{"full_name":"Asymptotics.IsLittleO.isBigO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[193,8],"def_end_pos":[193,24]},{"full_name":"isLittleO_exp_neg_mul_rpow_atTop","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean","def_pos":[304,8],"def_end_pos":[304,40]}]},{"state_before":"a : UnitAddCircle\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ (ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x","state_after":"no goals","tactic":"simp_rw [Function.comp_apply, one_mul, smul_eq_mul, ← ofReal_mul,\n    oddKernel_functional_equation a, one_div x, one_div x⁻¹, inv_rpow (le_of_lt hx), one_div,\n    inv_inv]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.ofReal_mul","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[214,8],"def_end_pos":[214,18]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"HurwitzZeta.oddKernel_functional_equation","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[184,6],"def_end_pos":[184,35]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real.inv_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[430,8],"def_end_pos":[430,16]},{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
{"url":"Mathlib/Data/Rat/Lemmas.lean","commit":"","full_name":"Rat.den_div_intCast_eq_one_iff","start":[214,0],"end":[219,35],"file_path":"Mathlib/Data/Rat/Lemmas.lean","tactics":[{"state_before":"m n : ℤ\nhn : n ≠ 0\n⊢ (↑m / ↑n).den = 1 ↔ n ∣ m","state_after":"m n : ℤ\nhn : ↑n ≠ 0\n⊢ (↑m / ↑n).den = 1 ↔ n ∣ m","tactic":"replace hn : (n : ℚ) ≠ 0 := num_ne_zero.mp hn","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Rat.num_ne_zero","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[96,6],"def_end_pos":[96,17]}]},{"state_before":"m n : ℤ\nhn : ↑n ≠ 0\n⊢ (↑m / ↑n).den = 1 ↔ n ∣ m","state_after":"case mp\nm n : ℤ\nhn : ↑n ≠ 0\n⊢ (↑m / ↑n).den = 1 → n ∣ m\n\ncase mpr\nm n : ℤ\nhn : ↑n ≠ 0\n⊢ n ∣ m → (↑m / ↑n).den = 1","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Types.lean","commit":"","full_name":"CategoryTheory.Limits.Types.pushoutCocone_inl_eq_inr_imp_of_iso","start":[906,0],"end":[911,55],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Types.lean","tactics":[{"state_before":"S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nc c' : PushoutCocone f g\ne : c ≅ c'\nx₁ : X₁\nx₂ : X₂\nh : c.inl x₁ = c.inr x₂\n⊢ c'.inl x₁ = c'.inr x₂","state_after":"case h.e'_2\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nc c' : PushoutCocone f g\ne : c ≅ c'\nx₁ : X₁\nx₂ : X₂\nh : c.inl x₁ = c.inr x₂\n⊢ c'.inl x₁ = e.hom.hom (c.inl x₁)\n\ncase h.e'_3\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nc c' : PushoutCocone f g\ne : c ≅ c'\nx₁ : X₁\nx₂ : X₂\nh : c.inl x₁ = c.inr x₂\n⊢ c'.inr x₂ = e.hom.hom (c.inr x₂)","tactic":"convert congr_arg e.hom.hom h","premises":[{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Limits.CoconeMorphism.hom","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[458,2],"def_end_pos":[458,5]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","commit":"","full_name":"Nat.not_coprime_of_dvd_of_dvd","start":[61,0],"end":[63,45],"file_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","tactics":[{"state_before":"d m n : Nat\ndgt1 : 1 < d\nHm : d ∣ m\nHn : d ∣ n\nco : m.Coprime n\n⊢ d ∣ 1","state_after":"d m n : Nat\ndgt1 : 1 < d\nHm : d ∣ m\nHn : d ∣ n\nco : m.Coprime n\n⊢ d ∣ m.gcd n","tactic":"rw [← co.gcd_eq_one]","premises":[{"full_name":"Nat.Coprime.gcd_eq_one","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[26,8],"def_end_pos":[26,26]}]},{"state_before":"d m n : Nat\ndgt1 : 1 < d\nHm : d ∣ m\nHn : d ∣ n\nco : m.Coprime n\n⊢ d ∣ m.gcd n","state_after":"no goals","tactic":"exact dvd_gcd Hm Hn","premises":[{"full_name":"Nat.dvd_gcd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[94,8],"def_end_pos":[94,15]}]}]}
{"url":"Mathlib/RingTheory/WittVector/WittPolynomial.lean","commit":"","full_name":"wittPolynomial_one","start":[132,0],"end":[135,58],"file_path":"Mathlib/RingTheory/WittVector/WittPolynomial.lean","tactics":[{"state_before":"p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nS : Type u_2\ninst✝ : CommRing S\n⊢ W_ R 1 = C ↑p * X 1 + X 0 ^ p","state_after":"no goals","tactic":"simp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton,\n    one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero]","premises":[{"full_name":"Finset.range_one","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2462,8],"def_end_pos":[2462,17]},{"full_name":"Finset.sum_range_succ_comm","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1291,2],"def_end_pos":[1291,13]},{"full_name":"Finset.sum_singleton","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[353,2],"def_end_pos":[353,13]},{"full_name":"MvPolynomial.C_1","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[197,8],"def_end_pos":[197,11]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]},{"full_name":"tsub_self","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[282,8],"def_end_pos":[282,17]},{"full_name":"tsub_zero","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[384,8],"def_end_pos":[384,17]},{"full_name":"wittPolynomial_eq_sum_C_mul_X_pow","def_path":"Mathlib/RingTheory/WittVector/WittPolynomial.lean","def_pos":[78,8],"def_end_pos":[78,41]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","commit":"","full_name":"CategoryTheory.Limits.coprodComparisonNatTrans_app","start":[1163,0],"end":[1170,52],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","tactics":[{"state_before":"C : Type u\ninst✝⁷ : Category.{v, u} C\nX Y : C\nD : Type u₂\ninst✝⁶ : Category.{w, u₂} D\nF✝ : C ⥤ D\nA✝ A' B B' : C\ninst✝⁵ : HasBinaryCoproduct A✝ B\ninst✝⁴ : HasBinaryCoproduct A' B'\ninst✝³ : HasBinaryCoproduct (F✝.obj A✝) (F✝.obj B)\ninst✝² : HasBinaryCoproduct (F✝.obj A') (F✝.obj B')\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasBinaryCoproducts D\nF : C ⥤ D\nA f : C\n⊢ ∀ ⦃Y : C⦄ (f_1 : f ⟶ Y),\n    (F ⋙ coprod.functor.obj (F.obj A)).map f_1 ≫ (fun B => coprodComparison F A B) Y =\n      (fun B => coprodComparison F A B) f ≫ (coprod.functor.obj A ⋙ F).map f_1","state_after":"no goals","tactic":"simp [coprodComparison_natural]","premises":[{"full_name":"CategoryTheory.Limits.coprodComparison_natural","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[1157,8],"def_end_pos":[1157,32]}]}]}
{"url":"Mathlib/MeasureTheory/Function/AEEqFun.lean","commit":"","full_name":"MeasureTheory.AEEqFun.inf_le_left","start":[517,0],"end":[521,19],"file_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ f ⊓ g ≤ f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ ↑(f ⊓ g) ≤ᶠ[ae μ] ↑f","tactic":"rw [← coeFn_le]","premises":[{"full_name":"MeasureTheory.AEEqFun.coeFn_le","def_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","def_pos":[471,8],"def_end_pos":[471,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ ↑(f ⊓ g) ≤ᶠ[ae μ] ↑f","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝","tactic":"filter_upwards [coeFn_inf f g] with _ ha","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"MeasureTheory.AEEqFun.coeFn_inf","def_path":"Mathlib/MeasureTheory/Function/AEEqFun.lean","def_pos":[514,8],"def_end_pos":[514,17]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝","state_after":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝","tactic":"rw [ha]","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝","state_after":"no goals","tactic":"exact inf_le_left","premises":[{"full_name":"inf_le_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[306,8],"def_end_pos":[306,19]}]}]}
{"url":"Mathlib/LinearAlgebra/FiniteSpan.lean","commit":"","full_name":"LinearEquiv.isOfFinOrder_of_finite_of_span_eq_top_of_mapsTo","start":[19,0],"end":[37,76],"file_path":"Mathlib/LinearAlgebra/FiniteSpan.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : MapsTo (⇑e) Φ Φ\n⊢ IsOfFinOrder e","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\n⊢ IsOfFinOrder e","tactic":"replace he : BijOn e Φ Φ := (hΦ₁.injOn_iff_bijOn_of_mapsTo he).mp e.injective.injOn","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LinearEquiv.injective","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[488,18],"def_end_pos":[488,27]},{"full_name":"Set.BijOn","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[259,4],"def_end_pos":[259,9]},{"full_name":"Set.Finite.injOn_iff_bijOn_of_mapsTo","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[803,8],"def_end_pos":[803,40]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\n⊢ IsOfFinOrder e","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\n⊢ IsOfFinOrder e","tactic":"let e' := he.equiv","premises":[{"full_name":"Set.BijOn.equiv","def_path":"Mathlib/Logic/Equiv/Set.lean","def_pos":[601,18],"def_end_pos":[601,33]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\n⊢ IsOfFinOrder e","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\n⊢ IsOfFinOrder e","tactic":"have : Finite Φ := finite_coe_iff.mpr hΦ₁","premises":[{"full_name":"Finite","def_path":"Mathlib/Data/Finite/Defs.lean","def_pos":[79,16],"def_end_pos":[79,22]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.finite_coe_iff","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[68,8],"def_end_pos":[68,22]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\n⊢ IsOfFinOrder e","state_after":"case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\n⊢ IsOfFinOrder e","tactic":"obtain ⟨k, hk₀, hk⟩ := isOfFinOrder_of_finite e'","premises":[{"full_name":"isOfFinOrder_of_finite","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[704,6],"def_end_pos":[704,28]}]},{"state_before":"case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\n⊢ IsOfFinOrder e","state_after":"case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\n⊢ IsPeriodicPt (fun x => e * x) k 1","tactic":"refine ⟨k, hk₀, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\n⊢ IsPeriodicPt (fun x => e * x) k 1","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\n⊢ ((fun x => e * x)^[k] 1) m = 1 m","tactic":"ext m","premises":[]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\n⊢ ((fun x => e * x)^[k] 1) m = 1 m","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\n⊢ ((fun x => e * x)^[k] 1) m = 1 m","tactic":"have hm : m ∈ span R Φ := hΦ₂ ▸ Submodule.mem_top","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Submodule.mem_top","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[144,8],"def_end_pos":[144,15]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]}]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\n⊢ ((fun x => e * x)^[k] 1) m = 1 m","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\n⊢ (e ^ k) m = m","tactic":"simp only [mul_left_iterate, mul_one, LinearEquiv.coe_one, id_eq]","premises":[{"full_name":"LinearEquiv.coe_one","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[87,6],"def_end_pos":[87,13]},{"full_name":"id_eq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[297,16],"def_end_pos":[297,21]},{"full_name":"mul_left_iterate","def_path":"Mathlib/Algebra/GroupPower/IterateHom.lean","def_pos":[96,8],"def_end_pos":[96,24]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\n⊢ (e ^ k) m = m","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\n⊢ (e ^ k) x = x","tactic":"refine Submodule.span_induction hm (fun x hx ↦ ?_) (by simp)\n    (fun x y hx hy ↦ by simp [map_add, hx, hy]) (fun t x hx ↦ by simp [map_smul, hx])","premises":[{"full_name":"LinearEquiv.map_smul","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[445,8],"def_end_pos":[445,16]},{"full_name":"Submodule.span_induction","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[158,8],"def_end_pos":[158,22]},{"full_name":"map_add","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[280,2],"def_end_pos":[280,13]}]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\n⊢ (e ^ k) x = x","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\n⊢ ↑((MapsTo.restrict (⇑e) Φ Φ ⋯)^[k] ⟨x, hx⟩) = x","tactic":"rw [LinearEquiv.pow_apply, ← he.1.coe_iterate_restrict ⟨x, hx⟩ k]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"LinearEquiv.pow_apply","def_path":"Mathlib/Algebra/Module/Equiv/Basic.lean","def_pos":[99,8],"def_end_pos":[99,17]},{"full_name":"Set.MapsTo.coe_iterate_restrict","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[304,8],"def_end_pos":[304,35]}]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nhk : IsPeriodicPt (fun x => e' * x) k 1\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\n⊢ ↑((MapsTo.restrict (⇑e) Φ Φ ⋯)^[k] ⟨x, hx⟩) = x","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\nhk : e' ^ k = 1\n⊢ ↑((MapsTo.restrict (⇑e) Φ Φ ⋯)^[k] ⟨x, hx⟩) = x","tactic":"replace hk : (e') ^ k = 1 := by simpa [IsPeriodicPt, IsFixedPt] using hk","premises":[{"full_name":"Function.IsFixedPt","def_path":"Mathlib/Dynamics/FixedPoints/Basic.lean","def_pos":[37,4],"def_end_pos":[37,13]},{"full_name":"Function.IsPeriodicPt","def_path":"Mathlib/Dynamics/PeriodicPts.lean","def_pos":[56,4],"def_end_pos":[56,16]}]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\nhk : e' ^ k = 1\n⊢ ↑((MapsTo.restrict (⇑e) Φ Φ ⋯)^[k] ⟨x, hx⟩) = x","state_after":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\nhk : (e' ^ k) ⟨x, hx⟩ = 1 ⟨x, hx⟩\n⊢ ↑((MapsTo.restrict (⇑e) Φ Φ ⋯)^[k] ⟨x, hx⟩) = x","tactic":"replace hk := Equiv.congr_fun hk ⟨x, hx⟩","premises":[{"full_name":"Equiv.congr_fun","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[124,18],"def_end_pos":[124,27]}]},{"state_before":"case intro.intro.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\ne : M ≃ₗ[R] M\nhe : BijOn (⇑e) Φ Φ\ne' : ↑Φ ≃ ↑Φ := BijOn.equiv (⇑e) he\nthis : Finite ↑Φ\nk : ℕ\nhk₀ : k > 0\nm : M\nhm : m ∈ span R Φ\nx : M\nhx : x ∈ Φ\nhk : (e' ^ k) ⟨x, hx⟩ = 1 ⟨x, hx⟩\n⊢ ↑((MapsTo.restrict (⇑e) Φ Φ ⋯)^[k] ⟨x, hx⟩) = x","state_after":"no goals","tactic":"rwa [Equiv.Perm.coe_one, id_eq, Subtype.ext_iff, Equiv.Perm.coe_pow] at hk","premises":[{"full_name":"Equiv.Perm.coe_one","def_path":"Mathlib/GroupTheory/Perm/Basic.lean","def_pos":[88,25],"def_end_pos":[88,32]},{"full_name":"Equiv.Perm.coe_pow","def_path":"Mathlib/GroupTheory/Perm/Basic.lean","def_pos":[92,19],"def_end_pos":[92,26]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]},{"full_name":"id_eq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[297,16],"def_end_pos":[297,21]}]}]}
{"url":"Mathlib/RingTheory/PowerBasis.lean","commit":"","full_name":"PowerBasis.aeval_minpolyGen","start":[165,0],"end":[169,71],"file_path":"Mathlib/RingTheory/PowerBasis.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_4\nB : Type u_5\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type u_6\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\n⊢ (aeval pb.gen) pb.minpolyGen = 0","state_after":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_4\nB : Type u_5\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type u_6\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\n⊢ pb.gen ^ pb.dim - ∑ x : Fin pb.dim, (pb.basis.repr (pb.gen ^ pb.dim)) x • pb.gen ^ ↑x = 0","tactic":"simp_rw [minpolyGen, map_sub, map_sum, map_mul, map_pow, aeval_C, ← Algebra.smul_def, aeval_X]","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Polynomial.aeval_C","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[254,8],"def_end_pos":[254,15]},{"full_name":"Polynomial.aeval_X","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[250,8],"def_end_pos":[250,15]},{"full_name":"PowerBasis.minpolyGen","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[162,18],"def_end_pos":[162,28]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]},{"full_name":"map_pow","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[423,8],"def_end_pos":[423,15]},{"full_name":"map_sub","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[414,2],"def_end_pos":[414,13]},{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]}]},{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_4\nB : Type u_5\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type u_6\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\n⊢ pb.gen ^ pb.dim - ∑ x : Fin pb.dim, (pb.basis.repr (pb.gen ^ pb.dim)) x • pb.gen ^ ↑x = 0","state_after":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_4\nB : Type u_5\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type u_6\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\n⊢ (Finsupp.total (Fin pb.dim) S A ⇑pb.basis) (pb.basis.repr (pb.gen ^ pb.dim)) =\n    ∑ x : Fin pb.dim, (pb.basis.repr (pb.gen ^ pb.dim)) x • pb.gen ^ ↑x","tactic":"refine sub_eq_zero.mpr ((pb.basis.total_repr (pb.gen ^ pb.dim)).symm.trans ?_)","premises":[{"full_name":"Basis.total_repr","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[156,8],"def_end_pos":[156,18]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"PowerBasis.basis","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[63,2],"def_end_pos":[63,7]},{"full_name":"PowerBasis.dim","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[62,2],"def_end_pos":[62,5]},{"full_name":"PowerBasis.gen","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[61,2],"def_end_pos":[61,5]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_4\nB : Type u_5\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type u_6\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\n⊢ (Finsupp.total (Fin pb.dim) S A ⇑pb.basis) (pb.basis.repr (pb.gen ^ pb.dim)) =\n    ∑ x : Fin pb.dim, (pb.basis.repr (pb.gen ^ pb.dim)) x • pb.gen ^ ↑x","state_after":"no goals","tactic":"rw [Finsupp.total_apply, Finsupp.sum_fintype] <;>\n    simp only [pb.coe_basis, zero_smul, eq_self_iff_true, imp_true_iff]","premises":[{"full_name":"Finsupp.sum_fintype","def_path":"Mathlib/Algebra/BigOperators/Finsupp.lean","def_pos":[54,2],"def_end_pos":[54,13]},{"full_name":"Finsupp.total_apply","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[602,8],"def_end_pos":[602,19]},{"full_name":"PowerBasis.coe_basis","def_path":"Mathlib/RingTheory/PowerBasis.lean","def_pos":[73,8],"def_end_pos":[73,17]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]},{"full_name":"imp_true_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1412,8],"def_end_pos":[1412,20]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","commit":"","full_name":"String.revFind_of_valid","start":[309,0],"end":[311,50],"file_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","tactics":[{"state_before":"p : Char → Bool\ns : String\n⊢ s.revFind p = Option.map (fun x => { byteIdx := utf8Len x }) (List.dropWhile (fun x => !p x) s.data.reverse).tail?","state_after":"no goals","tactic":"simpa using revFindAux_of_valid p s.1.reverse []","premises":[{"full_name":"List.nil","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2286,4],"def_end_pos":[2286,7]},{"full_name":"List.reverse","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[441,4],"def_end_pos":[441,11]},{"full_name":"String.data","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2376,2],"def_end_pos":[2376,6]},{"full_name":"String.revFindAux_of_valid","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[296,8],"def_end_pos":[296,27]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.disjoint_cons_right","start":[884,0],"end":[885,73],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α✝ : Type u_1\nl₁ : List α✝\na : α✝\nl₂ : List α✝\n⊢ (a :: l₂).Disjoint l₁ ↔ ¬a ∈ l₁ ∧ l₁.Disjoint l₂","state_after":"α✝ : Type u_1\nl₁ : List α✝\na : α✝\nl₂ : List α✝\n⊢ ¬a ∈ l₁ ∧ l₂.Disjoint l₁ ↔ ¬a ∈ l₁ ∧ l₁.Disjoint l₂","tactic":"rw [disjoint_cons_left]","premises":[{"full_name":"List.disjoint_cons_left","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[881,16],"def_end_pos":[881,34]}]},{"state_before":"α✝ : Type u_1\nl₁ : List α✝\na : α✝\nl₂ : List α✝\n⊢ ¬a ∈ l₁ ∧ l₂.Disjoint l₁ ↔ ¬a ∈ l₁ ∧ l₁.Disjoint l₂","state_after":"no goals","tactic":"simp [disjoint_comm]","premises":[{"full_name":"List.disjoint_comm","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[844,8],"def_end_pos":[844,21]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","commit":"","full_name":"AffineMap.lineMap_symm","start":[537,0],"end":[540,6],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","tactics":[{"state_before":"k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\n⊢ lineMap p₀ p₁ = (lineMap p₁ p₀).comp (lineMap 1 0)","state_after":"k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\n⊢ lineMap p₀ p₁ = lineMap ((lineMap p₁ p₀) 1) ((lineMap p₁ p₀) 0)","tactic":"rw [comp_lineMap]","premises":[{"full_name":"AffineMap.comp_lineMap","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[525,8],"def_end_pos":[525,20]}]},{"state_before":"k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\n⊢ lineMap p₀ p₁ = lineMap ((lineMap p₁ p₀) 1) ((lineMap p₁ p₀) 0)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Order/UpperLower/Basic.lean","commit":"","full_name":"LowerSet.mem_iSup₂_iff","start":[768,0],"end":[769,24],"file_path":"Mathlib/Order/UpperLower/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\nf : (i : ι) → κ i → LowerSet α\n⊢ a ∈ ⨆ i, ⨆ j, f i j ↔ ∃ i j, a ∈ f i j","state_after":"no goals","tactic":"simp_rw [mem_iSup_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LowerSet.mem_iSup_iff","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[758,8],"def_end_pos":[758,20]}]}]}
{"url":"Mathlib/Topology/Instances/ENNReal.lean","commit":"","full_name":"ENNReal.tsum_sub","start":[873,0],"end":[876,72],"file_path":"Mathlib/Topology/Instances/ENNReal.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g✝ : α → ℝ≥0∞\nf g : ℕ → ℝ≥0∞\nh₁ : ∑' (i : ℕ), g i ≠ ⊤\nh₂ : g ≤ f\nthis : ∀ (i : ℕ), f i - g i + g i = f i\n⊢ ∑' (i : ℕ), (f i - g i) + ∑' (i : ℕ), g i = ∑' (i : ℕ), f i","state_after":"no goals","tactic":"simp only [← ENNReal.tsum_add, this]","premises":[{"full_name":"ENNReal.tsum_add","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[742,18],"def_end_pos":[742,26]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean","commit":"","full_name":"Nat.recDiag_zero_succ","start":[74,0],"end":[79,25],"file_path":".lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean","tactics":[{"state_before":"motive : Nat → Nat → Sort u_1\nzero_zero : motive 0 0\nzero_succ : (n : Nat) → motive 0 n → motive 0 (n + 1)\nsucc_zero : (m : Nat) → motive m 0 → motive (m + 1) 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nn : Nat\n⊢ Nat.recDiag zero_zero zero_succ succ_zero succ_succ 0 (n + 1) =\n    zero_succ n (Nat.recDiag zero_zero zero_succ succ_zero succ_succ 0 n)","state_after":"motive : Nat → Nat → Sort u_1\nzero_zero : motive 0 0\nzero_succ : (n : Nat) → motive 0 n → motive 0 (n + 1)\nsucc_zero : (m : Nat) → motive m 0 → motive (m + 1) 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nn : Nat\n⊢ recDiag.left zero_zero zero_succ (n + 1) = zero_succ n (recDiag.left zero_zero zero_succ n)","tactic":"simp [Nat.recDiag]","premises":[{"full_name":"Nat.recDiag","def_path":".lake/packages/batteries/Batteries/Data/Nat/Basic.lean","def_pos":[57,14],"def_end_pos":[57,21]}]},{"state_before":"motive : Nat → Nat → Sort u_1\nzero_zero : motive 0 0\nzero_succ : (n : Nat) → motive 0 n → motive 0 (n + 1)\nsucc_zero : (m : Nat) → motive m 0 → motive (m + 1) 0\nsucc_succ : (m n : Nat) → motive m n → motive (m + 1) (n + 1)\nn : Nat\n⊢ recDiag.left zero_zero zero_succ (n + 1) = zero_succ n (recDiag.left zero_zero zero_succ n)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","commit":"","full_name":"Padic.const_equiv","start":[481,0],"end":[483,13],"file_path":"Mathlib/NumberTheory/Padics/PadicNumbers.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq r : ℚ\nheq : q = r\n⊢ const (padicNorm p) q ≈ const (padicNorm p) r","state_after":"no goals","tactic":"rw [heq]","premises":[]}]}
{"url":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","commit":"","full_name":"contMDiffOn_iff","start":[488,0],"end":[514,16],"file_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx : M\nm n : ℕ∞\n⊢ ContMDiffOn I I' n f s ↔\n    ContinuousOn f s ∧\n      ∀ (x : M) (y : M'),\n        ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)\n          ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))","state_after":"case mp\n𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx : M\nm n : ℕ∞\n⊢ ContMDiffOn I I' n f s →\n    ContinuousOn f s ∧\n      ∀ (x : M) (y : M'),\n        ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)\n          ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))\n\ncase mpr\n𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx : M\nm n : ℕ∞\n⊢ (ContinuousOn f s ∧\n      ∀ (x : M) (y : M'),\n        ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)\n          ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))) →\n    ContMDiffOn I I' n f s","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/NumberTheory/Padics/PadicIntegers.lean","commit":"","full_name":"PadicInt.norm_intCast_eq_padic_norm","start":[271,0],"end":[271,96],"file_path":"Mathlib/NumberTheory/Padics/PadicIntegers.lean","tactics":[{"state_before":"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nz : ℤ\n⊢ ‖↑z‖ = ‖↑z‖","state_after":"no goals","tactic":"simp [norm_def]","premises":[{"full_name":"PadicInt.norm_def","def_path":"Mathlib/NumberTheory/Padics/PadicIntegers.lean","def_pos":[210,8],"def_end_pos":[210,16]}]}]}
{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.sdiff_singleton_eq_erase","start":[1906,0],"end":[1908,52],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b a : α\ns : Finset α\n⊢ s \\ {a} = s.erase a","state_after":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝¹ b a : α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ s \\ {a} ↔ a✝ ∈ s.erase a","tactic":"ext","premises":[]},{"state_before":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝¹ b a : α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ s \\ {a} ↔ a✝ ∈ s.erase a","state_after":"no goals","tactic":"rw [mem_erase, mem_sdiff, mem_singleton, and_comm]","premises":[{"full_name":"Finset.mem_erase","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1582,8],"def_end_pos":[1582,17]},{"full_name":"Finset.mem_sdiff","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1753,8],"def_end_pos":[1753,17]},{"full_name":"Finset.mem_singleton","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[584,8],"def_end_pos":[584,21]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]}]}]}
{"url":"Mathlib/Topology/Order/LawsonTopology.lean","commit":"","full_name":"Topology.IsLawson.isTopologicalBasis","start":[86,0],"end":[101,95],"file_path":"Mathlib/Topology/Order/LawsonTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\n⊢ IsTopologicalBasis (lawsonBasis α)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ IsTopologicalBasis (lawsonBasis α)","tactic":"have lawsonBasis_image2 : lawsonBasis α =\n      (image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1)\n        (IsLower.lowerBasis (WithLower α)) {U | IsOpen[scott α] U}) := by\n    rw [lawsonBasis, image2, IsLower.lowerBasis]\n    simp_rw [diff_eq_compl_inter]\n    aesop","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"IsOpen","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[82,4],"def_end_pos":[82,10]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.diff_eq_compl_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1427,8],"def_end_pos":[1427,27]},{"full_name":"Set.image2","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[275,4],"def_end_pos":[275,10]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"Topology.IsLawson.lawsonBasis","def_path":"Mathlib/Topology/Order/LawsonTopology.lean","def_pos":[83,4],"def_end_pos":[83,15]},{"full_name":"Topology.IsLower.lowerBasis","def_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","def_pos":[186,4],"def_end_pos":[186,14]},{"full_name":"Topology.WithLower","def_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","def_pos":[69,4],"def_end_pos":[69,13]},{"full_name":"Topology.WithLower.toLower","def_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","def_pos":[76,21],"def_end_pos":[76,28]},{"full_name":"Topology.WithScott.toScott","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[331,21],"def_end_pos":[331,28]},{"full_name":"Topology.scott","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[193,4],"def_end_pos":[193,9]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ IsTopologicalBasis (lawsonBasis α)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ IsTopologicalBasis\n    (image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U})","tactic":"rw [lawsonBasis_image2]","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ IsTopologicalBasis\n    (image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U})","state_after":"case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ inst✝¹ =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","tactic":"convert IsTopologicalBasis.inf_induced IsLower.isTopologicalBasis\n    (isTopologicalBasis_opens (α := WithScott α))\n    WithLower.toLower WithScott.toScott","premises":[{"full_name":"TopologicalSpace.IsTopologicalBasis.inf_induced","def_path":"Mathlib/Topology/Bases.lean","def_pos":[258,8],"def_end_pos":[258,38]},{"full_name":"TopologicalSpace.isTopologicalBasis_opens","def_path":"Mathlib/Topology/Bases.lean","def_pos":[236,8],"def_end_pos":[236,32]},{"full_name":"Topology.IsLower.isTopologicalBasis","def_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","def_pos":[244,18],"def_end_pos":[244,36]},{"full_name":"Topology.WithLower.toLower","def_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","def_pos":[76,21],"def_end_pos":[76,28]},{"full_name":"Topology.WithScott","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[326,4],"def_end_pos":[326,13]},{"full_name":"Topology.WithScott.toScott","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[331,21],"def_end_pos":[331,28]}]},{"state_before":"case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ inst✝¹ =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","state_after":"case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ lawson α =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","tactic":"erw [@topology_eq_lawson α _ _ _]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Topology.IsLawson.topology_eq_lawson","def_path":"Mathlib/Topology/Order/LawsonTopology.lean","def_pos":[73,2],"def_end_pos":[73,20]}]},{"state_before":"case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ lawson α =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","state_after":"case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ lower α ⊓ scott α =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","tactic":"rw [lawson]","premises":[{"full_name":"Topology.lawson","def_path":"Mathlib/Topology/Order/LawsonTopology.lean","def_pos":[64,4],"def_end_pos":[64,10]}]},{"state_before":"case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ lower α ⊓ scott α =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ lower α = induced (⇑WithLower.toLower) WithLower.instTopologicalSpace\n\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","tactic":"apply (congrArg₂ Inf.inf _) _","premises":[{"full_name":"Inf.inf","def_path":"Mathlib/Order/Notation.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"congrArg₂","def_path":".lake/packages/batteries/Batteries/Logic.lean","def_pos":[42,8],"def_end_pos":[42,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\nx✝ : TopologicalSpace α := scott α\n⊢ scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","tactic":"letI _ := scott α","premises":[{"full_name":"Topology.scott","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[193,4],"def_end_pos":[193,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\nx✝ : TopologicalSpace α := scott α\n⊢ scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace","state_after":"no goals","tactic":"exact @IsScott.withScottHomeomorph α _ (scott α) ⟨rfl⟩ |>.inducing.induced","premises":[{"full_name":"Homeomorph.inducing","def_path":"Mathlib/Topology/Homeomorph.lean","def_pos":[208,18],"def_end_pos":[208,26]},{"full_name":"Inducing.induced","def_path":"Mathlib/Topology/Defs/Induced.lean","def_pos":[103,2],"def_end_pos":[103,9]},{"full_name":"Topology.IsScott.withScottHomeomorph","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[377,4],"def_end_pos":[377,31]},{"full_name":"Topology.scott","def_path":"Mathlib/Topology/Order/ScottTopology.lean","def_pos":[193,4],"def_end_pos":[193,9]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","commit":"","full_name":"CategoryTheory.ShortComplex.Homotopy.sub_h₂","start":[454,0],"end":[463,68],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ - φ₃).τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁ - (S₁.f ≫ h'.h₁ + h'.h₀ + φ₄.τ₁) =\n    S₁.f ≫ h.h₁ - S₁.f ≫ h'.h₁ + (h.h₀ - h'.h₀) + (φ₂.τ₁ - φ₄.τ₁)","tactic":"rw [sub_τ₁, sub_τ₁, h.comm₁, h'.comm₁, comp_sub]","premises":[{"full_name":"CategoryTheory.Preadditive.comp_sub","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"CategoryTheory.ShortComplex.Homotopy.comm₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[374,2],"def_end_pos":[374,7]},{"full_name":"CategoryTheory.ShortComplex.sub_τ₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[61,14],"def_end_pos":[61,20]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁ - (S₁.f ≫ h'.h₁ + h'.h₀ + φ₄.τ₁) =\n    S₁.f ≫ h.h₁ - S₁.f ≫ h'.h₁ + (h.h₀ - h'.h₀) + (φ₂.τ₁ - φ₄.τ₁)","state_after":"no goals","tactic":"abel","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ - φ₃).τ₂ = S₁.g ≫ (h.h₂ - h'.h₂) + (h.h₁ - h'.h₁) ≫ S₂.f + (φ₂ - φ₄).τ₂","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ - (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) =\n    S₁.g ≫ h.h₂ - S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f - h'.h₁ ≫ S₂.f) + (φ₂.τ₂ - φ₄.τ₂)","tactic":"rw [sub_τ₂, sub_τ₂, h.comm₂, h'.comm₂, comp_sub, sub_comp]","premises":[{"full_name":"CategoryTheory.Preadditive.comp_sub","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[133,8],"def_end_pos":[133,16]},{"full_name":"CategoryTheory.Preadditive.sub_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[128,8],"def_end_pos":[128,16]},{"full_name":"CategoryTheory.ShortComplex.Homotopy.comm₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[375,2],"def_end_pos":[375,7]},{"full_name":"CategoryTheory.ShortComplex.sub_τ₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[62,14],"def_end_pos":[62,20]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ - (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) =\n    S₁.g ≫ h.h₂ - S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f - h'.h₁ ≫ S₂.f) + (φ₂.τ₂ - φ₄.τ₂)","state_after":"no goals","tactic":"abel","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ - φ₃).τ₃ = h.h₃ - h'.h₃ + (h.h₂ - h'.h₂) ≫ S₂.g + (φ₂ - φ₄).τ₃","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ - (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) =\n    h.h₃ - h'.h₃ + (h.h₂ ≫ S₂.g - h'.h₂ ≫ S₂.g) + (φ₂.τ₃ - φ₄.τ₃)","tactic":"rw [sub_τ₃, sub_τ₃, h.comm₃, h'.comm₃, sub_comp]","premises":[{"full_name":"CategoryTheory.Preadditive.sub_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[128,8],"def_end_pos":[128,16]},{"full_name":"CategoryTheory.ShortComplex.Homotopy.comm₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[376,2],"def_end_pos":[376,7]},{"full_name":"CategoryTheory.ShortComplex.sub_τ₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[63,14],"def_end_pos":[63,20]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ - (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) =\n    h.h₃ - h'.h₃ + (h.h₂ ≫ S₂.g - h'.h₂ ≫ S₂.g) + (φ₂.τ₃ - φ₄.τ₃)","state_after":"no goals","tactic":"abel","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","commit":"","full_name":"map_one_ne_zero","start":[59,0],"end":[66,45],"file_path":"Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\n⊢ f 1 ≠ 0","state_after":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nh1 : f 1 = 0\n⊢ False","tactic":"intro h1","premises":[]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nh1 : f 1 = 0\n⊢ False","state_after":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y\n⊢ False","tactic":"specialize f_mul 1","premises":[]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y\n⊢ False","state_after":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f y ≤ 0\n⊢ False","tactic":"simp_rw [h1, one_mul, mul_zero, zero_mul] at f_mul","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f y ≤ 0\n⊢ False","state_after":"case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f y ≤ 0\nz : R\nhz : f z ≠ 0 z\n⊢ False","tactic":"obtain ⟨z, hz⟩ := Function.ne_iff.mp f_ne_zero","premises":[{"full_name":"Function.ne_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[57,8],"def_end_pos":[57,14]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f y ≤ 0\nz : R\nhz : f z ≠ 0 z\n⊢ False","state_after":"no goals","tactic":"exact hz <| (f_mul z).antisymm (f_nonneg z)","premises":[]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"mem_extChartAt_source","start":[1048,0],"end":[1049,49],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx : M\n⊢ x ∈ (extChartAt I x).source","state_after":"no goals","tactic":"simp only [extChartAt_source, mem_chart_source]","premises":[{"full_name":"extChartAt_source","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[1042,8],"def_end_pos":[1042,25]},{"full_name":"mem_chart_source","def_path":"Mathlib/Geometry/Manifold/ChartedSpace.lean","def_pos":[568,6],"def_end_pos":[568,22]}]}]}
{"url":"Mathlib/Algebra/Polynomial/RingDivision.lean","commit":"","full_name":"Polynomial.add_modByMonic","start":[53,0],"end":[64,45],"file_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Semiring S\np₁ p₂ : R[X]\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q","state_after":"case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Semiring S\np₁ p₂ : R[X]\nhq : q.Monic\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q\n\ncase neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Semiring S\np₁ p₂ : R[X]\nhq : ¬q.Monic\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q","tactic":"by_cases hq : q.Monic","premises":[{"full_name":"Polynomial.Monic","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[69,4],"def_end_pos":[69,9]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Nat/Factorization/PrimePow.lean","commit":"","full_name":"Nat.Coprime.isPrimePow_dvd_mul","start":[108,0],"end":[129,43],"file_path":"Mathlib/Data/Nat/Factorization/PrimePow.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b","state_after":"case inl\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n b : ℕ\nhn : IsPrimePow n\nhab : Coprime 0 b\n⊢ n ∣ 0 * b ↔ n ∣ 0 ∨ n ∣ b\n\ncase inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b","tactic":"rcases eq_or_ne a 0 with (rfl | ha)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b","state_after":"case inr.inl\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a : ℕ\nhn : IsPrimePow n\nha : a ≠ 0\nhab : a.Coprime 0\n⊢ n ∣ a * 0 ↔ n ∣ a ∨ n ∣ 0\n\ncase inr.inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b","tactic":"rcases eq_or_ne b 0 with (rfl | hb)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b","state_after":"case inr.inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b → n ∣ a ∨ n ∣ b","tactic":"refine\n    ⟨?_, fun h =>\n      Or.elim h (fun i => i.trans ((@dvd_mul_right a b a hab).mpr (dvd_refl a)))\n          fun i => i.trans ((@dvd_mul_left a b b hab.symm).mpr (dvd_refl b))⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.Coprime.dvd_mul_left","def_path":"Mathlib/Data/Nat/GCD/Basic.lean","def_pos":[142,8],"def_end_pos":[142,28]},{"full_name":"Nat.Coprime.dvd_mul_right","def_path":"Mathlib/Data/Nat/GCD/Basic.lean","def_pos":[139,8],"def_end_pos":[139,29]},{"full_name":"Nat.Coprime.symm","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[28,8],"def_end_pos":[28,20]},{"full_name":"Or.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[550,8],"def_end_pos":[550,15]},{"full_name":"dvd_refl","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[122,8],"def_end_pos":[122,16]}]},{"state_before":"case inr.inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b → n ∣ a ∨ n ∣ b","state_after":"case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b","tactic":"obtain ⟨p, k, hp, _, rfl⟩ := (isPrimePow_nat_iff _).1 hn","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"isPrimePow_nat_iff","def_path":"Mathlib/Algebra/IsPrimePow.lean","def_pos":[63,8],"def_end_pos":[63,26]}]},{"state_before":"case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b","state_after":"case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p","tactic":"simp only [hp.pow_dvd_iff_le_factorization (mul_ne_zero ha hb), Nat.factorization_mul ha hb,\n    hp.pow_dvd_iff_le_factorization ha, hp.pow_dvd_iff_le_factorization hb, Pi.add_apply,\n    Finsupp.coe_add]","premises":[{"full_name":"Finsupp.coe_add","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[888,25],"def_end_pos":[888,32]},{"full_name":"Nat.Prime.pow_dvd_iff_le_factorization","def_path":"Mathlib/Data/Nat/Factorization/Basic.lean","def_pos":[166,8],"def_end_pos":[166,42]},{"full_name":"Nat.factorization_mul","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[144,8],"def_end_pos":[144,25]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"mul_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[80,8],"def_end_pos":[80,19]}]},{"state_before":"case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p","state_after":"case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nthis : a.factorization p = 0 ∨ b.factorization p = 0\n⊢ k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p","tactic":"have : a.factorization p = 0 ∨ b.factorization p = 0 := by\n    rw [← Finsupp.not_mem_support_iff, ← Finsupp.not_mem_support_iff, ← not_and_or, ←\n      Finset.mem_inter]\n    intro t -- Porting note: used to be `exact` below, but the definition of `∈` has changed.\n    simpa using hab.disjoint_primeFactors.le_bot t","premises":[{"full_name":"Disjoint.le_bot","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[119,8],"def_end_pos":[119,23]},{"full_name":"Finset.mem_inter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1353,8],"def_end_pos":[1353,17]},{"full_name":"Finsupp.not_mem_support_iff","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[157,8],"def_end_pos":[157,27]},{"full_name":"Nat.Coprime.disjoint_primeFactors","def_path":"Mathlib/Data/Nat/PrimeFin.lean","def_pos":[102,16],"def_end_pos":[102,45]},{"full_name":"Nat.factorization","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[46,4],"def_end_pos":[46,17]},{"full_name":"Or","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[532,10],"def_end_pos":[532,12]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]}]},{"state_before":"case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nthis : a.factorization p = 0 ∨ b.factorization p = 0\n⊢ k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p","state_after":"no goals","tactic":"cases' this with h h <;> simp [h, imp_or]","premises":[{"full_name":"imp_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[311,8],"def_end_pos":[311,14]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","commit":"","full_name":"Complex.Gamma_ne_zero_of_re_pos","start":[455,0],"end":[459,90],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","tactics":[{"state_before":"s : ℂ\nhs : 0 < s.re\n⊢ Gamma s ≠ 0","state_after":"s : ℂ\nhs : 0 < s.re\nm : ℕ\n⊢ s ≠ -↑m","tactic":"refine Gamma_ne_zero fun m => ?_","premises":[{"full_name":"Complex.Gamma_ne_zero","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean","def_pos":[430,8],"def_end_pos":[430,21]}]},{"state_before":"s : ℂ\nhs : 0 < s.re\nm : ℕ\n⊢ s ≠ -↑m","state_after":"s : ℂ\nm : ℕ\nhs : s = -↑m\n⊢ s.re ≤ 0","tactic":"contrapose! hs","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"s : ℂ\nm : ℕ\nhs : s = -↑m\n⊢ s.re ≤ 0","state_after":"no goals","tactic":"simpa only [hs, neg_re, ← ofReal_natCast, ofReal_re, neg_nonpos] using Nat.cast_nonneg _","premises":[{"full_name":"Complex.neg_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[188,8],"def_end_pos":[188,14]},{"full_name":"Complex.ofReal_natCast","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[424,25],"def_end_pos":[424,39]},{"full_name":"Complex.ofReal_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]}]}]}
{"url":"Mathlib/Order/WellFoundedSet.lean","commit":"","full_name":"Set.PartiallyWellOrderedOn.iff_forall_not_isBadSeq","start":[643,0],"end":[645,43],"file_path":"Mathlib/Order/WellFoundedSet.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nr✝ r : α → α → Prop\ns : Set α\nf : ℕ → α\n⊢ ((∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ r (f m) (f n)) ↔ ¬IsBadSeq r s f","state_after":"no goals","tactic":"simp [IsBadSeq]","premises":[{"full_name":"Set.PartiallyWellOrderedOn.IsBadSeq","def_path":"Mathlib/Order/WellFoundedSet.lean","def_pos":[640,4],"def_end_pos":[640,12]}]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"PartialHomeomorph.contDiffWithinAt_extend_coord_change'","start":[1017,0],"end":[1022,38],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : M\nhxf : x ∈ f.source\nhxf' : x ∈ f'.source\n⊢ ContDiffWithinAt 𝕜 ⊤ (↑(f.extend I) ∘ ↑(f'.extend I).symm) (range ↑I) (↑(f'.extend I) x)","state_after":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : M\nhxf : x ∈ f.source\nhxf' : x ∈ f'.source\n⊢ ↑(f'.extend I) x ∈ ((f'.extend I).symm ≫ f.extend I).source","tactic":"refine contDiffWithinAt_extend_coord_change I hf hf' ?_","premises":[{"full_name":"PartialHomeomorph.contDiffWithinAt_extend_coord_change","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[1009,8],"def_end_pos":[1009,44]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : M\nhxf : x ∈ f.source\nhxf' : x ∈ f'.source\n⊢ ↑(f'.extend I) x ∈ ((f'.extend I).symm ≫ f.extend I).source","state_after":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : M\nhxf : x ∈ f.source\nhxf' : x ∈ f'.source\n⊢ ↑(f'.extend I) x ∈ ↑(f'.extend I) '' (f'.source ∩ f.source)","tactic":"rw [← extend_image_source_inter]","premises":[{"full_name":"PartialHomeomorph.extend_image_source_inter","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[979,8],"def_end_pos":[979,33]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : M\nhxf : x ∈ f.source\nhxf' : x ∈ f'.source\n⊢ ↑(f'.extend I) x ∈ ↑(f'.extend I) '' (f'.source ∩ f.source)","state_after":"no goals","tactic":"exact mem_image_of_mem _ ⟨hxf', hxf⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]}]}]}
{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Int.fract_neg_eq_zero","start":[932,0],"end":[935,59],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na x : α\n⊢ fract (-x) = 0 ↔ fract x = 0","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na x : α\n⊢ (∃ z, -x = ↑z) ↔ ∃ z, x = ↑z","tactic":"simp only [fract_eq_iff, le_refl, zero_lt_one, tsub_zero, true_and_iff]","premises":[{"full_name":"Int.fract_eq_iff","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[890,8],"def_end_pos":[890,20]},{"full_name":"le_refl","def_path":"Mathlib/Order/Defs.lean","def_pos":[39,8],"def_end_pos":[39,15]},{"full_name":"true_and_iff","def_path":"Mathlib/Init/Logic.lean","def_pos":[94,8],"def_end_pos":[94,20]},{"full_name":"tsub_zero","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[384,8],"def_end_pos":[384,17]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na x : α\n⊢ (∃ z, -x = ↑z) ↔ ∃ z, x = ↑z","state_after":"no goals","tactic":"constructor <;> rintro ⟨z, hz⟩ <;> use -z <;> simp [← hz]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]}]}
{"url":"Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean","commit":"","full_name":"NonUnitalSubalgebra.iSupLift_mk","start":[981,0],"end":[986,25],"file_path":"Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean","tactics":[{"state_before":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ T\n⊢ (iSupLift K dir f hf T hT) ⟨↑x, hx⟩ = (f i) x","state_after":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ (iSupLift K dir f hf (iSup K) ⋯) ⟨↑x, hx⟩ = (f i) x","tactic":"subst hT","premises":[]},{"state_before":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ (iSupLift K dir f hf (iSup K) ⋯) ⟨↑x, hx⟩ = (f i) x","state_after":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ ⟨↑x, hx⟩ = (f i) x","tactic":"dsimp [iSupLift]","premises":[{"full_name":"NonUnitalSubalgebra.iSupLift","def_path":"Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean","def_pos":[933,18],"def_end_pos":[933,26]}]},{"state_before":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ ⟨↑x, hx⟩ = (f i) x","state_after":"no goals","tactic":"apply Set.iUnionLift_mk","premises":[{"full_name":"Set.iUnionLift_mk","def_path":"Mathlib/Data/Set/UnionLift.lean","def_pos":[62,8],"def_end_pos":[62,21]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Preserves/Limits.lean","commit":"","full_name":"CategoryTheory.preservesColimitNatIso_inv_app","start":[141,0],"end":[154,23],"file_path":"Mathlib/CategoryTheory/Limits/Preserves/Limits.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\n⊢ ∀ {X Y : J ⥤ C} (f : X ⟶ Y),\n    (colim ⋙ G).map f ≫ ((fun F => preservesColimitIso G F) Y).hom =\n      ((fun F => preservesColimitIso G F) X).hom ≫ ((whiskeringRight J C D).obj G ⋙ colim).map f","state_after":"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\n⊢ (colim ⋙ G).map f ≫ ((fun F => preservesColimitIso G F) Y✝).hom =\n    ((fun F => preservesColimitIso G F) X✝).hom ≫ ((whiskeringRight J C D).obj G ⋙ colim).map f","tactic":"intro _ _ f","premises":[]},{"state_before":"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\n⊢ (colim ⋙ G).map f ≫ ((fun F => preservesColimitIso G F) Y✝).hom =\n    ((fun F => preservesColimitIso G F) X✝).hom ≫ ((whiskeringRight J C D).obj G ⋙ colim).map f","state_after":"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\n⊢ ((fun F => preservesColimitIso G F) X✝).inv ≫ (colim ⋙ G).map f =\n    ((whiskeringRight J C D).obj G ⋙ colim).map f ≫ ((fun F => preservesColimitIso G F) Y✝).inv","tactic":"rw [← Iso.inv_comp_eq, ← Category.assoc, ← Iso.eq_comp_inv]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Iso.eq_comp_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[187,8],"def_end_pos":[187,19]},{"full_name":"CategoryTheory.Iso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[178,8],"def_end_pos":[178,19]}]},{"state_before":"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\n⊢ ((fun F => preservesColimitIso G F) X✝).inv ≫ (colim ⋙ G).map f =\n    ((whiskeringRight J C D).obj G ⋙ colim).map f ≫ ((fun F => preservesColimitIso G F) Y✝).inv","state_after":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\n⊢ ∀ (j : J),\n    colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n        ((fun F => preservesColimitIso G F) X✝).inv ≫ (colim ⋙ G).map f =\n      colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n        ((whiskeringRight J C D).obj G ⋙ colim).map f ≫ ((fun F => preservesColimitIso G F) Y✝).inv","tactic":"apply colimit.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.colimit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[757,8],"def_end_pos":[757,23]}]},{"state_before":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\n⊢ ∀ (j : J),\n    colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n        ((fun F => preservesColimitIso G F) X✝).inv ≫ (colim ⋙ G).map f =\n      colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n        ((whiskeringRight J C D).obj G ⋙ colim).map f ≫ ((fun F => preservesColimitIso G F) Y✝).inv","state_after":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n      ((fun F => preservesColimitIso G F) X✝).inv ≫ (colim ⋙ G).map f =\n    colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n      ((whiskeringRight J C D).obj G ⋙ colim).map f ≫ ((fun F => preservesColimitIso G F) Y✝).inv","tactic":"intro j","premises":[]},{"state_before":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n      ((fun F => preservesColimitIso G F) X✝).inv ≫ (colim ⋙ G).map f =\n    colimit.ι (((whiskeringRight J C D).obj G).obj X✝) j ≫\n      ((whiskeringRight J C D).obj G ⋙ colim).map f ≫ ((fun F => preservesColimitIso G F) Y✝).inv","state_after":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ colimit.ι (X✝ ⋙ G) j ≫ (preservesColimitIso G X✝).inv ≫ G.map (colimMap f) =\n    colimit.ι (X✝ ⋙ G) j ≫ colimMap (whiskerRight f G) ≫ (preservesColimitIso G Y✝).inv","tactic":"dsimp","premises":[]},{"state_before":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ colimit.ι (X✝ ⋙ G) j ≫ (preservesColimitIso G X✝).inv ≫ G.map (colimMap f) =\n    colimit.ι (X✝ ⋙ G) j ≫ colimMap (whiskerRight f G) ≫ (preservesColimitIso G Y✝).inv","state_after":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ colimit.ι (X✝ ⋙ G) j ≫ (preservesColimitIso G X✝).inv ≫ G.map (colimMap f) =\n    (whiskerRight f G).app j ≫ colimit.ι (Y✝ ⋙ G) j ≫ (preservesColimitIso G Y✝).inv","tactic":"erw [ι_colimMap_assoc]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ colimit.ι (X✝ ⋙ G) j ≫ (preservesColimitIso G X✝).inv ≫ G.map (colimMap f) =\n    (whiskerRight f G).app j ≫ colimit.ι (Y✝ ⋙ G) j ≫ (preservesColimitIso G Y✝).inv","state_after":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ G.map (colimit.ι X✝ j ≫ colimMap f) = G.map (f.app j ≫ colimit.ι Y✝ j)","tactic":"simp only [ι_preservesColimitsIso_inv, whiskerRight_app, Category.assoc,\n        ι_preservesColimitsIso_inv_assoc, ← G.map_comp]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.whiskerRight_app","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[53,2],"def_end_pos":[53,7]},{"full_name":"CategoryTheory.ι_preservesColimitsIso_inv","def_path":"Mathlib/CategoryTheory/Limits/Preserves/Limits.lean","def_pos":[120,8],"def_end_pos":[120,34]}]},{"state_before":"case w\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝⁵ : Category.{w', w} J\nF : J ⥤ C\ninst✝⁴ : PreservesColimit F G\ninst✝³ : HasColimit F\ninst✝² : PreservesColimitsOfShape J G\ninst✝¹ : HasColimitsOfShape J D\ninst✝ : HasColimitsOfShape J C\nX✝ Y✝ : J ⥤ C\nf : X✝ ⟶ Y✝\nj : J\n⊢ G.map (colimit.ι X✝ j ≫ colimMap f) = G.map (f.app j ≫ colimit.ι Y✝ j)","state_after":"no goals","tactic":"erw [ι_colimMap]","premises":[{"full_name":"CategoryTheory.Limits.ι_colimMap","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[705,8],"def_end_pos":[705,18]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
{"url":"Mathlib/SetTheory/Game/Basic.lean","commit":"","full_name":"SetTheory.Game.not_lf","start":[112,0],"end":[116,20],"file_path":"Mathlib/SetTheory/Game/Basic.lean","tactics":[{"state_before":"⊢ ∀ {x y : Game}, ¬x ⧏ y ↔ y ≤ x","state_after":"case mk.mk\nx✝ : Game\nx : PGame\ny✝ : Game\ny : PGame\n⊢ ¬Quot.mk Setoid.r x ⧏ Quot.mk Setoid.r y ↔ Quot.mk Setoid.r y ≤ Quot.mk Setoid.r x","tactic":"rintro ⟨x⟩ ⟨y⟩","premises":[]},{"state_before":"case mk.mk\nx✝ : Game\nx : PGame\ny✝ : Game\ny : PGame\n⊢ ¬Quot.mk Setoid.r x ⧏ Quot.mk Setoid.r y ↔ Quot.mk Setoid.r y ≤ Quot.mk Setoid.r x","state_after":"no goals","tactic":"exact PGame.not_lf","premises":[{"full_name":"SetTheory.PGame.not_lf","def_path":"Mathlib/SetTheory/Game/PGame.lean","def_pos":[368,8],"def_end_pos":[368,14]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","commit":"","full_name":"EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two","start":[323,0],"end":[329,74],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\n⊢ dist p1 p3 * dist p1 p3 = dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 ↔ ∠ p1 p2 p3 = π / 2","state_after":"no goals","tactic":"erw [dist_comm p3 p2, dist_eq_norm_vsub V p1 p3, dist_eq_norm_vsub V p1 p2,\n    dist_eq_norm_vsub V p2 p3, ← norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two,\n    vsub_sub_vsub_cancel_right p1, ← neg_vsub_eq_vsub_rev p2 p3, norm_neg]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[52,8],"def_end_pos":[52,66]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"dist_comm","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[166,8],"def_end_pos":[166,17]},{"full_name":"dist_eq_norm_vsub","def_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","def_pos":[68,8],"def_end_pos":[68,25]},{"full_name":"neg_vsub_eq_vsub_rev","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[138,8],"def_end_pos":[138,28]},{"full_name":"norm_neg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[375,29],"def_end_pos":[375,37]},{"full_name":"vsub_sub_vsub_cancel_right","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[153,8],"def_end_pos":[153,34]}]}]}
{"url":"Mathlib/Algebra/Vertex/HVertexOperator.lean","commit":"","full_name":"HVertexOperator.of_coeff_apply","start":[81,0],"end":[88,31],"file_path":"Mathlib/Algebra/Vertex/HVertexOperator.lean","tactics":[{"state_before":"Γ : Type u_1\ninst✝⁵ : PartialOrder Γ\nR : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nf : Γ → V →ₗ[R] W\nhf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO\nx✝¹ x✝ : V\n⊢ (fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) (x✝¹ + x✝) =\n    (fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) x✝¹ +\n      (fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) x✝","state_after":"case h.h\nΓ : Type u_1\ninst✝⁵ : PartialOrder Γ\nR : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nf : Γ → V →ₗ[R] W\nhf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO\nx✝² x✝¹ : V\nx✝ : Γ\n⊢ ((of R).symm ((fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) (x✝² + x✝¹))).coeff x✝ =\n    ((of R).symm\n          ((fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) x✝² +\n            (fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) x✝¹)).coeff\n      x✝","tactic":"ext","premises":[]},{"state_before":"case h.h\nΓ : Type u_1\ninst✝⁵ : PartialOrder Γ\nR : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nf : Γ → V →ₗ[R] W\nhf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO\nx✝² x✝¹ : V\nx✝ : Γ\n⊢ ((of R).symm ((fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) (x✝² + x✝¹))).coeff x✝ =\n    ((of R).symm\n          ((fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) x✝² +\n            (fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }) x✝¹)).coeff\n      x✝","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"Γ : Type u_1\ninst✝⁵ : PartialOrder Γ\nR : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nf : Γ → V →ₗ[R] W\nhf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO\nx✝¹ : R\nx✝ : V\n⊢ { toFun := fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯ }.toFun (x✝¹ • x✝) =\n    (RingHom.id R) x✝¹ •\n      { toFun := fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯ }.toFun x✝","state_after":"case h.h\nΓ : Type u_1\ninst✝⁵ : PartialOrder Γ\nR : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nf : Γ → V →ₗ[R] W\nhf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO\nx✝² : R\nx✝¹ : V\nx✝ : Γ\n⊢ ((of R).symm\n          ({ toFun := fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯ }.toFun\n            (x✝² • x✝¹))).coeff\n      x✝ =\n    ((of R).symm\n          ((RingHom.id R) x✝² •\n            { toFun := fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯ }.toFun\n              x✝¹)).coeff\n      x✝","tactic":"ext","premises":[]},{"state_before":"case h.h\nΓ : Type u_1\ninst✝⁵ : PartialOrder Γ\nR : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nf : Γ → V →ₗ[R] W\nhf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO\nx✝² : R\nx✝¹ : V\nx✝ : Γ\n⊢ ((of R).symm\n          ({ toFun := fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯ }.toFun\n            (x✝² • x✝¹))).coeff\n      x✝ =\n    ((of R).symm\n          ((RingHom.id R) x✝² •\n            { toFun := fun x => (of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }, map_add' := ⋯ }.toFun\n              x✝¹)).coeff\n      x✝","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","commit":"","full_name":"Left.min_le_max_of_add_le_add","start":[287,0],"end":[290,94],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b c d : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nh : a * b ≤ c * d\n⊢ min a b ≤ max c d","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b c d : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nh : a * b ≤ c * d\n⊢ (a ≤ c ∨ a ≤ d) ∨ b ≤ c ∨ b ≤ d","tactic":"simp_rw [min_le_iff, le_max_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"le_max_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[35,8],"def_end_pos":[35,18]},{"full_name":"min_le_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[39,8],"def_end_pos":[39,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b c d : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nh : a * b ≤ c * d\n⊢ (a ≤ c ∨ a ≤ d) ∨ b ≤ c ∨ b ≤ d","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b c d : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nh : (c < a ∧ d < a) ∧ c < b ∧ d < b\n⊢ c * d < a * b","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b c d : α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nh : (c < a ∧ d < a) ∧ c < b ∧ d < b\n⊢ c * d < a * b","state_after":"no goals","tactic":"exact mul_lt_mul_of_le_of_lt h.1.1.le h.2.2","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"mul_lt_mul_of_le_of_lt","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[156,8],"def_end_pos":[156,30]}]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.length_erase_add_one","start":[2334,0],"end":[2336,68],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ (l.erase a).length + 1 = l.length","state_after":"no goals","tactic":"rw [erase_eq_eraseP, length_eraseP_add_one h (decide_eq_true rfl)]","premises":[{"full_name":"List.erase_eq_eraseP","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[592,8],"def_end_pos":[592,23]},{"full_name":"List.length_eraseP_add_one","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2320,8],"def_end_pos":[2320,29]},{"full_name":"decide_eq_true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[876,8],"def_end_pos":[876,22]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/RepresentationTheory/Basic.lean","commit":"","full_name":"Representation.smul_one_tprod_asModule","start":[394,0],"end":[402,74],"file_path":"Mathlib/RepresentationTheory/Basic.lean","tactics":[{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\ninst✝¹ : AddCommMonoid W\ninst✝ : Module k W\nρV : Representation k G V\nρW : Representation k G W\nr : MonoidAlgebra k G\nx : V\ny : W\n⊢ let y' := y;\n  let z := x ⊗ₜ[k] y;\n  r • z = x ⊗ₜ[k] (r • y')","state_after":"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\ninst✝¹ : AddCommMonoid W\ninst✝ : Module k W\nρV : Representation k G V\nρW : Representation k G W\nr : MonoidAlgebra k G\nx : V\ny : W\n⊢ ((1 ⊗ ρW).asAlgebraHom r) (x ⊗ₜ[k] y) = x ⊗ₜ[k] (ρW.asAlgebraHom r) y","tactic":"show asAlgebraHom (1 ⊗ ρW) _ _ = _ ⊗ₜ asAlgebraHom ρW _ _","premises":[{"full_name":"Representation.asAlgebraHom","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[91,18],"def_end_pos":[91,30]},{"full_name":"Representation.tprod","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[372,18],"def_end_pos":[372,23]},{"full_name":"TensorProduct.tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[122,4],"def_end_pos":[122,8]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\ninst✝¹ : AddCommMonoid W\ninst✝ : Module k W\nρV : Representation k G V\nρW : Representation k G W\nr : MonoidAlgebra k G\nx : V\ny : W\n⊢ ((1 ⊗ ρW).asAlgebraHom r) (x ⊗ₜ[k] y) = x ⊗ₜ[k] (ρW.asAlgebraHom r) y","state_after":"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\ninst✝¹ : AddCommMonoid W\ninst✝ : Module k W\nρV : Representation k G V\nρW : Representation k G W\nr : MonoidAlgebra k G\nx : V\ny : W\n⊢ (Finsupp.sum r fun i d => d • x ⊗ₜ[k] (ρW i) y) = x ⊗ₜ[k] Finsupp.sum r fun i d => d • (ρW i) y","tactic":"simp only [asAlgebraHom_def, MonoidAlgebra.lift_apply, tprod_apply, MonoidHom.one_apply,\n    LinearMap.finsupp_sum_apply, LinearMap.smul_apply, TensorProduct.map_tmul, LinearMap.one_apply]","premises":[{"full_name":"LinearMap.finsupp_sum_apply","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[1297,8],"def_end_pos":[1297,25]},{"full_name":"LinearMap.one_apply","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[54,8],"def_end_pos":[54,17]},{"full_name":"LinearMap.smul_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[748,8],"def_end_pos":[748,18]},{"full_name":"MonoidAlgebra.lift_apply","def_path":"Mathlib/Algebra/MonoidAlgebra/Basic.lean","def_pos":[829,8],"def_end_pos":[829,18]},{"full_name":"MonoidHom.one_apply","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[911,8],"def_end_pos":[911,27]},{"full_name":"Representation.asAlgebraHom_def","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[94,8],"def_end_pos":[94,24]},{"full_name":"Representation.tprod_apply","def_path":"Mathlib/RepresentationTheory/Basic.lean","def_pos":[380,8],"def_end_pos":[380,19]},{"full_name":"TensorProduct.map_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[733,8],"def_end_pos":[733,16]}]},{"state_before":"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\ninst✝¹ : AddCommMonoid W\ninst✝ : Module k W\nρV : Representation k G V\nρW : Representation k G W\nr : MonoidAlgebra k G\nx : V\ny : W\n⊢ (Finsupp.sum r fun i d => d • x ⊗ₜ[k] (ρW i) y) = x ⊗ₜ[k] Finsupp.sum r fun i d => d • (ρW i) y","state_after":"no goals","tactic":"simp only [Finsupp.sum, TensorProduct.tmul_sum, TensorProduct.tmul_smul]","premises":[{"full_name":"Finsupp.sum","def_path":"Mathlib/Algebra/BigOperators/Finsupp.lean","def_pos":[42,2],"def_end_pos":[42,13]},{"full_name":"TensorProduct.tmul_smul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[351,8],"def_end_pos":[351,17]},{"full_name":"TensorProduct.tmul_sum","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[439,8],"def_end_pos":[439,16]}]}]}
{"url":"Mathlib/RingTheory/LocalRing/RingHom/Basic.lean","commit":"","full_name":"RingHom.domain_localRing","start":[60,0],"end":[67,30],"file_path":"Mathlib/RingTheory/LocalRing/RingHom/Basic.lean","tactics":[{"state_before":"R✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\n⊢ LocalRing R","state_after":"R✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\n⊢ LocalRing R","tactic":"haveI : Nontrivial R := pullback_nonzero f f.map_zero f.map_one","premises":[{"full_name":"Nontrivial","def_path":"Mathlib/Logic/Nontrivial/Defs.lean","def_pos":[29,6],"def_end_pos":[29,16]},{"full_name":"RingHom.map_one","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[476,18],"def_end_pos":[476,25]},{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]},{"full_name":"pullback_nonzero","def_path":"Mathlib/Algebra/GroupWithZero/NeZero.lean","def_pos":[34,8],"def_end_pos":[34,24]}]},{"state_before":"R✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\n⊢ LocalRing R","state_after":"case h\nR✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\n⊢ ∀ (a b : R), a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R","tactic":"apply LocalRing.of_nonunits_add","premises":[{"full_name":"LocalRing.of_nonunits_add","def_path":"Mathlib/RingTheory/LocalRing/Basic.lean","def_pos":[30,8],"def_end_pos":[30,23]}]},{"state_before":"case h\nR✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\n⊢ ∀ (a b : R), a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R","state_after":"case h\nR✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\na b : R\n⊢ a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R","tactic":"intro a b","premises":[]},{"state_before":"case h\nR✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\na b : R\n⊢ a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R","state_after":"case h\nR✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\na b : R\n⊢ f a ∈ nonunits S → f b ∈ nonunits S → f a + f b ∈ nonunits S","tactic":"simp_rw [← map_mem_nonunits_iff f, f.map_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"RingHom.map_add","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[480,18],"def_end_pos":[480,25]},{"full_name":"map_mem_nonunits_iff","def_path":"Mathlib/RingTheory/LocalRing/RingHom/Basic.lean","def_pos":[33,8],"def_end_pos":[33,28]}]},{"state_before":"case h\nR✝ : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁵ : Semiring R✝\ninst✝⁴ : Semiring S✝\ninst✝³ : Semiring T\nR : Type u_4\nS : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\nH : LocalRing S\nf : R →+* S\ninst✝ : IsLocalRingHom f\nthis : Nontrivial R\na b : R\n⊢ f a ∈ nonunits S → f b ∈ nonunits S → f a + f b ∈ nonunits S","state_after":"no goals","tactic":"exact LocalRing.nonunits_add","premises":[{"full_name":"LocalRing.nonunits_add","def_path":"Mathlib/RingTheory/LocalRing/Basic.lean","def_pos":[65,8],"def_end_pos":[65,20]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","commit":"","full_name":"EuclideanSpace.single_apply","start":[236,0],"end":[239,81],"file_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : DecidableEq ι\ni : ι\na : 𝕜\nj : ι\n⊢ single i a j = if j = i then a else 0","state_after":"no goals","tactic":"rw [EuclideanSpace.single, WithLp.equiv_symm_pi_apply, ← Pi.single_apply i a j]","premises":[{"full_name":"EuclideanSpace.single","def_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","def_pos":[223,4],"def_end_pos":[223,25]},{"full_name":"Pi.single_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[290,2],"def_end_pos":[290,13]},{"full_name":"WithLp.equiv_symm_pi_apply","def_path":"Mathlib/Analysis/Normed/Lp/PiLp.lean","def_pos":[134,9],"def_end_pos":[134,42]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","commit":"","full_name":"HurwitzZeta.completedHurwitzZetaEven_residue_one","start":[471,0],"end":[478,98],"file_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","tactics":[{"state_before":"a : UnitAddCircle\n⊢ Tendsto (fun s => (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)","state_after":"a : UnitAddCircle\nh1 : Tendsto (fun s => (s - ↑(1 / 2)) * (hurwitzEvenFEPair a).Λ s) (𝓝[≠] ↑(1 / 2)) (𝓝 (1 * 1))\n⊢ Tendsto (fun s => (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)","tactic":"have h1 : Tendsto (fun s : ℂ ↦ (s - ↑(1  / 2 : ℝ)) * _) (𝓝[≠] ↑(1  / 2 : ℝ))\n    (𝓝 ((1 : ℂ) * (1 : ℂ))) := (hurwitzEvenFEPair a).Λ_residue_k","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"HurwitzZeta.hurwitzEvenFEPair","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","def_pos":[271,4],"def_end_pos":[271,21]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.instSingletonSet","def_path":"Mathlib/Init/Set.lean","def_pos":[172,9],"def_end_pos":[172,25]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"WeakFEPair.Λ_residue_k","def_path":"Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean","def_pos":[435,8],"def_end_pos":[435,19]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]}]},{"state_before":"a : UnitAddCircle\nh1 : Tendsto (fun s => (s - ↑(1 / 2)) * (hurwitzEvenFEPair a).Λ s) (𝓝[≠] ↑(1 / 2)) (𝓝 (1 * 1))\n⊢ Tendsto (fun s => (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)","state_after":"a : UnitAddCircle\nh1 : Tendsto (fun s => (s - 1 / 2) * (hurwitzEvenFEPair a).Λ s) (𝓝[≠] (1 / 2)) (𝓝 1)\n⊢ Tendsto (fun s => (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)","tactic":"simp only [push_cast, one_mul] at h1","premises":[{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]},{"state_before":"a : UnitAddCircle\nh1 : Tendsto (fun s => (s - 1 / 2) * (hurwitzEvenFEPair a).Λ s) (𝓝[≠] (1 / 2)) (𝓝 1)\n⊢ Tendsto (fun s => (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)","state_after":"a : UnitAddCircle\nh1 : Tendsto (fun s => (s - 1 / 2) * (hurwitzEvenFEPair a).Λ s) (𝓝[≠] (1 / 2)) (𝓝 1)\ns : ℂ\n⊢ ((fun s => (s - 1 / 2) * (hurwitzEvenFEPair a).Λ s) ∘ fun s => s / 2) s = (s - 1) * completedHurwitzZetaEven a s","tactic":"refine (h1.comp <| tendsto_div_two_punctured_nhds 1).congr (fun s ↦ ?_)","premises":[{"full_name":"Filter.Tendsto.comp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2652,8],"def_end_pos":[2652,20]},{"full_name":"Filter.Tendsto.congr","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2642,8],"def_end_pos":[2642,21]},{"full_name":"_private.Mathlib.NumberTheory.LSeries.HurwitzZetaEven.0.HurwitzZeta.tendsto_div_two_punctured_nhds","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","def_pos":[467,14],"def_end_pos":[467,44]}]},{"state_before":"a : UnitAddCircle\nh1 : Tendsto (fun s => (s - 1 / 2) * (hurwitzEvenFEPair a).Λ s) (𝓝[≠] (1 / 2)) (𝓝 1)\ns : ℂ\n⊢ ((fun s => (s - 1 / 2) * (hurwitzEvenFEPair a).Λ s) ∘ fun s => s / 2) s = (s - 1) * completedHurwitzZetaEven a s","state_after":"no goals","tactic":"rw [completedHurwitzZetaEven, Function.comp_apply, ← sub_div, div_mul_eq_mul_div, mul_div_assoc]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"HurwitzZeta.completedHurwitzZetaEven","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","def_pos":[317,4],"def_end_pos":[317,28]},{"full_name":"div_mul_eq_mul_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[558,8],"def_end_pos":[558,26]},{"full_name":"mul_div_assoc","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[330,8],"def_end_pos":[330,21]},{"full_name":"sub_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[143,8],"def_end_pos":[143,15]}]}]}
{"url":"Mathlib/Combinatorics/Quiver/Path.lean","commit":"","full_name":"Quiver.Path.cons_ne_nil","start":[41,0],"end":[42,25],"file_path":"Mathlib/Combinatorics/Quiver/Path.lean","tactics":[{"state_before":"V : Type u\ninst✝ : Quiver V\na b c d : V\np : Path a b\ne : b ⟶ a\nh : p.cons e = nil\n⊢ False","state_after":"no goals","tactic":"injection h","premises":[]}]}
{"url":"Mathlib/Data/Set/Function.lean","commit":"","full_name":"Set.range_extend_subset","start":[113,0],"end":[119,67],"file_path":"Mathlib/Data/Set/Function.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nf : α → β\ng : α → γ\ng' : β → γ\n⊢ range (extend f g g') ⊆ range g ∪ g' '' (range f)ᶜ","state_after":"no goals","tactic":"classical\n  rintro _ ⟨y, rfl⟩\n  rw [extend_def]\n  split_ifs with h\n  exacts [Or.inl (mem_range_self _), Or.inr (mem_image_of_mem _ h)]","premises":[{"full_name":"Function.extend_def","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[609,8],"def_end_pos":[609,18]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]},{"full_name":"Set.mem_range_self","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[148,22],"def_end_pos":[148,36]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean","commit":"","full_name":"PiTensorProduct.mapL_id","start":[374,0],"end":[379,28],"file_path":"Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean","tactics":[{"state_before":"ι : Type uι\ninst✝⁹ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE' : ι → Type u_1\nE'' : ι → Type u_2\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E' i)\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)\ng : (i : ι) → E' i →L[𝕜] E'' i\nf : (i : ι) → E i →L[𝕜] E' i\n⊢ (mapL fun i => ContinuousLinearMap.id 𝕜 (E i)) = ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i)","state_after":"case a\nι : Type uι\ninst✝⁹ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE' : ι → Type u_1\nE'' : ι → Type u_2\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E' i)\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)\ng : (i : ι) → E' i →L[𝕜] E'' i\nf : (i : ι) → E i →L[𝕜] E' i\n⊢ ↑(mapL fun i => ContinuousLinearMap.id 𝕜 (E i)) = ↑(ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i))","tactic":"apply ContinuousLinearMap.coe_injective","premises":[{"full_name":"ContinuousLinearMap.coe_injective","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[369,8],"def_end_pos":[369,21]}]},{"state_before":"case a\nι : Type uι\ninst✝⁹ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE' : ι → Type u_1\nE'' : ι → Type u_2\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E' i)\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)\ng : (i : ι) → E' i →L[𝕜] E'' i\nf : (i : ι) → E i →L[𝕜] E' i\n⊢ ↑(mapL fun i => ContinuousLinearMap.id 𝕜 (E i)) = ↑(ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i))","state_after":"case a.H.H\nι : Type uι\ninst✝⁹ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE' : ι → Type u_1\nE'' : ι → Type u_2\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E' i)\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)\ng : (i : ι) → E' i →L[𝕜] E'' i\nf : (i : ι) → E i →L[𝕜] E' i\nx✝ : (i : ι) → E i\n⊢ ((↑(mapL fun i => ContinuousLinearMap.id 𝕜 (E i))).compMultilinearMap (tprod 𝕜)) x✝ =\n    ((↑(ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i))).compMultilinearMap (tprod 𝕜)) x✝","tactic":"ext","premises":[]},{"state_before":"case a.H.H\nι : Type uι\ninst✝⁹ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE' : ι → Type u_1\nE'' : ι → Type u_2\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E' i)\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)\ng : (i : ι) → E' i →L[𝕜] E'' i\nf : (i : ι) → E i →L[𝕜] E' i\nx✝ : (i : ι) → E i\n⊢ ((↑(mapL fun i => ContinuousLinearMap.id 𝕜 (E i))).compMultilinearMap (tprod 𝕜)) x✝ =\n    ((↑(ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i))).compMultilinearMap (tprod 𝕜)) x✝","state_after":"no goals","tactic":"simp only [mapL_coe, ContinuousLinearMap.coe_id, map_id, LinearMap.compMultilinearMap_apply,\n    LinearMap.id_coe, id_eq]","premises":[{"full_name":"ContinuousLinearMap.coe_id","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[605,8],"def_end_pos":[605,14]},{"full_name":"LinearMap.compMultilinearMap_apply","def_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","def_pos":[795,8],"def_end_pos":[795,32]},{"full_name":"LinearMap.id_coe","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[282,8],"def_end_pos":[282,14]},{"full_name":"PiTensorProduct.mapL_coe","def_path":"Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean","def_pos":[338,8],"def_end_pos":[338,16]},{"full_name":"PiTensorProduct.map_id","def_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","def_pos":[525,8],"def_end_pos":[525,14]},{"full_name":"id_eq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[297,16],"def_end_pos":[297,21]}]}]}
{"url":"Mathlib/NumberTheory/PellMatiyasevic.lean","commit":"","full_name":"Pell.matiyasevic","start":[757,0],"end":[842,84],"file_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","tactics":[{"state_before":"a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n⊢ 1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])","state_after":"a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n⊢ 1 < a ∧\n    k ≤ yn a1 k ∧\n      (xn a1 k = 1 ∧ yn a1 k = 0 ∨\n        ∃ u v s t b,\n          xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 k] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])","tactic":"rw [← hx, ← hy]","premises":[]},{"state_before":"a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n⊢ 1 < a ∧\n    k ≤ yn a1 k ∧\n      (xn a1 k = 1 ∧ yn a1 k = 0 ∨\n        ∃ u v s t b,\n          xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 k] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])","state_after":"a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nkpos : k > 0\n⊢ k ≤ yn a1 k ∧\n    (xn a1 k = 1 ∧ yn a1 k = 0 ∨\n      ∃ u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧\n          u * u - (a * a - 1) * v * v = 1 ∧\n            s * s - (b * b - 1) * t * t = 1 ∧\n              1 < b ∧\n                b ≡ 1 [MOD 4 * yn a1 k] ∧\n                  b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])","tactic":"refine ⟨a1,\n        (Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact ⟨le_rfl, Or.inl ⟨rfl, rfl⟩⟩)\n          fun kpos => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]},{"full_name":"Or.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[550,8],"def_end_pos":[550,15]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nkpos : k > 0\n⊢ k ≤ yn a1 k ∧\n    (xn a1 k = 1 ∧ yn a1 k = 0 ∨\n      ∃ u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧\n          u * u - (a * a - 1) * v * v = 1 ∧\n            s * s - (b * b - 1) * t * t = 1 ∧\n              1 < b ∧\n                b ≡ 1 [MOD 4 * yn a1 k] ∧\n                  b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])","state_after":"no goals","tactic":"exact\n      let x := xn a1 k\n      let y := yn a1 k\n      let m := 2 * (k * y)\n      let u := xn a1 m\n      let v := yn a1 m\n      have ky : k ≤ y := yn_ge_n a1 k\n      have yv : y * y ∣ v := (ysq_dvd_yy a1 k).trans <| (y_dvd_iff _ _ _).2 <| dvd_mul_left _ _\n      have uco : Nat.Coprime u (4 * y) :=\n        have : 2 ∣ v :=\n          modEq_zero_iff_dvd.1 <| (yn_modEq_two _ _).trans (dvd_mul_right _ _).modEq_zero_nat\n        have : Nat.Coprime u 2 := (xy_coprime a1 m).coprime_dvd_right this\n        (this.mul_right this).mul_right <|\n          (xy_coprime _ _).coprime_dvd_right (dvd_of_mul_left_dvd yv)\n      let ⟨b, ba, bm1⟩ := chineseRemainder uco a 1\n      have m1 : 1 < m :=\n        have : 0 < k * y := mul_pos kpos (strictMono_y a1 kpos)\n        Nat.mul_le_mul_left 2 this\n      have vp : 0 < v := strictMono_y a1 (lt_trans zero_lt_one m1)\n      have b1 : 1 < b :=\n        have : xn a1 1 < u := strictMono_x a1 m1\n        have : a < u := by simpa using this\n        lt_of_lt_of_le a1 <| by\n          delta ModEq at ba; rw [Nat.mod_eq_of_lt this] at ba; rw [← ba]\n          apply Nat.mod_le\n      let s := xn b1 k\n      let t := yn b1 k\n      have sx : s ≡ x [MOD u] := (xy_modEq_of_modEq b1 a1 ba k).left\n      have tk : t ≡ k [MOD 4 * y] :=\n        have : 4 * y ∣ b - 1 :=\n          Int.natCast_dvd_natCast.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd\n        (yn_modEq_a_sub_one _ _).of_dvd this\n      ⟨ky,\n        Or.inr\n          ⟨u, v, s, t, b, pell_eq _ _, pell_eq _ _, pell_eq _ _, b1, bm1, ba, vp, yv, sx, tk⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Dvd.dvd.modEq_zero_nat","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[72,8],"def_end_pos":[72,37]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.natCast_dvd_natCast","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[445,19],"def_end_pos":[445,38]},{"full_name":"Int.ofNat_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[302,21],"def_end_pos":[302,30]},{"full_name":"Nat.Coprime","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[20,17],"def_end_pos":[20,24]},{"full_name":"Nat.Coprime.coprime_dvd_right","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[93,8],"def_end_pos":[93,33]},{"full_name":"Nat.Coprime.mul_right","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[84,8],"def_end_pos":[84,25]},{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"Nat.ModEq.of_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[93,6],"def_end_pos":[93,12]},{"full_name":"Nat.ModEq.symm","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[55,18],"def_end_pos":[55,22]},{"full_name":"Nat.ModEq.trans","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[59,18],"def_end_pos":[59,23]},{"full_name":"Nat.chineseRemainder","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[321,4],"def_end_pos":[321,20]},{"full_name":"Nat.modEq_zero_iff_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[70,8],"def_end_pos":[70,26]},{"full_name":"Nat.mod_eq_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[131,8],"def_end_pos":[131,20]},{"full_name":"Nat.mod_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[160,8],"def_end_pos":[160,14]},{"full_name":"Nat.mul_le_mul_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[713,8],"def_end_pos":[713,23]},{"full_name":"Or.inr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[536,4],"def_end_pos":[536,7]},{"full_name":"Pell.pell_eq","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[207,8],"def_end_pos":[207,15]},{"full_name":"Pell.strictMono_x","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[363,8],"def_end_pos":[363,20]},{"full_name":"Pell.strictMono_y","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[353,8],"def_end_pos":[353,20]},{"full_name":"Pell.xn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[103,4],"def_end_pos":[103,6]},{"full_name":"Pell.xy_coprime","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[347,8],"def_end_pos":[347,18]},{"full_name":"Pell.xy_modEq_of_modEq","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[745,8],"def_end_pos":[745,25]},{"full_name":"Pell.y_dvd_iff","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[383,8],"def_end_pos":[383,17]},{"full_name":"Pell.yn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[107,4],"def_end_pos":[107,6]},{"full_name":"Pell.yn_ge_n","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[373,8],"def_end_pos":[373,15]},{"full_name":"Pell.yn_modEq_a_sub_one","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[480,8],"def_end_pos":[480,26]},{"full_name":"Pell.yn_modEq_two","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[488,8],"def_end_pos":[488,20]},{"full_name":"Pell.ysq_dvd_yy","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[433,8],"def_end_pos":[433,18]},{"full_name":"dvd_mul_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[176,8],"def_end_pos":[176,20]},{"full_name":"dvd_mul_right","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[75,8],"def_end_pos":[75,21]},{"full_name":"dvd_of_mul_left_dvd","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[189,8],"def_end_pos":[189,27]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]},{"full_name":"lt_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[74,8],"def_end_pos":[74,16]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]},{"state_before":"a k x y : ℕ\nx✝ :\n  1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])\na1 : 1 < a\nky : k ≤ y\no :\n  x = 1 ∧ y = 0 ∨\n    ∃ u v s t b,\n      x * x - (a * a - 1) * y * y = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n⊢ xn a1 k = x ∧ yn a1 k = y","state_after":"a k x y : ℕ\nx✝ :\n  1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])\na1 : 1 < a\nky : k ≤ 0\no :\n  x = 1 ∧ y = 0 ∨\n    ∃ u v s t b,\n      x * x - (a * a - 1) * y * y = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n⊢ xn a1 k = x ∧ yn a1 k = y","tactic":"rw [y0] at ky","premises":[]},{"state_before":"a k x y : ℕ\nx✝ :\n  1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])\na1 : 1 < a\nky : k ≤ 0\no :\n  x = 1 ∧ y = 0 ∨\n    ∃ u v s t b,\n      x * x - (a * a - 1) * y * y = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n⊢ xn a1 k = x ∧ yn a1 k = y","state_after":"a k x y : ℕ\nx✝ :\n  1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])\na1 : 1 < a\nky : k ≤ 0\no :\n  x = 1 ∧ y = 0 ∨\n    ∃ u v s t b,\n      x * x - (a * a - 1) * y * y = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n⊢ xn a1 0 = 1 ∧ yn a1 0 = 0","tactic":"rw [Nat.eq_zero_of_le_zero ky, x1, y0]","premises":[{"full_name":"Nat.eq_zero_of_le_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[367,8],"def_end_pos":[367,26]}]},{"state_before":"a k x y : ℕ\nx✝ :\n  1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])\na1 : 1 < a\nky : k ≤ 0\no :\n  x = 1 ∧ y = 0 ∨\n    ∃ u v s t b,\n      x * x - (a * a - 1) * y * y = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n⊢ xn a1 0 = 1 ∧ yn a1 0 = 0","state_after":"no goals","tactic":"exact ⟨rfl, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\n⊢ xn a1 k = xn a1 i ∧ yn a1 k = yn a1 i","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n⊢ xn a1 k = xn a1 i ∧ yn a1 k = yn a1 i","tactic":"simp [i0] at ky","premises":[]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n⊢ xn a1 k = xn a1 i ∧ yn a1 k = yn a1 i","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n⊢ xn a1 0 = xn a1 0 ∧ yn a1 0 = yn a1 0","tactic":"rw [i0, ky]","premises":[]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n⊢ xn a1 0 = xn a1 0 ∧ yn a1 0 = yn a1 0","state_after":"no goals","tactic":"exact ⟨rfl, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n⊢ xn a1 k = xn a1 i ∧ yn a1 k = yn a1 i","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n⊢ i = k","tactic":"suffices i = k by rw [this]; exact ⟨rfl, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 ∧ yn a1 i = 0 ∨\n    ∃ u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧\n              b ≡ 1 [MOD 4 * yn a1 i] ∧\n                b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\n⊢ i = k","tactic":"clear o rem xy uv st","premises":[]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\n⊢ i = k","tactic":"have iln : i ≤ n :=\n              le_of_not_gt fun hin =>\n                not_lt_of_ge (Nat.le_of_dvd vp (dvd_of_mul_left_dvd yv)) (strictMono_y a1 hin)","premises":[{"full_name":"Nat.le_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[46,8],"def_end_pos":[46,17]},{"full_name":"Pell.strictMono_y","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[353,8],"def_end_pos":[353,20]},{"full_name":"dvd_of_mul_left_dvd","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[189,8],"def_end_pos":[189,27]},{"full_name":"le_of_not_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[281,8],"def_end_pos":[281,20]},{"full_name":"not_lt_of_ge","def_path":"Mathlib/Order/Defs.lean","def_pos":[125,8],"def_end_pos":[125,20]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\n⊢ i = k","tactic":"have yd : 4 * yn a1 i ∣ 4 * n := mul_dvd_mul_left _ <| dvd_of_ysq_dvd a1 yv","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Pell.dvd_of_ysq_dvd","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[438,8],"def_end_pos":[438,22]},{"full_name":"Pell.yn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[107,4],"def_end_pos":[107,6]},{"full_name":"mul_dvd_mul_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[148,8],"def_end_pos":[148,24]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\n⊢ i = k","tactic":"have jk : j ≡ k [MOD 4 * yn a1 i] :=\n              have : 4 * yn a1 i ∣ b - 1 :=\n                Int.natCast_dvd_natCast.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd\n              ((yn_modEq_a_sub_one b1 _).of_dvd this).symm.trans tk","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Int.natCast_dvd_natCast","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[445,19],"def_end_pos":[445,38]},{"full_name":"Int.ofNat_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[302,21],"def_end_pos":[302,30]},{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"Nat.ModEq.of_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[93,6],"def_end_pos":[93,12]},{"full_name":"Nat.ModEq.symm","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[55,18],"def_end_pos":[55,22]},{"full_name":"Nat.ModEq.trans","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[59,18],"def_end_pos":[59,23]},{"full_name":"Pell.yn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[107,4],"def_end_pos":[107,6]},{"full_name":"Pell.yn_modEq_a_sub_one","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[480,8],"def_end_pos":[480,26]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\n⊢ i = k","tactic":"have ki : k + i < 4 * yn a1 i :=\n              lt_of_le_of_lt (_root_.add_le_add ky (yn_ge_n a1 i)) <| by\n                rw [← two_mul]\n                exact Nat.mul_lt_mul_of_pos_right (by decide) (strictMono_y a1 ipos)","premises":[{"full_name":"Nat.mul_lt_mul_of_pos_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[728,18],"def_end_pos":[728,41]},{"full_name":"Pell.strictMono_y","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[353,8],"def_end_pos":[353,20]},{"full_name":"Pell.yn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[107,4],"def_end_pos":[107,6]},{"full_name":"Pell.yn_ge_n","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[373,8],"def_end_pos":[373,15]},{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j ≡ i [MOD 4 * n]\n⊢ i = k","tactic":"have ji : j ≡ i [MOD 4 * n] :=\n              have : xn a1 j ≡ xn a1 i [MOD xn a1 n] :=\n                (xy_modEq_of_modEq b1 a1 ba j).left.symm.trans sx\n              (modEq_of_xn_modEq a1 ipos iln this).resolve_right\n                fun ji : j + i ≡ 0 [MOD 4 * n] =>\n                not_le_of_gt ki <|\n                  Nat.le_of_dvd (lt_of_lt_of_le ipos <| Nat.le_add_left _ _) <|\n                    modEq_zero_iff_dvd.1 <| (jk.symm.add_right i).trans <| ji.of_dvd yd","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"Nat.ModEq.add_right","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[132,18],"def_end_pos":[132,27]},{"full_name":"Nat.ModEq.of_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[93,6],"def_end_pos":[93,12]},{"full_name":"Nat.ModEq.symm","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[55,18],"def_end_pos":[55,22]},{"full_name":"Nat.ModEq.trans","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[59,18],"def_end_pos":[59,23]},{"full_name":"Nat.le_add_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[399,8],"def_end_pos":[399,19]},{"full_name":"Nat.le_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[46,8],"def_end_pos":[46,17]},{"full_name":"Nat.modEq_zero_iff_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[70,8],"def_end_pos":[70,26]},{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]},{"full_name":"Pell.modEq_of_xn_modEq","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[721,8],"def_end_pos":[721,25]},{"full_name":"Pell.xn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[103,4],"def_end_pos":[103,6]},{"full_name":"Pell.xy_modEq_of_modEq","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[745,8],"def_end_pos":[745,25]},{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]},{"full_name":"not_le_of_gt","def_path":"Mathlib/Order/Defs.lean","def_pos":[122,8],"def_end_pos":[122,20]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j ≡ i [MOD 4 * n]\n⊢ i = k","state_after":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j ≡ i [MOD 4 * n]\nthis : i % (4 * yn a1 i) = k % (4 * yn a1 i)\n⊢ i = k","tactic":"have : i % (4 * yn a1 i) = k % (4 * yn a1 i) := (ji.of_dvd yd).symm.trans jk","premises":[{"full_name":"Nat.ModEq.of_dvd","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[93,6],"def_end_pos":[93,12]},{"full_name":"Nat.ModEq.symm","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[55,18],"def_end_pos":[55,22]},{"full_name":"Nat.ModEq.trans","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[59,18],"def_end_pos":[59,23]},{"full_name":"Pell.yn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[107,4],"def_end_pos":[107,6]}]},{"state_before":"a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j ≡ i [MOD 4 * n]\nthis : i % (4 * yn a1 i) = k % (4 * yn a1 i)\n⊢ i = k","state_after":"no goals","tactic":"rwa [Nat.mod_eq_of_lt (lt_of_le_of_lt (Nat.le_add_left _ _) ki),\n              Nat.mod_eq_of_lt (lt_of_le_of_lt (Nat.le_add_right _ _) ki)] at this","premises":[{"full_name":"Nat.le_add_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[399,8],"def_end_pos":[399,19]},{"full_name":"Nat.le_add_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[395,8],"def_end_pos":[395,20]},{"full_name":"Nat.mod_eq_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[131,8],"def_end_pos":[131,20]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.negAddY_of_Z_ne_zero","start":[967,0],"end":[972,62],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 ≠ Q x * P z ^ 2\n⊢ W.negAddY P Q / addZ P Q ^ 3 =\n    W.toAffine.negAddY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)\n      (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3))","state_after":"no goals","tactic":"rw [negAddY_eq hPz hQz, addX_eq' hP hQ, div_div, ← mul_pow _ _ 3, toAffine_slope_of_ne hPz hQz hx,\n    toAffine_negAddY_of_ne hPz hQz <| addZ_ne_zero_of_X_ne hx]","premises":[{"full_name":"WeierstrassCurve.Jacobian.addX_eq'","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[848,6],"def_end_pos":[848,14]},{"full_name":"WeierstrassCurve.Jacobian.addZ_ne_zero_of_X_ne","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[819,6],"def_end_pos":[819,26]},{"full_name":"WeierstrassCurve.Jacobian.negAddY_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[926,6],"def_end_pos":[926,16]},{"full_name":"_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.0.WeierstrassCurve.Jacobian.toAffine_negAddY_of_ne","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[957,14],"def_end_pos":[957,36]},{"full_name":"_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.0.WeierstrassCurve.Jacobian.toAffine_slope_of_ne","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[826,14],"def_end_pos":[826,34]},{"full_name":"div_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[543,8],"def_end_pos":[543,15]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Cofinality.lean","commit":"","full_name":"Ordinal.IsFundamentalSequence.succ","start":[520,0],"end":[525,22],"file_path":"Mathlib/SetTheory/Cardinal/Cofinality.lean","tactics":[{"state_before":"α : Type u_1\nr : α → α → Prop\na o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\n⊢ (succ o).IsFundamentalSequence 1 fun x x => o","state_after":"case refine_1\nα : Type u_1\nr : α → α → Prop\na o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\n⊢ 1 ≤ (succ o).cof.ord\n\ncase refine_2\nα : Type u_1\nr : α → α → Prop\na o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\ni j : Ordinal.{u}\nhi : i < 1\nhj : j < 1\nh : i < j\n⊢ (fun x x => o) i hi < (fun x x => o) j hj","tactic":"refine ⟨?_, @fun i j hi hj h => ?_, blsub_const Ordinal.one_ne_zero o⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Ordinal.blsub_const","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[1627,8],"def_end_pos":[1627,19]},{"full_name":"Ordinal.one_ne_zero","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[256,18],"def_end_pos":[256,29]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Spectrum.lean","commit":"","full_name":"LinearMap.IsSymmetric.orthogonalFamily_eigenspaces","start":[78,0],"end":[87,74],"file_path":"Mathlib/Analysis/InnerProductSpace/Spectrum.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\n⊢ OrthogonalFamily 𝕜 (fun μ => ↥(eigenspace T μ)) fun μ => (eigenspace T μ).subtypeₗᵢ","state_after":"case mk.mk\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0","tactic":"rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩","premises":[]},{"state_before":"case mk.mk\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0","state_after":"case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0\n\ncase neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : ¬v = 0\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0","tactic":"by_cases hv' : v = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : ¬v = 0\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0","state_after":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0","tactic":"have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"LinearMap.IsSymmetric.conj_eigenvalue_eq_self","def_path":"Mathlib/Analysis/InnerProductSpace/Spectrum.lean","def_pos":[73,8],"def_end_pos":[73,31]},{"full_name":"Module.End.hasEigenvalue_of_hasEigenvector","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[92,8],"def_end_pos":[92,39]}]},{"state_before":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0","state_after":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0","tactic":"rw [mem_eigenspace_iff] at hv hw","premises":[{"full_name":"Module.End.mem_eigenspace_iff","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[97,8],"def_end_pos":[97,26]}]},{"state_before":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0","state_after":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ν = μ ∨ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0","tactic":"refine Or.resolve_left ?_ hμν.symm","premises":[{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Or.resolve_left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[555,8],"def_end_pos":[555,23]}]},{"state_before":"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ν = μ ∨ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0","state_after":"no goals","tactic":"simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"inner_smul_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[413,8],"def_end_pos":[413,23]},{"full_name":"inner_smul_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[422,8],"def_end_pos":[422,24]}]}]}
{"url":"Mathlib/CategoryTheory/GradedObject/Monoidal.lean","commit":"","full_name":"CategoryTheory.GradedObject.Monoidal.leftUnitor_naturality","start":[469,0],"end":[473,41],"file_path":"Mathlib/CategoryTheory/GradedObject/Monoidal.lean","tactics":[{"state_before":"I : Type u\ninst✝⁵ : AddMonoid I\nC : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : MonoidalCategory C\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\ninst✝ : (X₂ : C) → PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)\nX X' : GradedObject I C\nφ : X ⟶ X'\n⊢ tensorHom (𝟙 tensorUnit) φ ≫ (leftUnitor X').hom = (leftUnitor X).hom ≫ φ","state_after":"no goals","tactic":"apply mapBifunctorLeftUnitor_naturality","premises":[{"full_name":"CategoryTheory.GradedObject.mapBifunctorLeftUnitor_naturality","def_path":"Mathlib/CategoryTheory/GradedObject/Unitor.lean","def_pos":[131,6],"def_end_pos":[131,39]}]}]}
{"url":"Mathlib/GroupTheory/Complement.lean","commit":"","full_name":"Subgroup.IsComplement.equiv_snd_eq_iff_rightCosetEquivalence","start":[347,0],"end":[361,48],"file_path":"Mathlib/GroupTheory/Complement.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (↑H) T\nhSK : IsComplement S ↑K\ng₁ g₂ : G\n⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ RightCosetEquivalence (↑H) g₁ g₂","state_after":"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (↑H) T\nhSK : IsComplement S ↑K\ng₁ g₂ : G\n⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ g₂ * g₁⁻¹ ∈ H","tactic":"rw [RightCosetEquivalence, rightCoset_eq_iff]","premises":[{"full_name":"RightCosetEquivalence","def_path":"Mathlib/GroupTheory/Coset.lean","def_pos":[72,4],"def_end_pos":[72,25]},{"full_name":"rightCoset_eq_iff","def_path":"Mathlib/GroupTheory/Coset.lean","def_pos":[207,8],"def_end_pos":[207,25]}]},{"state_before":"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (↑H) T\nhSK : IsComplement S ↑K\ng₁ g₂ : G\n⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ g₂ * g₁⁻¹ ∈ H","state_after":"case mp\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (↑H) T\nhSK : IsComplement S ↑K\ng₁ g₂ : G\n⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 → g₂ * g₁⁻¹ ∈ H\n\ncase mpr\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (↑H) T\nhSK : IsComplement S ↑K\ng₁ g₂ : G\n⊢ g₂ * g₁⁻¹ ∈ H → (hHT.equiv g₁).2 = (hHT.equiv g₂).2","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Sites/Localization.lean","commit":"","full_name":"CategoryTheory.GrothendieckTopology.W_adj_unit_app","start":[50,0],"end":[52,42],"file_path":"Mathlib/CategoryTheory/Sites/Localization.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{u_3, u_2} A\nG : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A\nadj : G ⊣ sheafToPresheaf J A\nP : Cᵒᵖ ⥤ A\n⊢ J.W (adj.unit.app P)","state_after":"C : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{u_3, u_2} A\nG : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A\nadj : G ⊣ sheafToPresheaf J A\nP : Cᵒᵖ ⥤ A\n⊢ LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) (adj.unit.app P)","tactic":"rw [W_eq_W_range_sheafToPresheaf_obj]","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.W_eq_W_range_sheafToPresheaf_obj","def_path":"Mathlib/CategoryTheory/Sites/Localization.lean","def_pos":[31,6],"def_end_pos":[31,38]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{u_3, u_2} A\nG : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A\nadj : G ⊣ sheafToPresheaf J A\nP : Cᵒᵖ ⥤ A\n⊢ LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) (adj.unit.app P)","state_after":"no goals","tactic":"exact LeftBousfield.W_adj_unit_app adj P","premises":[{"full_name":"CategoryTheory.Localization.LeftBousfield.W_adj_unit_app","def_path":"Mathlib/CategoryTheory/Localization/Bousfield.lean","def_pos":[112,6],"def_end_pos":[112,20]}]}]}
{"url":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","commit":"","full_name":"RingHom.isIntegralElem_leadingCoeff_mul","start":[395,0],"end":[414,60],"file_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf✝ f : R →+* S\np : R[X]\nx : S\nh : eval₂ f x p = 0\n⊢ f.IsIntegralElem (f p.leadingCoeff * x)","state_after":"case pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf✝ f : R →+* S\np : R[X]\nx : S\nh : eval₂ f x p = 0\nh' : 1 ≤ p.natDegree\n⊢ f.IsIntegralElem (f p.leadingCoeff * x)\n\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf✝ f : R →+* S\np : R[X]\nx : S\nh : eval₂ f x p = 0\nh' : ¬1 ≤ p.natDegree\n⊢ f.IsIntegralElem (f p.leadingCoeff * x)","tactic":"by_cases h' : 1 ≤ p.natDegree","premises":[{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/GroupTheory/Congruence/Basic.lean","commit":"","full_name":"AddCon.zsmul","start":[1055,0],"end":[1059,82],"file_path":"Mathlib/GroupTheory/Congruence/Basic.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ Int.ofNat n) (x ^ Int.ofNat n)","state_after":"no goals","tactic":"simpa only [zpow_natCast, Int.ofNat_eq_coe] using c.pow n h","premises":[{"full_name":"Con.pow","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[945,18],"def_end_pos":[945,21]},{"full_name":"Int.ofNat_eq_coe","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[71,16],"def_end_pos":[71,28]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]}]},{"state_before":"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ Int.negSucc n) (x ^ Int.negSucc n)","state_after":"no goals","tactic":"simpa only [zpow_negSucc] using c.inv (c.pow _ h)","premises":[{"full_name":"Con.inv","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[1047,18],"def_end_pos":[1047,21]},{"full_name":"Con.pow","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[945,18],"def_end_pos":[945,21]},{"full_name":"zpow_negSucc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[890,8],"def_end_pos":[890,20]}]}]}
{"url":"Mathlib/Data/Set/Image.lean","commit":"","full_name":"Function.Surjective.image_surjective","start":[1079,0],"end":[1082,24],"file_path":"Mathlib/Data/Set/Image.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\n⊢ Surjective (image f)","state_after":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\ns : Set β\n⊢ ∃ a, f '' a = s","tactic":"intro s","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\ns : Set β\n⊢ ∃ a, f '' a = s","state_after":"case h\nα : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\ns : Set β\n⊢ f '' (f ⁻¹' s) = s","tactic":"use f ⁻¹' s","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\ns : Set β\n⊢ f '' (f ⁻¹' s) = s","state_after":"no goals","tactic":"rw [hf.image_preimage]","premises":[{"full_name":"Function.Surjective.image_preimage","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[1076,8],"def_end_pos":[1076,33]}]}]}
{"url":"Mathlib/Analysis/Quaternion.lean","commit":"","full_name":"Quaternion.normSq_eq_norm_mul_self","start":[61,0],"end":[62,53],"file_path":"Mathlib/Analysis/Quaternion.lean","tactics":[{"state_before":"a : ℍ\n⊢ normSq a = ‖a‖ * ‖a‖","state_after":"no goals","tactic":"rw [← inner_self, real_inner_self_eq_norm_mul_norm]","premises":[{"full_name":"Quaternion.inner_self","def_path":"Mathlib/Analysis/Quaternion.lean","def_pos":[43,8],"def_end_pos":[43,18]},{"full_name":"real_inner_self_eq_norm_mul_norm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[867,8],"def_end_pos":[867,40]}]}]}
{"url":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","commit":"","full_name":"PMF.toMeasure_apply_eq_zero_iff","start":[231,0],"end":[233,87],"file_path":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : MeasurableSpace α\np : PMF α\ns t : Set α\nhs : MeasurableSet s\n⊢ p.toMeasure s = 0 ↔ Disjoint p.support s","state_after":"no goals","tactic":"rw [toMeasure_apply_eq_toOuterMeasure_apply p s hs, toOuterMeasure_apply_eq_zero_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"PMF.toMeasure_apply_eq_toOuterMeasure_apply","def_path":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","def_pos":[220,8],"def_end_pos":[220,47]},{"full_name":"PMF.toOuterMeasure_apply_eq_zero_iff","def_path":"Mathlib/Probability/ProbabilityMassFunction/Basic.lean","def_pos":[169,8],"def_end_pos":[169,40]}]}]}
{"url":"Mathlib/Algebra/DirectSum/Module.lean","commit":"","full_name":"DirectSum.isInternal_submodule_iff_isCompl","start":[396,0],"end":[403,84],"file_path":"Mathlib/Algebra/DirectSum/Module.lean","tactics":[{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\n⊢ IsInternal A ↔ IsCompl (A i) (A j)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ IsInternal A ↔ IsCompl (A i) (A j)","tactic":"have : ∀ k, k = i ∨ k = j := fun k ↦ by simpa using Set.ext_iff.mp h k","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Or","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[532,10],"def_end_pos":[532,12]},{"full_name":"Set.ext_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[190,18],"def_end_pos":[190,25]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ IsInternal A ↔ IsCompl (A i) (A j)","state_after":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤ ↔ IsCompl (A i) (A j)","tactic":"rw [isInternal_submodule_iff_independent_and_iSup_eq_top, iSup, ← Set.image_univ, h,\n    Set.image_insert_eq, Set.image_singleton, sSup_pair, CompleteLattice.independent_pair hij this]","premises":[{"full_name":"CompleteLattice.independent_pair","def_path":"Mathlib/Order/SupIndep.lean","def_pos":[403,8],"def_end_pos":[403,24]},{"full_name":"DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top","def_path":"Mathlib/Algebra/DirectSum/Module.lean","def_pos":[391,8],"def_end_pos":[391,60]},{"full_name":"Set.image_insert_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[332,8],"def_end_pos":[332,23]},{"full_name":"Set.image_singleton","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[286,8],"def_end_pos":[286,23]},{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]},{"full_name":"sSup_pair","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[418,8],"def_end_pos":[418,17]}]},{"state_before":"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤ ↔ IsCompl (A i) (A j)","state_after":"no goals","tactic":"exact ⟨fun ⟨hd, ht⟩ ↦ ⟨hd, codisjoint_iff.mpr ht⟩, fun ⟨hd, ht⟩ ↦ ⟨hd, ht.eq_top⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Codisjoint.eq_top","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[281,8],"def_end_pos":[281,25]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"codisjoint_iff","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[275,8],"def_end_pos":[275,22]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"AddAction.isClosedMap_quotient","start":[1112,0],"end":[1120,5],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt : Set β\ninst✝ : CompactSpace α\n⊢ IsClosedMap Quotient.mk'","state_after":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ IsClosed (Quotient.mk' '' t)","tactic":"intro t ht","premises":[]},{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ IsClosed (Quotient.mk' '' t)","state_after":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ IsClosed (⋃ g, (fun x => g • x) '' t)","tactic":"rw [← quotientMap_quotient_mk'.isClosed_preimage, MulAction.quotient_preimage_image_eq_union_mul]","premises":[{"full_name":"MulAction.quotient_preimage_image_eq_union_mul","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[386,8],"def_end_pos":[386,44]},{"full_name":"QuotientMap.isClosed_preimage","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[270,18],"def_end_pos":[270,35]},{"full_name":"quotientMap_quotient_mk'","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[1106,8],"def_end_pos":[1106,32]}]},{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ IsClosed (⋃ g, (fun x => g • x) '' t)","state_after":"case h.e'_3\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ ⋃ g, (fun x => g • x) '' t = univ • t","tactic":"convert ht.smul_left_of_isCompact (isCompact_univ (X := α))","premises":[{"full_name":"IsClosed.smul_left_of_isCompact","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[1079,8],"def_end_pos":[1079,39]},{"full_name":"isCompact_univ","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[727,8],"def_end_pos":[727,22]}]},{"state_before":"case h.e'_3\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ ⋃ g, (fun x => g • x) '' t = univ • t","state_after":"case h.e'_3\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ ⋃ x ∈ univ, (fun x_1 => x • x_1) '' t = ⋃ a ∈ univ, a • t","tactic":"rw [← biUnion_univ, ← iUnion_smul_left_image]","premises":[{"full_name":"Set.biUnion_univ","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[785,8],"def_end_pos":[785,20]},{"full_name":"Set.iUnion_smul_left_image","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[174,8],"def_end_pos":[174,30]}]},{"state_before":"case h.e'_3\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ ⋃ x ∈ univ, (fun x_1 => x • x_1) '' t = ⋃ a ∈ univ, a • t","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_natCast","start":[105,0],"end":[110,59],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\nn : ℕ\n⊢ eval₂ f x ↑n = ↑n","state_after":"case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x ↑0 = ↑0\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\nn : ℕ\nih : eval₂ f x ↑n = ↑n\n⊢ eval₂ f x ↑(n + 1) = ↑(n + 1)","tactic":"induction' n with n ih","premises":[]}]}
{"url":"Mathlib/NumberTheory/Divisors.lean","commit":"","full_name":"Nat.Prime.sum_divisors","start":[391,0],"end":[394,77],"file_path":"Mathlib/NumberTheory/Divisors.lean","tactics":[{"state_before":"n : ℕ\nα : Type u_1\ninst✝ : CommMonoid α\np : ℕ\nf : ℕ → α\nh : Prime p\n⊢ ∏ x ∈ p.divisors, f x = f p * f 1","state_after":"no goals","tactic":"rw [← cons_self_properDivisors h.ne_zero, prod_cons, h.prod_properDivisors]","premises":[{"full_name":"Finset.prod_cons","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[325,8],"def_end_pos":[325,17]},{"full_name":"Nat.Prime.ne_zero","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[48,8],"def_end_pos":[48,21]},{"full_name":"Nat.Prime.prod_properDivisors","def_path":"Mathlib/NumberTheory/Divisors.lean","def_pos":[388,8],"def_end_pos":[388,33]},{"full_name":"Nat.cons_self_properDivisors","def_path":"Mathlib/NumberTheory/Divisors.lean","def_pos":[80,8],"def_end_pos":[80,32]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean","commit":"","full_name":"MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Iic","start":[876,0],"end":[898,53],"file_path":"Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean","tactics":[{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\n⊢ Tendsto f atBot (𝓝 0)","tactic":"let F : E →L[ℝ] Completion E := Completion.toComplL","premises":[{"full_name":"ContinuousLinearMap","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[224,10],"def_end_pos":[224,29]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"UniformSpace.Completion","def_path":"Mathlib/Topology/UniformSpace/Completion.lean","def_pos":[293,4],"def_end_pos":[293,14]},{"full_name":"UniformSpace.Completion.toComplL","def_path":"Mathlib/Analysis/Normed/Module/Completion.lean","def_pos":[56,4],"def_end_pos":[56,12]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\n⊢ Tendsto f atBot (𝓝 0)","tactic":"have Fderiv : ∀ x ∈ Iic a, HasDerivAt (F ∘ f) (F (f' x)) x :=\n    fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx)","premises":[{"full_name":"ContinuousLinearMap.hasFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Linear.lean","def_pos":[63,18],"def_end_pos":[63,49]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"HasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[121,4],"def_end_pos":[121,14]},{"full_name":"HasFDerivAt.comp_hasDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Comp.lean","def_pos":[347,8],"def_end_pos":[347,35]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)","tactic":"have Fint : IntegrableOn (F ∘ f) (Iic a) := by apply F.integrable_comp fint","premises":[{"full_name":"ContinuousLinearMap.integrable_comp","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1415,8],"def_end_pos":[1415,43]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"MeasureTheory.IntegrableOn","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[80,4],"def_end_pos":[80,16]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)","tactic":"have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.integrable_comp f'int","premises":[{"full_name":"ContinuousLinearMap.integrable_comp","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1415,8],"def_end_pos":[1415,43]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"MeasureTheory.IntegrableOn","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[80,4],"def_end_pos":[80,16]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))\n⊢ Tendsto f atBot (𝓝 0)","tactic":"have A : Tendsto (F ∘ f) atBot (𝓝 (limUnder atBot (F ∘ f))) := by\n    apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic Fderiv F'int","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.atBot","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[47,4],"def_end_pos":[47,9]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic","def_path":"Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean","def_pos":[858,8],"def_end_pos":[858,58]},{"full_name":"limUnder","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[230,18],"def_end_pos":[230,26]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))\nB : limUnder atBot (⇑F ∘ f) = F 0\n⊢ Tendsto f atBot (𝓝 0)","tactic":"have B : limUnder atBot (F ∘ f) = F 0 := by\n    have : IntegrableAtFilter (F ∘ f) atBot := by exact ⟨Iic a, Iic_mem_atBot _, Fint⟩\n    apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A\n    intro s hs\n    rcases mem_atBot_sets.1 hs with ⟨b, hb⟩\n    apply le_antisymm (le_top)\n    rw [← volume_Iic (a := b)]\n    exact measure_mono hb","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.Iic_mem_atBot","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[63,8],"def_end_pos":[63,21]},{"full_name":"Filter.atBot","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[47,4],"def_end_pos":[47,9]},{"full_name":"Filter.mem_atBot_sets","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[150,8],"def_end_pos":[150,22]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasureTheory.IntegrableAtFilter","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[343,4],"def_end_pos":[343,22]},{"full_name":"MeasureTheory.IntegrableAtFilter.eq_zero_of_tendsto","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[480,6],"def_end_pos":[480,43]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"Real.volume_Iic","def_path":"Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean","def_pos":[148,8],"def_end_pos":[148,18]},{"full_name":"Set.Iic","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[58,4],"def_end_pos":[58,7]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[62,8],"def_end_pos":[62,14]},{"full_name":"limUnder","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[230,18],"def_end_pos":[230,26]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))\nB : limUnder atBot (⇑F ∘ f) = F 0\n⊢ Tendsto f atBot (𝓝 0)","state_after":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (F 0))\nB : limUnder atBot (⇑F ∘ f) = F 0\n⊢ Embedding ⇑F","tactic":"rwa [B, ← Embedding.tendsto_nhds_iff] at A","premises":[{"full_name":"Embedding.tendsto_nhds_iff","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[200,8],"def_end_pos":[200,34]}]},{"state_before":"E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (F 0))\nB : limUnder atBot (⇑F ∘ f) = F 0\n⊢ Embedding ⇑F","state_after":"no goals","tactic":"exact (Completion.uniformEmbedding_coe E).embedding","premises":[{"full_name":"UniformEmbedding.embedding","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[218,18],"def_end_pos":[218,44]},{"full_name":"UniformSpace.Completion.uniformEmbedding_coe","def_path":"Mathlib/Topology/UniformSpace/Completion.lean","def_pos":[360,8],"def_end_pos":[360,28]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","commit":"","full_name":"MeasureTheory.Measure.haar.index_union_eq","start":[201,0],"end":[234,50],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V","state_after":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V","tactic":"apply le_antisymm (index_union_le K₁ K₂ hV)","premises":[{"full_name":"MeasureTheory.Measure.haar.index_union_le","def_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","def_pos":[190,8],"def_end_pos":[190,22]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V","state_after":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V","tactic":"rcases index_elim (K₁.2.union K₂.2) hV with ⟨s, h1s, h2s⟩","premises":[{"full_name":"IsCompact.union","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[482,8],"def_end_pos":[482,23]},{"full_name":"MeasureTheory.Measure.haar.index_elim","def_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","def_pos":[154,8],"def_end_pos":[154,18]},{"full_name":"TopologicalSpace.Compacts.isCompact'","def_path":"Mathlib/Topology/Sets/Compacts.lean","def_pos":[37,2],"def_end_pos":[37,12]}]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V","state_after":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\n⊢ index K₁.carrier V + index K₂.carrier V ≤ s.card","tactic":"rw [← h2s]","premises":[]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ index K₁.carrier V + index K₂.carrier V ≤ s.card","state_after":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s).card +\n      (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s).card ≤\n    s.card","tactic":"refine\n    le_trans\n      (add_le_add (this K₁.1 <| Subset.trans subset_union_left h1s)\n        (this K₂.1 <| Subset.trans subset_union_right h1s)) ?_","premises":[{"full_name":"Set.Subset.trans","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[296,8],"def_end_pos":[296,20]},{"full_name":"Set.subset_union_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[657,8],"def_end_pos":[657,25]},{"full_name":"Set.subset_union_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[660,8],"def_end_pos":[660,26]},{"full_name":"TopologicalSpace.Compacts.carrier","def_path":"Mathlib/Topology/Sets/Compacts.lean","def_pos":[36,2],"def_end_pos":[36,9]},{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s).card +\n      (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s).card ≤\n    s.card","state_after":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ (Finset.filter\n        (fun x => ((fun h => x * h) ⁻¹' V ∩ K₁.carrier).Nonempty ∨ ((fun h => x * h) ⁻¹' V ∩ K₂.carrier).Nonempty)\n        s).card ≤\n    s.card\n\ncase intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ Disjoint (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s)\n    (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s)","tactic":"rw [← Finset.card_union_of_disjoint, Finset.filter_union_right]","premises":[{"full_name":"Finset.filter_union_right","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2300,8],"def_end_pos":[2300,26]}]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ Disjoint (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s)\n    (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s)","state_after":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ ∀ x ∈ s, ((fun h => x * h) ⁻¹' V ∩ K₁.carrier).Nonempty → ¬((fun h => x * h) ⁻¹' V ∩ K₂.carrier).Nonempty","tactic":"apply Finset.disjoint_filter.mpr","premises":[{"full_name":"Finset.disjoint_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2250,8],"def_end_pos":[2250,23]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ ∀ x ∈ s, ((fun h => x * h) ⁻¹' V ∩ K₁.carrier).Nonempty → ¬((fun h => x * h) ⁻¹' V ∩ K₂.carrier).Nonempty","state_after":"case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₃ : g₃ ∈ K₂.carrier\n⊢ False","tactic":"rintro g₁ _ ⟨g₂, h1g₂, h2g₂⟩ ⟨g₃, h1g₃, h2g₃⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₃ : g₃ ∈ K₂.carrier\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ False","tactic":"simp only [mem_preimage] at h1g₃ h1g₂","premises":[{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ False","state_after":"case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹","tactic":"refine h.le_bot (?_ : g₁⁻¹ ∈ _)","premises":[{"full_name":"Disjoint.le_bot","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[119,8],"def_end_pos":[119,23]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹","state_after":"case intro.intro.intro.intro.intro.intro.left\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ ∃ x ∈ K₁.carrier, ∃ y, y⁻¹ ∈ V ∧ x * y = g₁⁻¹\n\ncase intro.intro.intro.intro.intro.intro.right\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ ∃ x ∈ K₂.carrier, ∃ y, y⁻¹ ∈ V ∧ x * y = g₁⁻¹","tactic":"constructor <;> simp only [Set.mem_inv, Set.mem_mul, exists_exists_and_eq_and, exists_and_left]","premises":[{"full_name":"Set.mem_inv","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[155,8],"def_end_pos":[155,15]},{"full_name":"Set.mem_mul","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[273,8],"def_end_pos":[273,15]},{"full_name":"exists_and_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[288,16],"def_end_pos":[288,31]},{"full_name":"exists_exists_and_eq_and","def_path":"Mathlib/Logic/Basic.lean","def_pos":[560,16],"def_end_pos":[560,40]}]}]}
{"url":"Mathlib/Data/List/Range.lean","commit":"","full_name":"List.length_finRange","start":[104,0],"end":[106,42],"file_path":"Mathlib/Data/List/Range.lean","tactics":[{"state_before":"α : Type u\nn : ℕ\n⊢ (finRange n).length = n","state_after":"no goals","tactic":"rw [finRange, length_pmap, length_range]","premises":[{"full_name":"List.finRange","def_path":"Mathlib/Data/List/Range.lean","def_pos":[87,4],"def_end_pos":[87,12]},{"full_name":"List.length_pmap","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[1922,8],"def_end_pos":[1922,19]},{"full_name":"List.length_range","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[1407,16],"def_end_pos":[1407,28]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.lift_natCast","start":[2069,0],"end":[2072,37],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"⊢ lift.{u, v} ↑0 = ↑0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"n : ℕ\n⊢ lift.{u, v} ↑(n + 1) = ↑(n + 1)","state_after":"no goals","tactic":"simp [lift_natCast n]","premises":[]}]}
{"url":"Mathlib/Order/Interval/Set/Basic.lean","commit":"","full_name":"Set.Ioo_inter_Ioc_of_left_le","start":[1526,0],"end":[1527,56],"file_path":"Mathlib/Order/Interval/Set/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\nh : b₁ ≤ b₂\n⊢ Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioo (max a₁ a₂) b₁","state_after":"no goals","tactic":"rw [inter_comm, Ioc_inter_Ioo_of_right_le h, max_comm]","premises":[{"full_name":"Set.Ioc_inter_Ioo_of_right_le","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[1521,8],"def_end_pos":[1521,33]},{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]},{"full_name":"max_comm","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[102,8],"def_end_pos":[102,16]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"DFinsupp.sum_inner","start":[468,0],"end":[471,92],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : _root_.RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nι : Type u_4\ninst✝² : DecidableEq ι\nα : ι → Type u_5\ninst✝¹ : (i : ι) → AddZeroClass (α i)\ninst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)\nf : (i : ι) → α i → E\nl : Π₀ (i : ι), α i\nx : E\n⊢ ⟪l.sum f, x⟫_𝕜 = l.sum fun i a => ⟪f i a, x⟫_𝕜","state_after":"no goals","tactic":"simp (config := { contextual := true }) only [DFinsupp.sum, _root_.sum_inner, smul_eq_mul]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"DFinsupp.sum","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[1497,2],"def_end_pos":[1497,13]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]},{"full_name":"sum_inner","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[447,8],"def_end_pos":[447,17]}]}]}
{"url":"Mathlib/Data/Finsupp/Basic.lean","commit":"","full_name":"Finsupp.filter_pos_add_filter_neg","start":[835,0],"end":[839,52],"file_path":"Mathlib/Data/Finsupp/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : AddZeroClass M\nf : α →₀ M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ (fun f => ⇑f) (filter p f + filter (fun a => ¬p a) f) = (fun f => ⇑f) f","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : AddZeroClass M\nf : α →₀ M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ {x | p x}.indicator ⇑f + {a | ¬p a}.indicator ⇑f = ⇑f","tactic":"simp only [coe_add, filter_eq_indicator]","premises":[{"full_name":"Finsupp.coe_add","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[888,25],"def_end_pos":[888,32]},{"full_name":"Finsupp.filter_eq_indicator","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[781,8],"def_end_pos":[781,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : AddZeroClass M\nf : α →₀ M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ {x | p x}.indicator ⇑f + {a | ¬p a}.indicator ⇑f = ⇑f","state_after":"no goals","tactic":"exact Set.indicator_self_add_compl { x | p x } f","premises":[{"full_name":"Set.indicator_self_add_compl","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[323,2],"def_end_pos":[323,13]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","commit":"","full_name":"MeasureTheory.Measure.mkMetric_mono","start":[456,0],"end":[460,90],"file_path":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","tactics":[{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝³ : EMetricSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nhle : m₁ ≤ᶠ[𝓝[≥] 0] m₂\n⊢ mkMetric m₁ ≤ mkMetric m₂","state_after":"no goals","tactic":"convert @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*]","premises":[{"full_name":"ENNReal.one_ne_top","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[308,16],"def_end_pos":[308,26]},{"full_name":"MeasureTheory.Measure.mkMetric_mono_smul","def_path":"Mathlib/MeasureTheory/Measure/Hausdorff.lean","def_pos":[446,8],"def_end_pos":[446,26]},{"full_name":"one_ne_zero","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[49,14],"def_end_pos":[49,25]}]}]}
{"url":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","commit":"","full_name":"TensorProduct.gradedMul_def","start":[172,0],"end":[181,38],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","tactics":[{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ","state_after":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ","tactic":"refine TensorProduct.curry ?_","premises":[{"full_name":"TensorProduct.curry","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[600,4],"def_end_pos":[600,9]}]},{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ","state_after":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n    ((⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), 𝒜 i) ⊗[R] (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i","tactic":"refine map (LinearMap.mul' R (⨁ i, 𝒜 i)) (LinearMap.mul' R (⨁ i, ℬ i)) ∘ₗ ?_","premises":[{"full_name":"DirectSum","def_path":"Mathlib/Algebra/DirectSum/Basic.lean","def_pos":[33,4],"def_end_pos":[33,13]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.mul'","def_path":"Mathlib/Algebra/Algebra/Bilinear.lean","def_pos":[35,18],"def_end_pos":[35,22]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"TensorProduct.map","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[729,4],"def_end_pos":[729,7]}]},{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]\n    ((⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), 𝒜 i) ⊗[R] (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i","state_after":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] (⨁ (i : ι), 𝒜 i) ⊗[R] (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i","tactic":"refine (assoc R _ _ _).symm.toLinearMap ∘ₗ .lTensor _ ?_ ∘ₗ (assoc R _ _ _).toLinearMap","premises":[{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.lTensor","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1055,4],"def_end_pos":[1055,11]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"TensorProduct.assoc","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[705,14],"def_end_pos":[705,19]}]},{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] (⨁ (i : ι), 𝒜 i) ⊗[R] (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i","state_after":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 →ₗ[R] (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i","tactic":"refine (assoc R _ _ _).toLinearMap ∘ₗ .rTensor _ ?_ ∘ₗ (assoc R _ _ _).symm.toLinearMap","premises":[{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.rTensor","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1060,4],"def_end_pos":[1060,11]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"TensorProduct.assoc","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[705,14],"def_end_pos":[705,19]}]},{"state_before":"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\n⊢ DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 →ₗ[R] (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i","state_after":"no goals","tactic":"exact (gradedComm _ _ _).toLinearMap","premises":[{"full_name":"TensorProduct.gradedComm","def_path":"Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean","def_pos":[103,4],"def_end_pos":[103,14]}]}]}
{"url":"Mathlib/LinearAlgebra/Dual.lean","commit":"","full_name":"Submodule.range_dualMap_mkQ_eq","start":[1190,0],"end":[1200,34],"file_path":"Mathlib/LinearAlgebra/Dual.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\n⊢ LinearMap.range W.mkQ.dualMap = W.dualAnnihilator","state_after":"case h\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ φ ∈ LinearMap.range W.mkQ.dualMap ↔ φ ∈ W.dualAnnihilator","tactic":"ext φ","premises":[]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ φ ∈ LinearMap.range W.mkQ.dualMap ↔ φ ∈ W.dualAnnihilator","state_after":"case h\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ (∃ y, W.mkQ.dualMap y = φ) ↔ φ ∈ W.dualAnnihilator","tactic":"rw [LinearMap.mem_range]","premises":[{"full_name":"LinearMap.mem_range","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[67,8],"def_end_pos":[67,17]}]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ (∃ y, W.mkQ.dualMap y = φ) ↔ φ ∈ W.dualAnnihilator","state_after":"case h.mp\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ (∃ y, W.mkQ.dualMap y = φ) → φ ∈ W.dualAnnihilator\n\ncase h.mpr\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ φ ∈ W.dualAnnihilator → ∃ y, W.mkQ.dualMap y = φ","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","commit":"","full_name":"PresheafOfModules.Sheafify.mul_smul","start":[263,0],"end":[273,55],"file_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)","tactic":"let S := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m","premises":[{"full_name":"CategoryTheory.Presheaf.imageSieve","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[45,4],"def_end_pos":[45,14]},{"full_name":"Inf.inf","def_path":"Mathlib/Order/Notation.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)","state_after":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)","tactic":"have hS : S ∈ J X.unop := by\n    refine J.intersection_covering (J.intersection_covering ?_ ?_) ?_\n    all_goals apply Presheaf.imageSieve_mem","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.intersection_covering","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[155,8],"def_end_pos":[155,29]},{"full_name":"CategoryTheory.Presheaf.imageSieve_mem","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[86,6],"def_end_pos":[86,20]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]}]},{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)","state_after":"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\n⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),\n    S.arrows f → (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))","tactic":"apply A.isSeparated _ _ hS","premises":[{"full_name":"CategoryTheory.Sheaf.isSeparated","def_path":"Mathlib/CategoryTheory/Sites/Whiskering.lean","def_pos":[144,6],"def_end_pos":[144,23]}]},{"state_before":"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\n⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),\n    S.arrows f → (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))","state_after":"case a.intro.intro.intro.intro.intro\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y✝ : Cᵒᵖ\nπ : X ⟶ Y✝\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\nY : C\nf : Y ⟶ Opposite.unop X\nr₀ : ↑(R₀.obj (Opposite.op Y))\nhr₀ : (α.app (Opposite.op Y)) r₀ = (R.val.map f.op) r\nr₀' : ↑(R₀.obj (Opposite.op Y))\nhr₀' : (α.app (Opposite.op Y)) r₀' = (R.val.map f.op) r'\nm₀ : ↑(M₀.presheaf.obj (Opposite.op Y))\nhm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m\n⊢ (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))","tactic":"rintro Y f ⟨⟨⟨r₀ : R₀.obj _, hr₀⟩, ⟨r₀' : R₀.obj _, hr₀'⟩⟩, ⟨m₀ : M₀.presheaf.obj _, hm₀⟩⟩","premises":[{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"PresheafOfModules.presheaf","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf.lean","def_pos":[42,2],"def_end_pos":[42,10]}]},{"state_before":"case a.intro.intro.intro.intro.intro\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y✝ : Cᵒᵖ\nπ : X ⟶ Y✝\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\nY : C\nf : Y ⟶ Opposite.unop X\nr₀ : ↑(R₀.obj (Opposite.op Y))\nhr₀ : (α.app (Opposite.op Y)) r₀ = (R.val.map f.op) r\nr₀' : ↑(R₀.obj (Opposite.op Y))\nhr₀' : (α.app (Opposite.op Y)) r₀' = (R.val.map f.op) r'\nm₀ : ↑(M₀.presheaf.obj (Opposite.op Y))\nhm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m\n⊢ (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))","state_after":"no goals","tactic":"erw [map_smul_eq α φ (r * r') m f.op (r₀ * r₀')\n    (by rw [map_mul, map_mul, hr₀, hr₀']) m₀ hm₀, mul_smul,\n    map_smul_eq α φ r (smul α φ r' m) f.op r₀ hr₀ (r₀' • m₀)\n      (map_smul_eq α φ r' m f.op r₀' hr₀' m₀ hm₀).symm]","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MulAction.mul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[99,2],"def_end_pos":[99,10]},{"full_name":"PresheafOfModules.Sheafify.map_smul_eq","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","def_pos":[214,6],"def_end_pos":[214,17]},{"full_name":"PresheafOfModules.Sheafify.smul","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","def_pos":[212,18],"def_end_pos":[212,22]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"smul_mem_nhds","start":[1060,0],"end":[1063,85],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\na : α\nx : β\nht : t ∈ 𝓝 x\n⊢ a • t ∈ 𝓝 (a • x)","state_after":"case intro.intro.intro\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\na : α\nx : β\nht : t ∈ 𝓝 x\nu : Set β\nut : u ⊆ t\nu_open : IsOpen u\nhu : x ∈ u\n⊢ a • t ∈ 𝓝 (a • x)","tactic":"rcases mem_nhds_iff.1 ht with ⟨u, ut, u_open, hu⟩","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mem_nhds_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[716,8],"def_end_pos":[716,20]}]},{"state_before":"case intro.intro.intro\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\na : α\nx : β\nht : t ∈ 𝓝 x\nu : Set β\nut : u ⊆ t\nu_open : IsOpen u\nhu : x ∈ u\n⊢ a • t ∈ 𝓝 (a • x)","state_after":"no goals","tactic":"exact mem_nhds_iff.2 ⟨a • u, smul_set_mono ut, u_open.smul a, smul_mem_smul_set hu⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IsOpen.smul","def_path":"Mathlib/Topology/Algebra/ConstMulAction.lean","def_pos":[220,8],"def_end_pos":[220,19]},{"full_name":"Set.smul_mem_smul_set","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[242,8],"def_end_pos":[242,25]},{"full_name":"Set.smul_set_mono","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[262,8],"def_end_pos":[262,21]},{"full_name":"mem_nhds_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[716,8],"def_end_pos":[716,20]}]}]}
{"url":"Mathlib/Algebra/Homology/Localization.lean","commit":"","full_name":"ComplexShape.QFactorsThroughHomotopy_of_exists_prev","start":[234,0],"end":[241,47],"file_path":"Mathlib/Algebra/Homology/Localization.lean","tactics":[{"state_before":"ι : Type u_1\nc : ComplexShape ι\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nC : Type u_2\ninst✝³ : Category.{u_3, u_2} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : CategoryWithHomology C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\n⊢ AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g","state_after":"ι : Type u_1\nc : ComplexShape ι\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nC : Type u_2\ninst✝³ : Category.{u_3, u_2} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : CategoryWithHomology C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\nthis : DecidableRel c.Rel\n⊢ AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g","tactic":"have : DecidableRel c.Rel := by classical infer_instance","premises":[{"full_name":"ComplexShape.Rel","def_path":"Mathlib/Algebra/Homology/ComplexShape.lean","def_pos":[62,2],"def_end_pos":[62,5]},{"full_name":"DecidableRel","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[861,7],"def_end_pos":[861,19]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"ι : Type u_1\nc : ComplexShape ι\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nC : Type u_2\ninst✝³ : Category.{u_3, u_2} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : CategoryWithHomology C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\nthis : DecidableRel c.Rel\n⊢ AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g","state_after":"no goals","tactic":"exact h.map_eq_of_inverts_homotopyEquivalences hc _\n      (MorphismProperty.IsInvertedBy.of_le _ _ _\n        (Localization.inverts _ (HomologicalComplex.quasiIso C _))\n        (homotopyEquivalences_le_quasiIso C _))","premises":[{"full_name":"CategoryTheory.Localization.inverts","def_path":"Mathlib/CategoryTheory/Localization/Predicate.lean","def_pos":[132,8],"def_end_pos":[132,15]},{"full_name":"CategoryTheory.MorphismProperty.IsInvertedBy.of_le","def_path":"Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean","def_pos":[36,6],"def_end_pos":[36,11]},{"full_name":"HomologicalComplex.quasiIso","def_path":"Mathlib/Algebra/Homology/QuasiIso.lean","def_pos":[270,4],"def_end_pos":[270,12]},{"full_name":"Homotopy.map_eq_of_inverts_homotopyEquivalences","def_path":"Mathlib/Algebra/Homology/HomotopyCofiber.lean","def_pos":[545,6],"def_end_pos":[545,60]},{"full_name":"homotopyEquivalences_le_quasiIso","def_path":"Mathlib/Algebra/Homology/QuasiIso.lean","def_pos":[298,6],"def_end_pos":[298,38]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","commit":"","full_name":"CategoryTheory.exact_f_d","start":[260,0],"end":[271,26],"file_path":"Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\n⊢ f ≫ d f = 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","tactic":"let α : ShortComplex.mk f (cokernel.π f) (by simp) ⟶ ShortComplex.mk f (d f) (by simp) :=\n    { τ₁ := 𝟙 _\n      τ₂ := 𝟙 _\n      τ₃ := Injective.ι _  }","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"CategoryTheory.Injective.d","def_path":"Mathlib/CategoryTheory/Preadditive/Injective.lean","def_pos":[234,7],"def_end_pos":[234,8]},{"full_name":"CategoryTheory.Injective.ι","def_path":"Mathlib/CategoryTheory/Preadditive/Injective.lean","def_pos":[210,4],"def_end_pos":[210,5]},{"full_name":"CategoryTheory.Limits.cokernel.π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[678,7],"def_end_pos":[678,17]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis : Epi α.τ₁\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","tactic":"have : Epi α.τ₁ := by dsimp; infer_instance","premises":[{"full_name":"CategoryTheory.Epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[241,6],"def_end_pos":[241,9]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[54,2],"def_end_pos":[54,4]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis : Epi α.τ₁\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝ : Epi α.τ₁\nthis : IsIso α.τ₂\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","tactic":"have : IsIso α.τ₂ := by dsimp; infer_instance","premises":[{"full_name":"CategoryTheory.IsIso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[229,6],"def_end_pos":[229,11]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[56,2],"def_end_pos":[56,4]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝ : Epi α.τ₁\nthis : IsIso α.τ₂\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","tactic":"have : Mono α.τ₃ := by dsimp; infer_instance","premises":[{"full_name":"CategoryTheory.Mono","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[250,6],"def_end_pos":[250,10]},{"full_name":"CategoryTheory.ShortComplex.Hom.τ₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[58,2],"def_end_pos":[58,4]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ (ShortComplex.mk f (d f) ⋯).Exact","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ (ShortComplex.mk f (cokernel.π f) ⋯).Exact","tactic":"rw [← ShortComplex.exact_iff_of_epi_of_isIso_of_mono α]","premises":[{"full_name":"CategoryTheory.ShortComplex.exact_iff_of_epi_of_isIso_of_mono","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[135,6],"def_end_pos":[135,39]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ (ShortComplex.mk f (cokernel.π f) ⋯).Exact","state_after":"case hS\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ IsColimit (CokernelCofork.ofπ (ShortComplex.mk f (cokernel.π f) ⋯).g ⋯)","tactic":"apply ShortComplex.exact_of_g_is_cokernel","premises":[{"full_name":"CategoryTheory.ShortComplex.exact_of_g_is_cokernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[415,6],"def_end_pos":[415,28]}]},{"state_before":"case hS\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ IsColimit (CokernelCofork.ofπ (ShortComplex.mk f (cokernel.π f) ⋯).g ⋯)","state_after":"no goals","tactic":"apply cokernelIsCokernel","premises":[{"full_name":"CategoryTheory.Limits.cokernelIsCokernel","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[690,4],"def_end_pos":[690,22]}]}]}
{"url":"Mathlib/Probability/Martingale/Centering.lean","commit":"","full_name":"MeasureTheory.predictablePart_bdd_difference","start":[153,0],"end":[157,77],"file_path":"Mathlib/Probability/Martingale/Centering.lean","tactics":[{"state_before":"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ✝ : Filtration ℕ m0\nn : ℕ\nR : ℝ≥0\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\n⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω| ≤ ↑R","state_after":"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ✝ : Filtration ℕ m0\nn : ℕ\nR : ℝ≥0\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\n⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(μ[f (i + 1) - f i|↑ℱ i]) ω| ≤ ↑R","tactic":"simp_rw [predictablePart, Finset.sum_apply, Finset.sum_range_succ_sub_sum]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.sum_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[32,2],"def_end_pos":[32,13]},{"full_name":"Finset.sum_range_succ_sub_sum","def_path":"Mathlib/Algebra/BigOperators/Intervals.lean","def_pos":[277,2],"def_end_pos":[277,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.predictablePart","def_path":"Mathlib/Probability/Martingale/Centering.lean","def_pos":[43,18],"def_end_pos":[43,33]}]},{"state_before":"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ✝ : Filtration ℕ m0\nn : ℕ\nR : ℝ≥0\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\n⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(μ[f (i + 1) - f i|↑ℱ i]) ω| ≤ ↑R","state_after":"no goals","tactic":"exact ae_all_iff.2 fun i => ae_bdd_condexp_of_ae_bdd <| ae_all_iff.1 hbdd i","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.ae_all_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[93,8],"def_end_pos":[93,18]},{"full_name":"MeasureTheory.ae_bdd_condexp_of_ae_bdd","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean","def_pos":[143,8],"def_end_pos":[143,32]}]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.tendsto_mul_const_atBot_iff","start":[1045,0],"end":[1049,55],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\n⊢ Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop","state_after":"no goals","tactic":"simp only [mul_comm _ r, tendsto_const_mul_atBot_iff]","premises":[{"full_name":"Filter.tendsto_const_mul_atBot_iff","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1041,8],"def_end_pos":[1041,35]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Topology/Category/CompHaus/Limits.lean","commit":"","full_name":"CompHaus.pullback_fst_eq","start":[130,0],"end":[133,93],"file_path":"Mathlib/Topology/Category/CompHaus/Limits.lean","tactics":[{"state_before":"X Y B : CompHaus\nf : X ⟶ B\ng : Y ⟶ B\n⊢ pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst f g","state_after":"X Y B : CompHaus\nf : X ⟶ B\ng : Y ⟶ B\n⊢ pullback.fst f g =\n    ((pullback.isLimit f g).conePointUniqueUpToIso (limit.isLimit (cospan f g))).hom ≫ Limits.pullback.fst f g","tactic":"dsimp [pullbackIsoPullback]","premises":[{"full_name":"CompHaus.pullbackIsoPullback","def_path":"Mathlib/Topology/Category/CompHaus/Limits.lean","def_pos":[122,4],"def_end_pos":[122,23]}]},{"state_before":"X Y B : CompHaus\nf : X ⟶ B\ng : Y ⟶ B\n⊢ pullback.fst f g =\n    ((pullback.isLimit f g).conePointUniqueUpToIso (limit.isLimit (cospan f g))).hom ≫ Limits.pullback.fst f g","state_after":"no goals","tactic":"simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]","premises":[{"full_name":"CategoryTheory.Limits.limit.conePointUniqueUpToIso_hom_comp","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[200,8],"def_end_pos":[200,45]},{"full_name":"CompHaus.pullback.cone_pt","def_path":"Mathlib/Topology/Category/CompHaus/Limits.lean","def_pos":[103,9],"def_end_pos":[103,11]},{"full_name":"CompHaus.pullback.cone_π","def_path":"Mathlib/Topology/Category/CompHaus/Limits.lean","def_pos":[103,12],"def_end_pos":[103,13]}]}]}
{"url":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","commit":"","full_name":"Real.fourierIntegral_fderiv","start":[672,0],"end":[678,67],"file_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","tactics":[{"state_before":"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : V → E\nhf : Integrable f volume\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) volume\n⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-innerSL ℝ) (𝓕 f)","state_after":"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : V → E\nhf : Integrable f volume\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) volume\n⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-(innerSL ℝ).flip) (𝓕 f)","tactic":"rw [← innerSL_real_flip V]","premises":[{"full_name":"innerSL_real_flip","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1589,14],"def_end_pos":[1589,31]}]},{"state_before":"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : V → E\nhf : Integrable f volume\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) volume\n⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-(innerSL ℝ).flip) (𝓕 f)","state_after":"no goals","tactic":"exact VectorFourier.fourierIntegral_fderiv (innerSL ℝ) hf h'f hf'","premises":[{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"VectorFourier.fourierIntegral_fderiv","def_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","def_pos":[251,8],"def_end_pos":[251,30]},{"full_name":"innerSL","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1550,4],"def_end_pos":[1550,11]}]}]}
{"url":"Mathlib/NumberTheory/Harmonic/Bounds.lean","commit":"","full_name":"log_add_one_le_harmonic","start":[17,0],"end":[24,81],"file_path":"Mathlib/NumberTheory/Harmonic/Bounds.lean","tactics":[{"state_before":"n : ℕ\n⊢ Real.log ↑(n + 1) ≤ ↑(harmonic n)","state_after":"case calc_1\nn : ℕ\n⊢ Real.log ↑(n + 1) = ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹\n\ncase calc_2\nn : ℕ\n⊢ ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹ ≤ ∑ d ∈ Finset.Icc 1 n, (↑d)⁻¹\n\ncase calc_3\nn : ℕ\n⊢ ∑ d ∈ Finset.Icc 1 n, (↑d)⁻¹ = ↑(harmonic n)","tactic":"calc _ = ∫ x in (1 : ℕ)..↑(n+1), x⁻¹ := ?_\n       _ ≤ ∑ d ∈ Finset.Icc 1 n, (d : ℝ)⁻¹ := ?_\n       _ = harmonic n := ?_","premises":[{"full_name":"Finset.Icc","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[276,4],"def_end_pos":[276,7]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"harmonic","def_path":"Mathlib/NumberTheory/Harmonic/Defs.lean","def_pos":[21,4],"def_end_pos":[21,12]},{"full_name":"intervalIntegral","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[414,4],"def_end_pos":[414,20]}]}]}
{"url":"Mathlib/Data/Seq/Computation.lean","commit":"","full_name":"Computation.tail_think","start":[153,0],"end":[155,59],"file_path":"Mathlib/Data/Seq/Computation.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\n⊢ s.think.tail = s","state_after":"case mk\nα : Type u\nβ : Type v\nγ : Type w\nf : Stream' (Option α)\nal : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a\n⊢ (think ⟨f, al⟩).tail = ⟨f, al⟩","tactic":"cases' s with f al","premises":[]},{"state_before":"case mk\nα : Type u\nβ : Type v\nγ : Type w\nf : Stream' (Option α)\nal : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a\n⊢ (think ⟨f, al⟩).tail = ⟨f, al⟩","state_after":"case mk.a\nα : Type u\nβ : Type v\nγ : Type w\nf : Stream' (Option α)\nal : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a\n⊢ ↑(think ⟨f, al⟩).tail = ↑⟨f, al⟩","tactic":"apply Subtype.eq","premises":[{"full_name":"Subtype.eq","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1110,18],"def_end_pos":[1110,20]}]},{"state_before":"case mk.a\nα : Type u\nβ : Type v\nγ : Type w\nf : Stream' (Option α)\nal : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a\n⊢ ↑(think ⟨f, al⟩).tail = ↑⟨f, al⟩","state_after":"no goals","tactic":"dsimp [tail, think]","premises":[{"full_name":"Computation.tail","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[71,4],"def_end_pos":[71,8]},{"full_name":"Computation.think","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[50,4],"def_end_pos":[50,9]}]}]}
{"url":"Mathlib/RingTheory/ReesAlgebra.lean","commit":"","full_name":"monomial_mem_adjoin_monomial","start":[76,0],"end":[89,38],"file_path":"Mathlib/RingTheory/ReesAlgebra.lean","tactics":[{"state_before":"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI✝ I : Ideal R\nn : ℕ\nr : R\nhr : r ∈ I ^ n\n⊢ (monomial n) r ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)","state_after":"case zero\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI✝ I : Ideal R\nr : R\nhr : r ∈ I ^ 0\n⊢ (monomial 0) r ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)\n\ncase succ\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI✝ I : Ideal R\nn : ℕ\nhn : ∀ {r : R}, r ∈ I ^ n → (monomial n) r ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)\nr : R\nhr : r ∈ I ^ (n + 1)\n⊢ (monomial (n + 1)) r ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)","tactic":"induction' n with n hn generalizing r","premises":[]}]}
{"url":"Mathlib/Data/Nat/Pairing.lean","commit":"","full_name":"Nat.pair_unpair'","start":[52,0],"end":[53,31],"file_path":"Mathlib/Data/Nat/Pairing.lean","tactics":[{"state_before":"n a b : ℕ\nH : unpair n = (a, b)\n⊢ pair a b = n","state_after":"no goals","tactic":"simpa [H] using pair_unpair n","premises":[{"full_name":"Nat.pair_unpair","def_path":"Mathlib/Data/Nat/Pairing.lean","def_pos":[43,8],"def_end_pos":[43,19]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","commit":"","full_name":"EuclideanGeometry.Sphere.dist_div_sin_oangle_div_two_eq_radius","start":[208,0],"end":[218,65],"file_path":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₁ p₃ / |(∡ p₁ p₂ p₃).sin| / 2 = s.radius","state_after":"case h.e'_2.h.e'_5.h.e'_6\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ |(∡ p₁ p₂ p₃).sin| = (∡ p₃ p₁ s.center).cos","tactic":"convert dist_div_cos_oangle_center_div_two_eq_radius hp₁ hp₃ hp₁p₃","premises":[{"full_name":"EuclideanGeometry.Sphere.dist_div_cos_oangle_center_div_two_eq_radius","def_path":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","def_pos":[176,8],"def_end_pos":[176,52]}]},{"state_before":"case h.e'_2.h.e'_5.h.e'_6\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ |(∡ p₁ p₂ p₃).sin| = (∡ p₃ p₁ s.center).cos","state_after":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ |(∡ p₃ p₁ s.center).toReal| ≤ π / 2","tactic":"rw [← Real.Angle.abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi\n    (two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi hp₁ hp₂ hp₃ hp₁p₂.symm hp₂p₃ hp₁p₃),\n    _root_.abs_of_nonneg (Real.Angle.cos_nonneg_iff_abs_toReal_le_pi_div_two.2 _)]","premises":[{"full_name":"EuclideanGeometry.Sphere.two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi","def_path":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","def_pos":[126,8],"def_end_pos":[126,58]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Real.Angle.abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[642,8],"def_end_pos":[642,59]},{"full_name":"Real.Angle.cos_nonneg_iff_abs_toReal_le_pi_div_two","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean","def_pos":[614,8],"def_end_pos":[614,47]},{"full_name":"abs_of_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[76,2],"def_end_pos":[76,13]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ |(∡ p₃ p₁ s.center).toReal| ≤ π / 2","state_after":"no goals","tactic":"exact (abs_oangle_center_right_toReal_lt_pi_div_two hp₁ hp₃).le","premises":[{"full_name":"EuclideanGeometry.Sphere.abs_oangle_center_right_toReal_lt_pi_div_two","def_path":"Mathlib/Geometry/Euclidean/Angle/Sphere.lean","def_pos":[139,8],"def_end_pos":[139,52]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean","commit":"","full_name":"nndist_pi_eq_iff","start":[273,0],"end":[279,35],"file_path":"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\n⊢ nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\n⊢ ((∃ x, ¬nndist (f x) (g x) < r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r) ↔\n    (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r","tactic":"rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]","premises":[{"full_name":"Classical.not_forall","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[137,20],"def_end_pos":[137,30]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"eq_iff_le_not_lt","def_path":"Mathlib/Order/Basic.lean","def_pos":[321,8],"def_end_pos":[321,24]},{"full_name":"nndist_pi_le_iff","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean","def_pos":[265,6],"def_end_pos":[265,22]},{"full_name":"nndist_pi_lt_iff","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean","def_pos":[268,6],"def_end_pos":[268,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\n⊢ ((∃ x, ¬nndist (f x) (g x) < r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r) ↔\n    (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\n⊢ (∀ (b : β), nndist (f b) (g b) ≤ r) →\n    ((∃ x, r ≤ nndist (f x) (g x)) ↔ ∃ i, nndist (f i) (g i) ≤ r ∧ r ≤ nndist (f i) (g i))","tactic":"simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"and_congr_left_iff","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[158,16],"def_end_pos":[158,34]},{"full_name":"le_antisymm_iff","def_path":"Mathlib/Order/Defs.lean","def_pos":[161,8],"def_end_pos":[161,23]},{"full_name":"not_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[312,8],"def_end_pos":[312,14]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\n⊢ (∀ (b : β), nndist (f b) (g b) ≤ r) →\n    ((∃ x, r ≤ nndist (f x) (g x)) ↔ ∃ i, nndist (f i) (g i) ≤ r ∧ r ≤ nndist (f i) (g i))","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\nh : ∀ (b : β), nndist (f b) (g b) ≤ r\n⊢ (∃ x, r ≤ nndist (f x) (g x)) ↔ ∃ i, nndist (f i) (g i) ≤ r ∧ r ≤ nndist (f i) (g i)","tactic":"intro h","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\nh : ∀ (b : β), nndist (f b) (g b) ≤ r\n⊢ (∃ x, r ≤ nndist (f x) (g x)) ↔ ∃ i, nndist (f i) (g i) ≤ r ∧ r ≤ nndist (f i) (g i)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\nh : ∀ (b : β), nndist (f b) (g b) ≤ r\nb : β\n⊢ r ≤ nndist (f b) (g b) ↔ nndist (f b) (g b) ≤ r ∧ r ≤ nndist (f b) (g b)","tactic":"refine exists_congr fun b => ?_","premises":[{"full_name":"exists_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[210,8],"def_end_pos":[210,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nπ : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (π b)\nf g : (b : β) → π b\nr : ℝ≥0\nhr : 0 < r\nh : ∀ (b : β), nndist (f b) (g b) ≤ r\nb : β\n⊢ r ≤ nndist (f b) (g b) ↔ nndist (f b) (g b) ≤ r ∧ r ≤ nndist (f b) (g b)","state_after":"no goals","tactic":"apply (and_iff_right <| h _).symm","premises":[{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"and_iff_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[70,8],"def_end_pos":[70,21]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Pairwise.lean","commit":"","full_name":"List.pairwise_iff_getElem","start":[237,0],"end":[248,28],"file_path":".lake/packages/batteries/Batteries/Data/List/Pairwise.lean","tactics":[{"state_before":"α✝ : Type u_1\nR : α✝ → α✝ → Prop\nl : List α✝\n⊢ Pairwise R l ↔ ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j]","state_after":"α✝ : Type u_1\nR : α✝ → α✝ → Prop\nl : List α✝\n⊢ (∀ {a b : α✝}, [a, b] <+ l → R a b) ↔ ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j]","tactic":"rw [pairwise_iff_forall_sublist]","premises":[{"full_name":"List.pairwise_iff_forall_sublist","def_path":".lake/packages/batteries/Batteries/Data/List/Pairwise.lean","def_pos":[173,8],"def_end_pos":[173,35]}]},{"state_before":"α✝ : Type u_1\nR : α✝ → α✝ → Prop\nl : List α✝\n⊢ (∀ {a b : α✝}, [a, b] <+ l → R a b) ↔ ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j]","state_after":"case mp\nα✝ : Type u_1\nR : α✝ → α✝ → Prop\nl : List α✝\nh : ∀ {a b : α✝}, [a, b] <+ l → R a b\n⊢ ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j]\n\ncase mpr\nα✝ : Type u_1\nR : α✝ → α✝ → Prop\nl : List α✝\nh : ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j]\n⊢ ∀ {a b : α✝}, [a, b] <+ l → R a b","tactic":"constructor <;> intro h","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Types.lean","commit":"","full_name":"CategoryTheory.Limits.Types.Pushout.inr_rel'_inr_iff","start":[806,0],"end":[813,19],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Types.lean","tactics":[{"state_before":"S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nx₂ y₂ : X₂\n⊢ Rel' f g (Sum.inr x₂) (Sum.inr y₂) ↔ x₂ = y₂","state_after":"case mp\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nx₂ y₂ : X₂\n⊢ Rel' f g (Sum.inr x₂) (Sum.inr y₂) → x₂ = y₂\n\ncase mpr\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nx₂ y₂ : X₂\n⊢ x₂ = y₂ → Rel' f g (Sum.inr x₂) (Sum.inr y₂)","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/CategoryTheory/NatIso.lean","commit":"","full_name":"CategoryTheory.NatIso.naturality_2","start":[150,0],"end":[151,6],"file_path":"Mathlib/CategoryTheory/NatIso.lean","tactics":[{"state_before":"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF G : C ⥤ D\nX Y : C\nα : F ≅ G\nf : X ⟶ Y\n⊢ α.hom.app X ≫ G.map f ≫ α.inv.app Y = F.map f","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Fintype/Option.lean","commit":"","full_name":"Fintype.induction_empty_option","start":[84,0],"end":[99,11],"file_path":"Mathlib/Data/Fintype/Option.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nP : (α : Type u) → [inst : Fintype α] → Prop\nof_equiv : ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β), P α → P β\nh_empty : P PEmpty.{u + 1}\nh_option : ∀ (α : Type u) [inst : Fintype α], P α → P (Option α)\nα : Type u\nh_fintype : Fintype α\n⊢ P α","state_after":"case mk\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nP : (α : Type u) → [inst : Fintype α] → Prop\nof_equiv : ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β), P α → P β\nh_empty : P PEmpty.{u + 1}\nh_option : ∀ (α : Type u) [inst : Fintype α], P α → P (Option α)\nα : Type u\nh_fintype : Fintype α\nx✝ : Trunc (∀ (h : Fintype α), P α)\np : ∀ (h : Fintype α), P α\n⊢ P α","tactic":"obtain ⟨p⟩ :=\n    let f_empty := fun i => by convert h_empty\n    let h_option : ∀ {α : Type u} [Fintype α] [DecidableEq α],\n          (∀ (h : Fintype α), P α) → ∀ (h : Fintype (Option α)), P (Option α)  := by\n      rintro α hα - Pα hα'\n      convert h_option α (Pα _)\n    @truncRecEmptyOption (fun α => ∀ h, @P α h) (@fun α β e hα hβ => @of_equiv α β hβ e (hα _))\n      f_empty h_option α _ (Classical.decEq α)","premises":[{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]},{"full_name":"DecidableEq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[868,7],"def_end_pos":[868,18]},{"full_name":"Fintype","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[55,6],"def_end_pos":[55,13]},{"full_name":"Fintype.truncRecEmptyOption","def_path":"Mathlib/Data/Fintype/Option.lean","def_pos":[51,4],"def_end_pos":[51,23]},{"full_name":"Option","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2240,10],"def_end_pos":[2240,16]}]},{"state_before":"case mk\nα✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nP : (α : Type u) → [inst : Fintype α] → Prop\nof_equiv : ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β), P α → P β\nh_empty : P PEmpty.{u + 1}\nh_option : ∀ (α : Type u) [inst : Fintype α], P α → P (Option α)\nα : Type u\nh_fintype : Fintype α\nx✝ : Trunc (∀ (h : Fintype α), P α)\np : ∀ (h : Fintype α), P α\n⊢ P α","state_after":"no goals","tactic":"exact p _","premises":[]}]}
{"url":"Mathlib/Analysis/Fourier/Inversion.lean","commit":"","full_name":"Continuous.fourier_inversion","start":[165,0],"end":[172,47],"file_path":"Mathlib/Analysis/Fourier/Inversion.lean","tactics":[{"state_before":"V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nh : Continuous f\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\n⊢ 𝓕⁻ (𝓕 f) = f","state_after":"case h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nh : Continuous f\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\n⊢ 𝓕⁻ (𝓕 f) v = f v","tactic":"ext v","premises":[]},{"state_before":"case h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nh : Continuous f\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\n⊢ 𝓕⁻ (𝓕 f) v = f v","state_after":"no goals","tactic":"exact hf.fourier_inversion h'f h.continuousAt","premises":[{"full_name":"Continuous.continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1424,8],"def_end_pos":[1424,31]},{"full_name":"MeasureTheory.Integrable.fourier_inversion","def_path":"Mathlib/Analysis/Fourier/Inversion.lean","def_pos":[159,8],"def_end_pos":[159,50]}]}]}
{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"mem_closure_iff_ultrafilter","start":[1163,0],"end":[1167,78],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\n⊢ x ∈ closure s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x","state_after":"no goals","tactic":"simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]","premises":[{"full_name":"ClusterPt","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[239,4],"def_end_pos":[239,13]},{"full_name":"Filter.exists_ultrafilter_iff","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[396,8],"def_end_pos":[396,30]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"closure_eq_cluster_pts","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1119,8],"def_end_pos":[1119,30]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Basic.lean","commit":"","full_name":"UniformSpace.toTopologicalSpace_sInf","start":[1191,0],"end":[1194,39],"file_path":"Mathlib/Topology/UniformSpace/Basic.lean","tactics":[{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ns : Set (UniformSpace α)\n⊢ toTopologicalSpace = ⨅ i ∈ s, toTopologicalSpace","state_after":"no goals","tactic":"simp only [← toTopologicalSpace_iInf]","premises":[{"full_name":"UniformSpace.toTopologicalSpace_iInf","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[1186,8],"def_end_pos":[1186,31]}]}]}
{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.iInter_eq_compl_iUnion_compl","start":[411,0],"end":[412,39],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ns : ι → Set β\n⊢ ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ","state_after":"no goals","tactic":"simp only [compl_iUnion, compl_compl]","premises":[{"full_name":"Set.compl_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[389,8],"def_end_pos":[389,20]},{"full_name":"compl_compl","def_path":"Mathlib/Order/BooleanAlgebra.lean","def_pos":[583,8],"def_end_pos":[583,19]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","commit":"","full_name":"CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_snd","start":[368,0],"end":[373,74],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y✝ X Y : C\nh : IsTerminal X\nc : BinaryFan X Y\n⊢ Nonempty (IsLimit c) ↔ IsIso c.snd","state_after":"C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y✝ X Y : C\nh : IsTerminal X\nc : BinaryFan X Y\n⊢ Nonempty (IsLimit c) ↔ Nonempty (IsLimit (mk c.snd c.fst))","tactic":"refine Iff.trans ?_ (BinaryFan.isLimit_iff_isIso_fst h (BinaryFan.mk c.snd c.fst))","premises":[{"full_name":"CategoryTheory.Limits.BinaryFan.fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[176,7],"def_end_pos":[176,20]},{"full_name":"CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[353,8],"def_end_pos":[353,39]},{"full_name":"CategoryTheory.Limits.BinaryFan.mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[259,4],"def_end_pos":[259,16]},{"full_name":"CategoryTheory.Limits.BinaryFan.snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[180,7],"def_end_pos":[180,20]},{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]}]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y✝ X Y : C\nh : IsTerminal X\nc : BinaryFan X Y\n⊢ Nonempty (IsLimit c) ↔ Nonempty (IsLimit (mk c.snd c.fst))","state_after":"no goals","tactic":"exact\n    ⟨fun h => ⟨BinaryFan.isLimitFlip h.some⟩, fun h =>\n      ⟨(BinaryFan.isLimitFlip h.some).ofIsoLimit (isoBinaryFanMk c).symm⟩⟩","premises":[{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.Limits.BinaryFan.isLimitFlip","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[345,4],"def_end_pos":[345,25]},{"full_name":"CategoryTheory.Limits.IsLimit.ofIsoLimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[157,4],"def_end_pos":[157,14]},{"full_name":"CategoryTheory.Limits.isoBinaryFanMk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[288,4],"def_end_pos":[288,18]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]},{"full_name":"Nonempty.some","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[81,31],"def_end_pos":[81,44]}]}]}
{"url":"Mathlib/Probability/Kernel/CondDistrib.lean","commit":"","full_name":"ProbabilityTheory.condDistrib_apply_of_ne_zero","start":[69,0],"end":[76,36],"file_path":"Mathlib/Probability/Kernel/CondDistrib.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns✝ : Set Ω\nt : Set β\nf : β × Ω → F\ninst✝ : MeasurableSingletonClass β\nhY : Measurable Y\nx : β\nhX : (Measure.map X μ) {x} ≠ 0\ns : Set Ω\n⊢ ((condDistrib Y X μ) x) s = ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)","state_after":"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns✝ : Set Ω\nt : Set β\nf : β × Ω → F\ninst✝ : MeasurableSingletonClass β\nhY : Measurable Y\nx : β\nhX : (Measure.map X μ) {x} ≠ 0\ns : Set Ω\n⊢ ((Measure.map (fun a => (X a, Y a)) μ).fst {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s) =\n    ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)\n\nα : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns✝ : Set Ω\nt : Set β\nf : β × Ω → F\ninst✝ : MeasurableSingletonClass β\nhY : Measurable Y\nx : β\nhX : (Measure.map X μ) {x} ≠ 0\ns : Set Ω\n⊢ (Measure.map (fun a => (X a, Y a)) μ).fst {x} ≠ 0","tactic":"rw [condDistrib, Measure.condKernel_apply_of_ne_zero _ s]","premises":[{"full_name":"MeasureTheory.Measure.condKernel_apply_of_ne_zero","def_path":"Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean","def_pos":[422,6],"def_end_pos":[422,62]},{"full_name":"ProbabilityTheory.condDistrib","def_path":"Mathlib/Probability/Kernel/CondDistrib.lean","def_pos":[60,30],"def_end_pos":[60,41]},{"full_name":"ProbabilityTheory.condDistrib_def","def_path":"Mathlib/Probability/Kernel/CondDistrib.lean","def_pos":[60,30],"def_end_pos":[60,41]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.subsingleton_iff_subsingleton","start":[676,0],"end":[679,19],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ Subsingleton R → Subsingleton R[X]","state_after":"R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\na✝ : Subsingleton R\n⊢ Subsingleton R[X]","tactic":"intro","premises":[]},{"state_before":"R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\na✝ : Subsingleton R\n⊢ Subsingleton R[X]","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.map₂Right_eq_map₂Right'","start":[2561,0],"end":[2562,58],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : Option α → β → γ\na : α\nas : List α\nb : β\nbs : List β\n⊢ map₂Right f as bs = (map₂Right' f as bs).1","state_after":"no goals","tactic":"simp only [map₂Right, map₂Right', map₂Left_eq_map₂Left']","premises":[{"full_name":"List.map₂Left_eq_map₂Left'","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2521,8],"def_end_pos":[2521,29]},{"full_name":"List.map₂Right","def_path":"Mathlib/Data/List/Defs.lean","def_pos":[363,4],"def_end_pos":[363,13]},{"full_name":"List.map₂Right'","def_path":"Mathlib/Data/List/Defs.lean","def_pos":[329,4],"def_end_pos":[329,14]}]}]}
{"url":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","commit":"","full_name":"LinearMap.continuous_of_finiteDimensional","start":[238,0],"end":[254,48],"file_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","tactics":[{"state_before":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\n⊢ Continuous ⇑f","state_after":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\n⊢ Continuous ⇑f","tactic":"let b := Basis.ofVectorSpace 𝕜 E","premises":[{"full_name":"Basis.ofVectorSpace","def_path":"Mathlib/LinearAlgebra/Basis/VectorSpace.lean","def_pos":[99,18],"def_end_pos":[99,31]}]},{"state_before":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\n⊢ Continuous ⇑f","state_after":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\n⊢ Continuous ⇑f","tactic":"have A : Continuous b.equivFun := continuous_equivFun_basis_aux b","premises":[{"full_name":"Basis.equivFun","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[817,4],"def_end_pos":[817,18]},{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"_private.Mathlib.Topology.Algebra.Module.FiniteDimension.0.continuous_equivFun_basis_aux","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[193,16],"def_end_pos":[193,45]}]},{"state_before":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\n⊢ Continuous ⇑f","state_after":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\n⊢ Continuous ⇑f","tactic":"have B : Continuous (f.comp (b.equivFun.symm : (Basis.ofVectorSpaceIndex 𝕜 E → 𝕜) →ₗ[𝕜] E)) :=\n    LinearMap.continuous_on_pi _","premises":[{"full_name":"Basis.equivFun","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[817,4],"def_end_pos":[817,18]},{"full_name":"Basis.ofVectorSpaceIndex","def_path":"Mathlib/LinearAlgebra/Basis/VectorSpace.lean","def_pos":[95,18],"def_end_pos":[95,36]},{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.continuous_on_pi","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[205,8],"def_end_pos":[205,34]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\n⊢ Continuous ⇑f","state_after":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\n⊢ Continuous ⇑f","tactic":"have :\n    Continuous\n      (f.comp (b.equivFun.symm : (Basis.ofVectorSpaceIndex 𝕜 E → 𝕜) →ₗ[𝕜] E) ∘ b.equivFun) :=\n    B.comp A","premises":[{"full_name":"Basis.equivFun","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[817,4],"def_end_pos":[817,18]},{"full_name":"Basis.ofVectorSpaceIndex","def_path":"Mathlib/LinearAlgebra/Basis/VectorSpace.lean","def_pos":[95,18],"def_end_pos":[95,36]},{"full_name":"Continuous","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[141,10],"def_end_pos":[141,20]},{"full_name":"Continuous.comp","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1389,8],"def_end_pos":[1389,23]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"LinearEquiv.symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[258,4],"def_end_pos":[258,8]},{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]}]},{"state_before":"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\n⊢ Continuous ⇑f","state_after":"case h.e'_5\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\n⊢ ⇑f = ⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun","tactic":"convert this","premises":[]},{"state_before":"case h.e'_5\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\n⊢ ⇑f = ⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun","state_after":"case h.e'_5.h\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\nx : E\n⊢ f x = (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun) x","tactic":"ext x","premises":[]},{"state_before":"case h.e'_5.h\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\nx : E\n⊢ f x = (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun) x","state_after":"case h.e'_5.h\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\nx : E\n⊢ f x = f (b.equivFun.symm ⇑(b.repr x))","tactic":"dsimp","premises":[]},{"state_before":"case h.e'_5.h\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\nx : E\n⊢ f x = f (b.equivFun.symm ⇑(b.repr x))","state_after":"no goals","tactic":"rw [Basis.equivFun_symm_apply, Basis.sum_repr]","premises":[{"full_name":"Basis.equivFun_symm_apply","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[840,8],"def_end_pos":[840,33]},{"full_name":"Basis.sum_repr","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[857,8],"def_end_pos":[857,22]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Lipschitz.lean","commit":"","full_name":"continuousAt_of_locally_lipschitz","start":[319,0],"end":[327,6],"file_path":"Mathlib/Topology/MetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ ContinuousAt f x","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ Tendsto (fun a => K * dist a x) (𝓝 x) (𝓝 0)","tactic":"refine tendsto_iff_dist_tendsto_zero.2 (squeeze_zero' (eventually_of_forall fun _ => dist_nonneg)\n    (mem_of_superset (ball_mem_nhds _ hr) h) ?_)","premises":[{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Metric.ball_mem_nhds","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[863,8],"def_end_pos":[863,21]},{"full_name":"dist_nonneg","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[234,8],"def_end_pos":[234,19]},{"full_name":"squeeze_zero'","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean","def_pos":[46,6],"def_end_pos":[46,19]},{"full_name":"tendsto_iff_dist_tendsto_zero","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[1224,8],"def_end_pos":[1224,37]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ Tendsto (fun a => K * dist a x) (𝓝 x) (𝓝 0)","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ K * dist (id x) x = 0","tactic":"refine (continuous_const.mul (continuous_id.dist continuous_const)).tendsto' _ _ ?_","premises":[{"full_name":"Continuous.dist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean","def_pos":[201,16],"def_end_pos":[201,31]},{"full_name":"Continuous.mul","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[94,8],"def_end_pos":[94,22]},{"full_name":"Continuous.tendsto'","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1419,8],"def_end_pos":[1419,27]},{"full_name":"continuous_const","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1436,8],"def_end_pos":[1436,24]},{"full_name":"continuous_id","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1382,8],"def_end_pos":[1382,21]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ K * dist (id x) x = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Complex/Angle.lean","commit":"","full_name":"Complex.angle_one_right","start":[47,0],"end":[47,90],"file_path":"Mathlib/Analysis/Complex/Angle.lean","tactics":[{"state_before":"a x y : ℂ\nhx : x ≠ 0\n⊢ angle x 1 = |x.arg|","state_after":"no goals","tactic":"simp [angle_eq_abs_arg, hx]","premises":[{"full_name":"Complex.angle_eq_abs_arg","def_path":"Mathlib/Analysis/Complex/Angle.lean","def_pos":[37,6],"def_end_pos":[37,22]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.ae_eq_bot","start":[1789,0],"end":[1791,73],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\n⊢ ae μ = ⊥ ↔ μ = 0","state_after":"no goals","tactic":"rw [← empty_mem_iff_bot, mem_ae_iff, compl_empty, measure_univ_eq_zero]","premises":[{"full_name":"Filter.empty_mem_iff_bot","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[620,8],"def_end_pos":[620,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"MeasureTheory.Measure.measure_univ_eq_zero","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1031,8],"def_end_pos":[1031,28]},{"full_name":"MeasureTheory.mem_ae_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[70,8],"def_end_pos":[70,18]},{"full_name":"Set.compl_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1314,8],"def_end_pos":[1314,19]}]}]}
{"url":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","commit":"","full_name":"mul_nonpos_iff_pos_imp_nonpos","start":[975,0],"end":[979,33],"file_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\na b : α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)","state_after":"α : Type u\nβ : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\na b : α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ (0 < a → 0 ≤ -b) ∧ (0 < -b → 0 ≤ a) ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)","tactic":"rw [← neg_nonneg, ← mul_neg, mul_nonneg_iff_pos_imp_nonneg (α := α)]","premises":[{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]},{"full_name":"mul_nonneg_iff_pos_imp_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[798,6],"def_end_pos":[798,35]}]},{"state_before":"α : Type u\nβ : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\na b : α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ (0 < a → 0 ≤ -b) ∧ (0 < -b → 0 ≤ a) ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)","state_after":"no goals","tactic":"simp only [neg_pos, neg_nonneg]","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Field/Basic.lean","commit":"","full_name":"div_le_iff'","start":[65,0],"end":[65,91],"file_path":"Mathlib/Algebra/Order/Field/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhb : 0 < b\n⊢ a / b ≤ c ↔ a ≤ b * c","state_after":"no goals","tactic":"rw [mul_comm, div_le_iff hb]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"div_le_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[51,8],"def_end_pos":[51,18]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Topology/UniformSpace/Ascoli.lean","commit":"","full_name":"Equicontinuous.inducing_uniformFun_iff_pi","start":[143,0],"end":[158,35],"file_path":"Mathlib/Topology/UniformSpace/Ascoli.lean","tactics":[{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ Inducing (⇑UniformFun.ofFun ∘ F) ↔ Inducing F","state_after":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ inst✝¹ = TopologicalSpace.induced (⇑UniformFun.ofFun ∘ F) (UniformFun.topologicalSpace X α) ↔\n    inst✝¹ = TopologicalSpace.induced F Pi.topologicalSpace","tactic":"rw [inducing_iff, inducing_iff]","premises":[{"full_name":"inducing_iff","def_path":"Mathlib/Topology/Defs/Induced.lean","def_pos":[100,2],"def_end_pos":[100,8]}]},{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ inst✝¹ = TopologicalSpace.induced (⇑UniformFun.ofFun ∘ F) (UniformFun.topologicalSpace X α) ↔\n    inst✝¹ = TopologicalSpace.induced F Pi.topologicalSpace","state_after":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ inst✝¹ = UniformSpace.toTopologicalSpace ↔ inst✝¹ = UniformSpace.toTopologicalSpace","tactic":"change (_ = (UniformFun.uniformSpace X α |>.comap F |>.toTopologicalSpace)) ↔\n         (_ = (Pi.uniformSpace _ |>.comap F |>.toTopologicalSpace))","premises":[{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Pi.uniformSpace","def_path":"Mathlib/Topology/UniformSpace/Pi.lean","def_pos":[22,9],"def_end_pos":[22,24]},{"full_name":"UniformFun.uniformSpace","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[285,9],"def_end_pos":[285,21]},{"full_name":"UniformSpace.comap","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[1100,7],"def_end_pos":[1100,25]}]},{"state_before":"ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ inst✝¹ = UniformSpace.toTopologicalSpace ↔ inst✝¹ = UniformSpace.toTopologicalSpace","state_after":"no goals","tactic":"rw [F_eqcont.comap_uniformFun_eq]","premises":[{"full_name":"Equicontinuous.comap_uniformFun_eq","def_path":"Mathlib/Topology/UniformSpace/Ascoli.lean","def_pos":[85,8],"def_end_pos":[85,42]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/GroupTheory/Coxeter/Length.lean","commit":"","full_name":"CoxeterSystem.exists_reduced_word'","start":[208,0],"end":[211,7],"file_path":"Mathlib/GroupTheory/Coxeter/Length.lean","tactics":[{"state_before":"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n⊢ ∃ ω, cs.IsReduced ω ∧ w = cs.wordProd ω","state_after":"case intro.intro\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : ω.length = cs.length (cs.wordProd ω)\n⊢ ∃ ω_1, cs.IsReduced ω_1 ∧ cs.wordProd ω = cs.wordProd ω_1","tactic":"rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩","premises":[{"full_name":"CoxeterSystem.exists_reduced_word","def_path":"Mathlib/GroupTheory/Coxeter/Length.lean","def_pos":[71,8],"def_end_pos":[71,27]}]},{"state_before":"case intro.intro\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : ω.length = cs.length (cs.wordProd ω)\n⊢ ∃ ω_1, cs.IsReduced ω_1 ∧ cs.wordProd ω = cs.wordProd ω_1","state_after":"case h\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : ω.length = cs.length (cs.wordProd ω)\n⊢ cs.IsReduced ω ∧ cs.wordProd ω = cs.wordProd ω","tactic":"use ω","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : ω.length = cs.length (cs.wordProd ω)\n⊢ cs.IsReduced ω ∧ cs.wordProd ω = cs.wordProd ω","state_after":"no goals","tactic":"tauto","premises":[{"full_name":"Classical.or_iff_not_imp_left","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[147,8],"def_end_pos":[147,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"congrArg","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[362,8],"def_end_pos":[362,16]},{"full_name":"congrFun","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[376,8],"def_end_pos":[376,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"trivial","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[645,34],"def_end_pos":[645,41]}]}]}
{"url":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","commit":"","full_name":"DiscreteValuationRing.of_ufd_of_unique_irreducible","start":[265,0],"end":[286,90],"file_path":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","tactics":[{"state_before":"R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ DiscreteValuationRing R","state_after":"R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime","tactic":"rw [iff_pid_with_one_nonzero_prime]","premises":[{"full_name":"DiscreteValuationRing.iff_pid_with_one_nonzero_prime","def_path":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","def_pos":[108,8],"def_end_pos":[108,38]}]},{"state_before":"R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime","state_after":"R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime","tactic":"haveI PID : IsPrincipalIdealRing R := aux_pid_of_ufd_of_unique_irreducible R h₁ h₂","premises":[{"full_name":"DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible","def_path":"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean","def_pos":[234,8],"def_end_pos":[234,44]},{"full_name":"IsPrincipalIdealRing","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[44,6],"def_end_pos":[44,26]}]},{"state_before":"R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime","state_after":"case intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime","tactic":"obtain ⟨p, hp⟩ := h₁","premises":[]},{"state_before":"case intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime","state_after":"case intro.refine_1\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n⊢ span {p} ≠ ⊥\n\ncase intro.refine_2\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n⊢ (span {p}).IsPrime\n\ncase intro.refine_3\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n⊢ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = span {p}","tactic":"refine ⟨PID, ⟨Ideal.span {p}, ⟨?_, ?_⟩, ?_⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.span","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[101,4],"def_end_pos":[101,8]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]}]}
{"url":"Mathlib/CategoryTheory/ComposableArrows.lean","commit":"","full_name":"CategoryTheory.ComposableArrows.Precomp.map_one_one","start":[322,0],"end":[323,62],"file_path":"Mathlib/CategoryTheory/ComposableArrows.lean","tactics":[{"state_before":"C : Type u_1\ninst✝ : Category.{?u.94923, u_1} C\nn m : ℕ\nF G : ComposableArrows C n\nX : C\nf : X ⟶ F.left\n⊢ 1 ≤ 1","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle","start":[354,0],"end":[363,90],"file_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\n⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C","state_after":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C","tactic":"obtain ⟨Z, g, h, H⟩ := Pretriangulated.distinguished_cocone_triangle₁ f.unop","premises":[{"full_name":"CategoryTheory.Pretriangulated.distinguished_cocone_triangle₁","def_path":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","def_pos":[168,6],"def_end_pos":[168,36]},{"full_name":"Quiver.Hom.unop","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[139,4],"def_end_pos":[139,12]}]},{"state_before":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C","state_after":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Triangle.mk f g.op\n      ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op) ∈\n    distinguishedTriangles C","tactic":"refine ⟨_, g.op, (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫\n    (shiftFunctor Cᵒᵖ (1 : ℤ)).map h.op, ?_⟩","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Equivalence.counitIso","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[87,2],"def_end_pos":[87,11]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CategoryTheory.Pretriangulated.opShiftFunctorEquivalence","def_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","def_pos":[139,18],"def_end_pos":[139,43]},{"full_name":"CategoryTheory.shiftFunctor","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[159,4],"def_end_pos":[159,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Opposite","def_path":"Mathlib/Data/Opposite.lean","def_pos":[33,10],"def_end_pos":[33,18]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]}]},{"state_before":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Triangle.mk f g.op\n      ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op) ∈\n    distinguishedTriangles C","state_after":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Opposite.unop\n      ((triangleOpEquivalence C).inverse.obj\n        (Triangle.mk f g.op\n          ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ∈\n    Pretriangulated.distinguishedTriangles","tactic":"simp only [mem_distinguishedTriangles_iff]","premises":[{"full_name":"CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff","def_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","def_pos":[294,6],"def_end_pos":[294,36]}]},{"state_before":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Opposite.unop\n      ((triangleOpEquivalence C).inverse.obj\n        (Triangle.mk f g.op\n          ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ∈\n    Pretriangulated.distinguishedTriangles","state_after":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Opposite.unop\n      ((triangleOpEquivalence C).inverse.obj\n        (Triangle.mk f g.op\n          ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ≅\n    Triangle.mk g f.unop h","tactic":"refine Pretriangulated.isomorphic_distinguished _ H _ ?_","premises":[{"full_name":"CategoryTheory.Pretriangulated.isomorphic_distinguished","def_path":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","def_pos":[65,2],"def_end_pos":[65,26]}]},{"state_before":"case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Opposite.unop\n      ((triangleOpEquivalence C).inverse.obj\n        (Triangle.mk f g.op\n          ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ≅\n    Triangle.mk g f.unop h","state_after":"no goals","tactic":"exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) (by aesop_cat) (by aesop_cat)\n    (Quiver.Hom.op_inj (by simp [shift_unop_opShiftFunctorEquivalence_counitIso_inv_app]))","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]},{"full_name":"CategoryTheory.Pretriangulated.Triangle.isoMk","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[185,4],"def_end_pos":[185,18]},{"full_name":"CategoryTheory.Pretriangulated.shift_unop_opShiftFunctorEquivalence_counitIso_inv_app","def_path":"Mathlib/CategoryTheory/Triangulated/Opposite.lean","def_pos":[182,6],"def_end_pos":[182,60]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]},{"full_name":"Quiver.Hom.op_inj","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[35,8],"def_end_pos":[35,25]}]}]}
{"url":"Mathlib/Computability/NFA.lean","commit":"","full_name":"NFA.pumping_lemma","start":[107,0],"end":[113,37],"file_path":"Mathlib/Computability/NFA.lean","tactics":[{"state_before":"α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ M.accepts\nhlen : Fintype.card (Set σ) ≤ x.length\n⊢ ∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts","state_after":"α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ M.toDFA.accepts\nhlen : Fintype.card (Set σ) ≤ x.length\n⊢ ∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.toDFA.accepts","tactic":"rw [← toDFA_correct] at hx ⊢","premises":[{"full_name":"NFA.toDFA_correct","def_path":"Mathlib/Computability/NFA.lean","def_pos":[102,8],"def_end_pos":[102,21]}]},{"state_before":"α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ M.toDFA.accepts\nhlen : Fintype.card (Set σ) ≤ x.length\n⊢ ∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.toDFA.accepts","state_after":"no goals","tactic":"exact M.toDFA.pumping_lemma hx hlen","premises":[{"full_name":"DFA.pumping_lemma","def_path":"Mathlib/Computability/DFA.lean","def_pos":[152,8],"def_end_pos":[152,21]},{"full_name":"NFA.toDFA","def_path":"Mathlib/Computability/NFA.lean","def_pos":[96,4],"def_end_pos":[96,9]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Cones.lean","commit":"","full_name":"CategoryTheory.Limits.Cones.whiskeringEquivalence_unitIso","start":[371,0],"end":[384,69],"file_path":"Mathlib/CategoryTheory/Limits/Cones.lean","tactics":[{"state_before":"J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u₃\ninst✝¹ : Category.{v₃, u₃} C\nD : Type u₄\ninst✝ : Category.{v₄, u₄} D\nF : J ⥤ C\ne : K ≌ J\ns : Cone (e.functor ⋙ F)\n⊢ ∀ (j : K),\n    (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).π.app j =\n      (Iso.refl (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).pt).hom ≫\n        ((𝟭 (Cone (e.functor ⋙ F))).obj s).π.app j","state_after":"J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u₃\ninst✝¹ : Category.{v₃, u₃} C\nD : Type u₄\ninst✝ : Category.{v₄, u₄} D\nF : J ⥤ C\ne : K ≌ J\ns : Cone (e.functor ⋙ F)\nk : K\n⊢ (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).π.app k =\n    (Iso.refl (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).pt).hom ≫\n      ((𝟭 (Cone (e.functor ⋙ F))).obj s).π.app k","tactic":"intro k","premises":[]},{"state_before":"J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u₃\ninst✝¹ : Category.{v₃, u₃} C\nD : Type u₄\ninst✝ : Category.{v₄, u₄} D\nF : J ⥤ C\ne : K ≌ J\ns : Cone (e.functor ⋙ F)\nk : K\n⊢ (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).π.app k =\n    (Iso.refl (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).pt).hom ≫\n      ((𝟭 (Cone (e.functor ⋙ F))).obj s).π.app k","state_after":"no goals","tactic":"simpa [e.counit_app_functor] using s.w (e.unitInv.app k)","premises":[{"full_name":"CategoryTheory.Equivalence.counit_app_functor","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[156,8],"def_end_pos":[156,26]},{"full_name":"CategoryTheory.Equivalence.unitInv","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[109,7],"def_end_pos":[109,14]},{"full_name":"CategoryTheory.Limits.Cone.w","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[133,8],"def_end_pos":[133,14]},{"full_name":"CategoryTheory.NatTrans.app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[48,2],"def_end_pos":[48,5]}]}]}
{"url":"Mathlib/Algebra/Order/Pointwise.lean","commit":"","full_name":"sInf_inv","start":[59,0],"end":[62,42],"file_path":"Mathlib/Algebra/Order/Pointwise.lean","tactics":[{"state_before":"α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ sInf s⁻¹ = (sSup s)⁻¹","state_after":"α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ ⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹","tactic":"rw [← image_inv, sInf_image]","premises":[{"full_name":"Set.image_inv","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[208,8],"def_end_pos":[208,17]},{"full_name":"sInf_image","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1148,8],"def_end_pos":[1148,18]}]},{"state_before":"α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ ⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹","state_after":"no goals","tactic":"exact ((OrderIso.inv α).map_sSup _).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"OrderIso.inv","def_path":"Mathlib/Algebra/Order/Group/OrderIso.lean","def_pos":[36,4],"def_end_pos":[36,16]},{"full_name":"OrderIso.map_sSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[788,8],"def_end_pos":[788,25]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","commit":"","full_name":"hasDerivWithinAt_iff_tendsto_slope'","start":[68,0],"end":[70,68],"file_path":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nhs : x ∉ s\n⊢ HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s] x) (𝓝 f')","state_after":"no goals","tactic":"rw [hasDerivWithinAt_iff_tendsto_slope, diff_singleton_eq_self hs]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.diff_singleton_eq_self","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1613,8],"def_end_pos":[1613,30]},{"full_name":"hasDerivWithinAt_iff_tendsto_slope","def_path":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","def_pos":[63,8],"def_end_pos":[63,42]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Comp.lean","commit":"","full_name":"HasDerivAt.scomp_hasDerivWithinAt_of_eq","start":[125,0],"end":[128,53],"file_path":"Mathlib/Analysis/Calculus/Deriv/Comp.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\ny : 𝕜'\nhg : HasDerivAt g₁ g₁' y\nhh : HasDerivWithinAt h h' s x\nhy : y = h x\n⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x","state_after":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\ny : 𝕜'\nhg : HasDerivAt g₁ g₁' (h x)\nhh : HasDerivWithinAt h h' s x\nhy : y = h x\n⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x","tactic":"rw [hy] at hg","premises":[]},{"state_before":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\ny : 𝕜'\nhg : HasDerivAt g₁ g₁' (h x)\nhh : HasDerivWithinAt h h' s x\nhy : y = h x\n⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x","state_after":"no goals","tactic":"exact hg.scomp_hasDerivWithinAt x hh","premises":[{"full_name":"HasDerivAt.scomp_hasDerivWithinAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Comp.lean","def_pos":[121,8],"def_end_pos":[121,41]}]}]}
{"url":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","commit":"","full_name":"Associates.pow_factors","start":[1684,0],"end":[1690,45],"file_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero α\ninst✝³ : UniqueFactorizationMonoid α\ninst✝² : DecidableEq (Associates α)\ninst✝¹ : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : Nontrivial α\na : Associates α\nk : ℕ\n⊢ (a ^ k).factors = k • a.factors","state_after":"case zero\nα : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero α\ninst✝³ : UniqueFactorizationMonoid α\ninst✝² : DecidableEq (Associates α)\ninst✝¹ : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : Nontrivial α\na : Associates α\n⊢ (a ^ 0).factors = 0 • a.factors\n\ncase succ\nα : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero α\ninst✝³ : UniqueFactorizationMonoid α\ninst✝² : DecidableEq (Associates α)\ninst✝¹ : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : Nontrivial α\na : Associates α\nn : ℕ\nh : (a ^ n).factors = n • a.factors\n⊢ (a ^ (n + 1)).factors = (n + 1) • a.factors","tactic":"induction' k with n h","premises":[]}]}
{"url":"Mathlib/RingTheory/Algebraic.lean","commit":"","full_name":"Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero","start":[422,0],"end":[432,42],"file_path":"Mathlib/RingTheory/Algebraic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ (↑x)⁻¹ ∈ A","state_after":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ (↑x)⁻¹ = (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX)","tactic":"suffices (x⁻¹ : L) = (-p.coeff 0)⁻¹ • aeval x (divX p) by\n    rw [this]\n    exact A.smul_mem (aeval x _).2 _","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Polynomial.aeval","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[227,4],"def_end_pos":[227,9]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.divX","def_path":"Mathlib/Algebra/Polynomial/Inductions.lean","def_pos":[37,4],"def_end_pos":[37,8]},{"full_name":"Subalgebra.smul_mem","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[105,8],"def_end_pos":[105,16]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ (↑x)⁻¹ = (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX)","state_after":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ (↑x)⁻¹ = (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX)","tactic":"have : aeval (x : L) p = 0 := by rw [Subalgebra.aeval_coe, aeval_eq, Subalgebra.coe_zero]","premises":[{"full_name":"Polynomial.aeval","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[227,4],"def_end_pos":[227,9]},{"full_name":"Subalgebra.aeval_coe","def_path":"Mathlib/RingTheory/Polynomial/Tower.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"Subalgebra.coe_zero","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[300,18],"def_end_pos":[300,26]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ (↑x)⁻¹ = (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX)","state_after":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ -(((algebraMap K L) (p.coeff 0))⁻¹ * (aeval ↑x) p.divX) = (algebraMap K L) (-p.coeff 0)⁻¹ * ↑((aeval x) p.divX)","tactic":"rw [inv_eq_of_root_of_coeff_zero_ne_zero this coeff_zero_ne, div_eq_inv_mul, Algebra.smul_def]","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"div_eq_inv_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[522,8],"def_end_pos":[522,22]},{"full_name":"inv_eq_of_root_of_coeff_zero_ne_zero","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[412,8],"def_end_pos":[412,44]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ -(((algebraMap K L) (p.coeff 0))⁻¹ * (aeval ↑x) p.divX) = (algebraMap K L) (-p.coeff 0)⁻¹ * ↑((aeval x) p.divX)","state_after":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ -(((algebraMap K L) (p.coeff 0))⁻¹ * ↑((aeval x) p.divX)) = (algebraMap K L) (-p.coeff 0)⁻¹ * ↑((aeval x) p.divX)","tactic":"simp only [aeval_coe, Submonoid.coe_mul, Subsemiring.coe_toSubmonoid, coe_toSubsemiring,\n    coe_algebraMap]","premises":[{"full_name":"Subalgebra.aeval_coe","def_path":"Mathlib/RingTheory/Polynomial/Tower.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"Subalgebra.coe_algebraMap","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[315,8],"def_end_pos":[315,22]},{"full_name":"Subalgebra.coe_toSubsemiring","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[59,8],"def_end_pos":[59,25]},{"full_name":"Submonoid.coe_mul","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[496,8],"def_end_pos":[496,15]},{"full_name":"Subsemiring.coe_toSubmonoid","def_path":"Mathlib/Algebra/Ring/Subsemiring/Basic.lean","def_pos":[337,8],"def_end_pos":[337,23]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ -(((algebraMap K L) (p.coeff 0))⁻¹ * ↑((aeval x) p.divX)) = (algebraMap K L) (-p.coeff 0)⁻¹ * ↑((aeval x) p.divX)","state_after":"no goals","tactic":"rw [map_inv₀, map_neg, inv_neg, neg_mul]","premises":[{"full_name":"inv_neg","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[112,8],"def_end_pos":[112,15]},{"full_name":"map_inv₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean","def_pos":[59,8],"def_end_pos":[59,16]},{"full_name":"map_neg","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[395,2],"def_end_pos":[395,13]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Content.lean","commit":"","full_name":"MeasureTheory.Content.outerMeasure_caratheodory","start":[294,0],"end":[300,28],"file_path":"Mathlib/MeasureTheory/Measure/Content.lean","tactics":[{"state_before":"G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ MeasurableSet A ↔ ∀ (U : Opens G), μ.outerMeasure (↑U ∩ A) + μ.outerMeasure (↑U \\ A) ≤ μ.outerMeasure ↑U","state_after":"G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ MeasurableSet A ↔\n    ∀ (U : Set G) (hU : IsOpen U),\n      μ.outerMeasure (↑{ carrier := U, is_open' := hU } ∩ A) + μ.outerMeasure (↑{ carrier := U, is_open' := hU } \\ A) ≤\n        μ.outerMeasure ↑{ carrier := U, is_open' := hU }","tactic":"rw [Opens.forall]","premises":[{"full_name":"TopologicalSpace.Opens.forall","def_path":"Mathlib/Topology/Sets/Opens.lean","def_pos":[78,8],"def_end_pos":[78,16]}]},{"state_before":"G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ MeasurableSet A ↔\n    ∀ (U : Set G) (hU : IsOpen U),\n      μ.outerMeasure (↑{ carrier := U, is_open' := hU } ∩ A) + μ.outerMeasure (↑{ carrier := U, is_open' := hU } \\ A) ≤\n        μ.outerMeasure ↑{ carrier := U, is_open' := hU }","state_after":"case msU\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),\n    μ.innerContent { carrier := ⋃ i, f i, is_open' := ⋯ } ≤ ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := ⋯ }\n\ncase m_mono\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ ∀ ⦃s₁ s₂ : Set G⦄ (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂),\n    s₁ ⊆ s₂ → μ.innerContent { carrier := s₁, is_open' := hs₁ } ≤ μ.innerContent { carrier := s₂, is_open' := hs₂ }\n\ncase PU\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ ∀ ⦃f : ℕ → Set G⦄, (∀ (i : ℕ), IsOpen (f i)) → IsOpen (⋃ i, f i)","tactic":"apply inducedOuterMeasure_caratheodory","premises":[{"full_name":"MeasureTheory.inducedOuterMeasure_caratheodory","def_path":"Mathlib/MeasureTheory/OuterMeasure/Induced.lean","def_pos":[218,8],"def_end_pos":[218,40]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Count.lean","commit":"","full_name":"List.countP_map","start":[100,0],"end":[103,79],"file_path":".lake/packages/batteries/Batteries/Data/List/Count.lean","tactics":[{"state_before":"α : Type u_2\np✝ q : α → Bool\nβ : Type u_1\np : β → Bool\nf : α → β\na : α\nl : List α\n⊢ countP p (map f (a :: l)) = countP (p ∘ f) (a :: l)","state_after":"α : Type u_2\np✝ q : α → Bool\nβ : Type u_1\np : β → Bool\nf : α → β\na : α\nl : List α\n⊢ (countP (p ∘ f) l + if p (f a) = true then 1 else 0) = countP (p ∘ f) l + if (p ∘ f) a = true then 1 else 0","tactic":"rw [map_cons, countP_cons, countP_cons, countP_map p f l]","premises":[{"full_name":"List.countP_cons","def_path":".lake/packages/batteries/Batteries/Data/List/Count.lean","def_pos":[44,8],"def_end_pos":[44,19]},{"full_name":"List.map_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[366,16],"def_end_pos":[366,24]}]},{"state_before":"α : Type u_2\np✝ q : α → Bool\nβ : Type u_1\np : β → Bool\nf : α → β\na : α\nl : List α\n⊢ (countP (p ∘ f) l + if p (f a) = true then 1 else 0) = countP (p ∘ f) l + if (p ∘ f) a = true then 1 else 0","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Finsupp/WellFounded.lean","commit":"","full_name":"Finsupp.Lex.acc","start":[32,0],"end":[40,43],"file_path":"Mathlib/Data/Finsupp/WellFounded.lean","tactics":[{"state_before":"α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ a ∈ x.support, Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (Finsupp.Lex r s) x","state_after":"α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ a ∈ x.support, Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (InvImage (DFinsupp.Lex r fun x => s) toDFinsupp) x","tactic":"rw [lex_eq_invImage_dfinsupp_lex]","premises":[{"full_name":"Finsupp.lex_eq_invImage_dfinsupp_lex","def_path":"Mathlib/Data/Finsupp/Lex.lean","def_pos":[42,8],"def_end_pos":[42,36]}]},{"state_before":"α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ a ∈ x.support, Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (InvImage (DFinsupp.Lex r fun x => s) toDFinsupp) x","state_after":"no goals","tactic":"classical\n    refine InvImage.accessible toDFinsupp (DFinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ ?_)\n    simpa only [toDFinsupp_support] using h","premises":[{"full_name":"DFinsupp.Lex.acc","def_path":"Mathlib/Data/DFinsupp/WellFounded.lean","def_pos":[148,8],"def_end_pos":[148,15]},{"full_name":"Finsupp.toDFinsupp","def_path":"Mathlib/Data/Finsupp/ToDFinsupp.lean","def_pos":[69,4],"def_end_pos":[69,22]},{"full_name":"InvImage.accessible","def_path":".lake/packages/lean4/src/lean/Init/WF.lean","def_pos":[139,8],"def_end_pos":[139,18]},{"full_name":"toDFinsupp_support","def_path":"Mathlib/Data/Finsupp/ToDFinsupp.lean","def_pos":[92,8],"def_end_pos":[92,26]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Category.lean","commit":"","full_name":"CategoryTheory.MonoidalCategory.associator_naturality_left","start":[451,0],"end":[453,73],"file_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","tactics":[{"state_before":"C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z✝ X X' : C\nf : X ⟶ X'\nY Z : C\n⊢ f ▷ Y ▷ Z ≫ (α_ X' Y Z).hom = (α_ X Y Z).hom ≫ f ▷ (Y ⊗ Z)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Topology/Algebra/InfiniteSum/Real.lean","commit":"","full_name":"summable_of_sum_range_le","start":[86,0],"end":[90,39],"file_path":"Mathlib/Topology/Algebra/InfiniteSum/Real.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : PseudoMetricSpace α\nf✝ : ℕ → α\na : α\nf : ℕ → ℝ\nc : ℝ\nhf : ∀ (n : ℕ), 0 ≤ f n\nh : ∀ (n : ℕ), ∑ i ∈ range n, f i ≤ c\n⊢ Summable f","state_after":"α : Type u_1\nβ : Type u_2\ninst✝ : PseudoMetricSpace α\nf✝ : ℕ → α\na : α\nf : ℕ → ℝ\nc : ℝ\nhf : ∀ (n : ℕ), 0 ≤ f n\nh : ∀ (n : ℕ), ∑ i ∈ range n, f i ≤ c\nH : Tendsto (fun n => ∑ i ∈ range n, f i) atTop atTop\n⊢ False","tactic":"refine (summable_iff_not_tendsto_nat_atTop_of_nonneg hf).2 fun H => ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"summable_iff_not_tendsto_nat_atTop_of_nonneg","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Real.lean","def_pos":[63,8],"def_end_pos":[63,52]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : PseudoMetricSpace α\nf✝ : ℕ → α\na : α\nf : ℕ → ℝ\nc : ℝ\nhf : ∀ (n : ℕ), 0 ≤ f n\nh : ∀ (n : ℕ), ∑ i ∈ range n, f i ≤ c\nH : Tendsto (fun n => ∑ i ∈ range n, f i) atTop atTop\n⊢ False","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝ : PseudoMetricSpace α\nf✝ : ℕ → α\na : α\nf : ℕ → ℝ\nc : ℝ\nhf : ∀ (n : ℕ), 0 ≤ f n\nh : ∀ (n : ℕ), ∑ i ∈ range n, f i ≤ c\nH : Tendsto (fun n => ∑ i ∈ range n, f i) atTop atTop\nn : ℕ\nhn : c < ∑ i ∈ range n, f i\n⊢ False","tactic":"rcases exists_lt_of_tendsto_atTop H 0 c with ⟨n, -, hn⟩","premises":[{"full_name":"Filter.exists_lt_of_tendsto_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[501,8],"def_end_pos":[501,34]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝ : PseudoMetricSpace α\nf✝ : ℕ → α\na : α\nf : ℕ → ℝ\nc : ℝ\nhf : ∀ (n : ℕ), 0 ≤ f n\nh : ∀ (n : ℕ), ∑ i ∈ range n, f i ≤ c\nH : Tendsto (fun n => ∑ i ∈ range n, f i) atTop atTop\nn : ℕ\nhn : c < ∑ i ∈ range n, f i\n⊢ False","state_after":"no goals","tactic":"exact lt_irrefl _ (hn.trans_le (h n))","premises":[{"full_name":"lt_irrefl","def_path":"Mathlib/Order/Defs.lean","def_pos":[65,8],"def_end_pos":[65,17]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.omega_isLimit","start":[2128,0],"end":[2131,39],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"o : Ordinal.{u_1}\nh : o < ω\n⊢ succ o < ω","state_after":"o : Ordinal.{u_1}\nh : o < ω\nn : ℕ\ne : o = ↑n\n⊢ succ o < ω","tactic":"let ⟨n, e⟩ := lt_omega.1 h","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Ordinal.lt_omega","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[2114,8],"def_end_pos":[2114,16]}]},{"state_before":"o : Ordinal.{u_1}\nh : o < ω\nn : ℕ\ne : o = ↑n\n⊢ succ o < ω","state_after":"o : Ordinal.{u_1}\nh : o < ω\nn : ℕ\ne : o = ↑n\n⊢ succ ↑n < ω","tactic":"rw [e]","premises":[]},{"state_before":"o : Ordinal.{u_1}\nh : o < ω\nn : ℕ\ne : o = ↑n\n⊢ succ ↑n < ω","state_after":"no goals","tactic":"exact nat_lt_omega (n + 1)","premises":[{"full_name":"Ordinal.nat_lt_omega","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[2117,8],"def_end_pos":[2117,20]}]}]}
{"url":"Mathlib/Data/Set/Pointwise/Interval.lean","commit":"","full_name":"Set.image_const_sub_Iic","start":[355,0],"end":[358,39],"file_path":"Mathlib/Data/Set/Pointwise/Interval.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a - x) '' Iic b = Ici (a - b)","state_after":"α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iic b = Ici (a - b)","tactic":"have := image_comp (fun x => a + x) fun x => -x","premises":[{"full_name":"Set.image_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[227,8],"def_end_pos":[227,18]}]},{"state_before":"α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iic b = Ici (a - b)","state_after":"α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iic b = Ici (a - b)","tactic":"dsimp [Function.comp_def] at this","premises":[{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]}]},{"state_before":"α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iic b = Ici (a - b)","state_after":"no goals","tactic":"simp [sub_eq_add_neg, this, add_comm]","premises":[{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","commit":"","full_name":"Complex.tendstoUniformlyOn_deriv_of_cthickening_subset","start":[108,0],"end":[119,80],"file_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","tactics":[{"state_before":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","state_after":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","tactic":"have h1 : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K) := by\n    filter_upwards [hF] with n h using h.continuousOn.mono hKU","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ContinuousOn","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[164,4],"def_end_pos":[164,16]},{"full_name":"ContinuousOn.mono","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[798,8],"def_end_pos":[798,25]},{"full_name":"DifferentiableOn.continuousOn","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[715,8],"def_end_pos":[715,37]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Metric.cthickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[177,4],"def_end_pos":[177,15]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","state_after":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","tactic":"have h2 : IsCompact (cthickening δ K) := hK.cthickening","premises":[{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.cthickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[279,18],"def_end_pos":[279,46]},{"full_name":"Metric.cthickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[177,4],"def_end_pos":[177,15]}]},{"state_before":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","state_after":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\nh3 : TendstoUniformlyOn F f φ (cthickening δ K)\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","tactic":"have h3 : TendstoUniformlyOn F f φ (cthickening δ K) :=\n    (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hf (cthickening δ K) hKU h2","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Metric.cthickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[177,4],"def_end_pos":[177,15]},{"full_name":"TendstoUniformlyOn","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[98,4],"def_end_pos":[98,22]},{"full_name":"tendstoLocallyUniformlyOn_iff_forall_isCompact","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[668,8],"def_end_pos":[668,54]}]},{"state_before":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\nh3 : TendstoUniformlyOn F f φ (cthickening δ K)\n⊢ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K","state_after":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\nh3 : TendstoUniformlyOn F f φ (cthickening δ K)\n⊢ ∀ᶠ (n : ι) in φ, EqOn ((cderiv δ ∘ F) n) ((deriv ∘ F) n) K","tactic":"apply (h3.cderiv hδ h1).congr","premises":[{"full_name":"TendstoUniformlyOn.cderiv","def_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","def_pos":[87,8],"def_end_pos":[87,40]},{"full_name":"TendstoUniformlyOn.congr","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[191,8],"def_end_pos":[191,32]}]},{"state_before":"E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\nh3 : TendstoUniformlyOn F f φ (cthickening δ K)\n⊢ ∀ᶠ (n : ι) in φ, EqOn ((cderiv δ ∘ F) n) ((deriv ∘ F) n) K","state_after":"case h\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz✝ : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\nh3 : TendstoUniformlyOn F f φ (cthickening δ K)\nn : ι\nh : DifferentiableOn ℂ (F n) U\nz : ℂ\nhz : z ∈ K\n⊢ (cderiv δ ∘ F) n z = (deriv ∘ F) n z","tactic":"filter_upwards [hF] with n h z hz","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz✝ : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nδ : ℝ\nhδ : 0 < δ\nhK : IsCompact K\nhU : IsOpen U\nhKU : cthickening δ K ⊆ U\nh1 : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nh2 : IsCompact (cthickening δ K)\nh3 : TendstoUniformlyOn F f φ (cthickening δ K)\nn : ι\nh : DifferentiableOn ℂ (F n) U\nz : ℂ\nhz : z ∈ K\n⊢ (cderiv δ ∘ F) n z = (deriv ∘ F) n z","state_after":"no goals","tactic":"exact cderiv_eq_deriv hU h hδ ((closedBall_subset_cthickening hz δ).trans hKU)","premises":[{"full_name":"Complex.cderiv_eq_deriv","def_path":"Mathlib/Analysis/Complex/LocallyUniformLimit.lean","def_pos":[42,8],"def_end_pos":[42,23]},{"full_name":"Metric.closedBall_subset_cthickening","def_path":"Mathlib/Topology/MetricSpace/Thickening.lean","def_pos":[517,8],"def_end_pos":[517,37]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.Triangle.isZero₃_of_isZero₁₂","start":[352,0],"end":[356,48],"file_path":"Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nh₁ : IsZero T.obj₁\nh₂ : IsZero T.obj₂\n⊢ IsZero T.rotate.obj₃","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nh₁ : IsZero T.obj₁\nh₂ : IsZero T.obj₂\n⊢ IsZero ((CategoryTheory.shiftFunctor C 1).obj T.obj₁)","tactic":"dsimp","premises":[]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nh₁ : IsZero T.obj₁\nh₂ : IsZero T.obj₂\n⊢ IsZero ((CategoryTheory.shiftFunctor C 1).obj T.obj₁)","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nh₂ : IsZero T.obj₂\nh₁ : 𝟙 T.obj₁ = 0\n⊢ 𝟙 ((CategoryTheory.shiftFunctor C 1).obj T.obj₁) = 0","tactic":"simp only [IsZero.iff_id_eq_zero] at h₁ ⊢","premises":[{"full_name":"CategoryTheory.Limits.IsZero.iff_id_eq_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[168,8],"def_end_pos":[168,22]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nh₂ : IsZero T.obj₂\nh₁ : 𝟙 T.obj₁ = 0\n⊢ 𝟙 ((CategoryTheory.shiftFunctor C 1).obj T.obj₁) = 0","state_after":"no goals","tactic":"rw [← Functor.map_id, h₁, Functor.map_zero]","premises":[{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Functor.map_zero","def_path":"Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean","def_pos":[49,18],"def_end_pos":[49,26]}]}]}
{"url":"Mathlib/Algebra/AddConstMap/Basic.lean","commit":"","full_name":"AddConstMapClass.map_nat'","start":[112,0],"end":[115,32],"file_path":"Mathlib/Algebra/AddConstMap/Basic.lean","tactics":[{"state_before":"F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddMonoidWithOne G\ninst✝¹ : AddMonoid H\ninst✝ : AddConstMapClass F G H 1 b\nf : F\nn : ℕ\n⊢ f ↑n = f 0 + n • b","state_after":"no goals","tactic":"simpa using map_add_nat' f 0 n","premises":[{"full_name":"AddConstMapClass.map_add_nat'","def_path":"Mathlib/Algebra/AddConstMap/Basic.lean","def_pos":[79,8],"def_end_pos":[79,20]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","commit":"","full_name":"Substring.Valid.toString_extract","start":[1026,0],"end":[1033,58],"file_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","tactics":[{"state_before":"b e : Pos\nx✝ : Substring\nh₁ : x✝.Valid\nh₂ : { str := x✝.toString, startPos := b, stopPos := e }.Valid\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","state_after":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nh₂ : { str := x✝.toString, startPos := b, stopPos := e }.Valid\nl m r : List Char\nh₁ : ValidFor l m r x✝\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","tactic":"let ⟨l, m, r, h₁⟩ := h₁.validFor","premises":[{"full_name":"Substring.Valid.validFor","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[939,8],"def_end_pos":[939,16]}]},{"state_before":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nh₂ : { str := x✝.toString, startPos := b, stopPos := e }.Valid\nl m r : List Char\nh₁ : ValidFor l m r x✝\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","state_after":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nl m : List Char\nh₂ : { str := { data := m }, startPos := b, stopPos := e }.Valid\nr : List Char\nh₁ : ValidFor l m r x✝\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","tactic":"rw [h₁.toString] at h₂","premises":[{"full_name":"Substring.ValidFor.toString","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[794,8],"def_end_pos":[794,16]}]},{"state_before":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nl m : List Char\nh₂ : { str := { data := m }, startPos := b, stopPos := e }.Valid\nr : List Char\nh₁ : ValidFor l m r x✝\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","state_after":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nl m : List Char\nh₂✝ : { str := { data := m }, startPos := b, stopPos := e }.Valid\nr : List Char\nh₁ : ValidFor l m r x✝\nml mm mr : List Char\nh₂ : ValidFor ml mm mr { str := { data := m }, startPos := b, stopPos := e }\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","tactic":"let ⟨ml, mm, mr, h₂⟩ := h₂.validFor","premises":[{"full_name":"Substring.Valid.validFor","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[939,8],"def_end_pos":[939,16]}]},{"state_before":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nl m : List Char\nh₂✝ : { str := { data := m }, startPos := b, stopPos := e }.Valid\nr : List Char\nh₁ : ValidFor l m r x✝\nml mm mr : List Char\nh₂ : ValidFor ml mm mr { str := { data := m }, startPos := b, stopPos := e }\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","state_after":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nl m : List Char\nh₂✝ : { str := { data := m }, startPos := b, stopPos := e }.Valid\nr : List Char\nh₁ : ValidFor l m r x✝\nml mm mr : List Char\nh₂ : ValidFor ml mm mr { str := { data := m }, startPos := b, stopPos := e }\nl' r' : List Char\nh₃ : ValidFor l' mm r' (x✝.extract b e)\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","tactic":"have ⟨l', r', h₃⟩ := h₁.extract h₂","premises":[{"full_name":"Substring.ValidFor.extract","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[887,8],"def_end_pos":[887,15]}]},{"state_before":"b e : Pos\nx✝ : Substring\nh₁✝ : x✝.Valid\nl m : List Char\nh₂✝ : { str := { data := m }, startPos := b, stopPos := e }.Valid\nr : List Char\nh₁ : ValidFor l m r x✝\nml mm mr : List Char\nh₂ : ValidFor ml mm mr { str := { data := m }, startPos := b, stopPos := e }\nl' r' : List Char\nh₃ : ValidFor l' mm r' (x✝.extract b e)\n⊢ (x✝.extract b e).toString = x✝.toString.extract b e","state_after":"no goals","tactic":"rw [h₃.toString, h₁.toString, ← h₂.toString, toString]","premises":[{"full_name":"Substring.ValidFor.toString","def_path":".lake/packages/batteries/Batteries/Data/String/Lemmas.lean","def_pos":[794,8],"def_end_pos":[794,16]},{"full_name":"Substring.toString","def_path":".lake/packages/lean4/src/lean/Init/Data/String/Basic.lean","def_pos":[835,14],"def_end_pos":[835,22]}]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","commit":"","full_name":"cfc_comp_star","start":[561,0],"end":[564,35],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : TopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R p\nf✝ g : R → R\na✝ : A\nha✝ : autoParam (p a✝) _auto✝\nhf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝\nhg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝\ninst✝ : UniqueContinuousFunctionalCalculus R A\nf : R → R\na : A\nhf : autoParam (ContinuousOn f (star '' spectrum R a)) _auto✝\nha : autoParam (p a) _auto✝\n⊢ cfc (fun x => f (star x)) a = cfc f (star a)","state_after":"no goals","tactic":"rw [cfc_comp' .., cfc_star_id ..]","premises":[{"full_name":"cfc_comp'","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","def_pos":[540,6],"def_end_pos":[540,15]},{"full_name":"cfc_star_id","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","def_pos":[491,6],"def_end_pos":[491,17]}]}]}
{"url":"Mathlib/NumberTheory/NumberField/Embeddings.lean","commit":"","full_name":"NumberField.InfinitePlace.card_add_two_mul_card_eq_rank","start":[581,0],"end":[585,33],"file_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","tactics":[{"state_before":"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ NrRealPlaces K + 2 * NrComplexPlaces K = finrank ℚ K","state_after":"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ card { φ // ComplexEmbedding.IsReal φ } ≤ card (K →+* ℂ)","tactic":"rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,\n    ← Embeddings.card K ℂ, Nat.add_sub_of_le]","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Fintype.card_subtype_compl","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[768,8],"def_end_pos":[768,34]},{"full_name":"Nat.add_sub_of_le","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[882,8],"def_end_pos":[882,21]},{"full_name":"NumberField.Embeddings.card","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[53,8],"def_end_pos":[53,12]},{"full_name":"NumberField.InfinitePlace.card_complex_embeddings","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[564,8],"def_end_pos":[564,31]},{"full_name":"NumberField.InfinitePlace.card_real_embeddings","def_path":"Mathlib/NumberTheory/NumberField/Embeddings.lean","def_pos":[555,8],"def_end_pos":[555,28]}]},{"state_before":"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ card { φ // ComplexEmbedding.IsReal φ } ≤ card (K →+* ℂ)","state_after":"no goals","tactic":"exact Fintype.card_subtype_le _","premises":[{"full_name":"Fintype.card_subtype_le","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[753,8],"def_end_pos":[753,31]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","commit":"","full_name":"CategoryTheory.ShortComplex.Homotopy.trans_h₁","start":[432,0],"end":[441,55],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ φ₁.τ₁ = S₁.f ≫ (h₁₂.h₁ + h₂₃.h₁) + (h₁₂.h₀ + h₂₃.h₀) + φ₃.τ₁","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ S₁.f ≫ h₁₂.h₁ + h₁₂.h₀ + (S₁.f ≫ h₂₃.h₁ + h₂₃.h₀ + φ₃.τ₁) = S₁.f ≫ h₁₂.h₁ + S₁.f ≫ h₂₃.h₁ + (h₁₂.h₀ + h₂₃.h₀) + φ₃.τ₁","tactic":"rw [h₁₂.comm₁, h₂₃.comm₁, comp_add]","premises":[{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]},{"full_name":"CategoryTheory.ShortComplex.Homotopy.comm₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[374,2],"def_end_pos":[374,7]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ S₁.f ≫ h₁₂.h₁ + h₁₂.h₀ + (S₁.f ≫ h₂₃.h₁ + h₂₃.h₀ + φ₃.τ₁) = S₁.f ≫ h₁₂.h₁ + S₁.f ≫ h₂₃.h₁ + (h₁₂.h₀ + h₂₃.h₀) + φ₃.τ₁","state_after":"no goals","tactic":"abel","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ φ₁.τ₂ = S₁.g ≫ (h₁₂.h₂ + h₂₃.h₂) + (h₁₂.h₁ + h₂₃.h₁) ≫ S₂.f + φ₃.τ₂","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ S₁.g ≫ h₁₂.h₂ + h₁₂.h₁ ≫ S₂.f + (S₁.g ≫ h₂₃.h₂ + h₂₃.h₁ ≫ S₂.f + φ₃.τ₂) =\n    S₁.g ≫ h₁₂.h₂ + S₁.g ≫ h₂₃.h₂ + (h₁₂.h₁ ≫ S₂.f + h₂₃.h₁ ≫ S₂.f) + φ₃.τ₂","tactic":"rw [h₁₂.comm₂, h₂₃.comm₂, comp_add, add_comp]","premises":[{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.Preadditive.comp_add","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[60,2],"def_end_pos":[60,10]},{"full_name":"CategoryTheory.ShortComplex.Homotopy.comm₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[375,2],"def_end_pos":[375,7]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ S₁.g ≫ h₁₂.h₂ + h₁₂.h₁ ≫ S₂.f + (S₁.g ≫ h₂₃.h₂ + h₂₃.h₁ ≫ S₂.f + φ₃.τ₂) =\n    S₁.g ≫ h₁₂.h₂ + S₁.g ≫ h₂₃.h₂ + (h₁₂.h₁ ≫ S₂.f + h₂₃.h₁ ≫ S₂.f) + φ₃.τ₂","state_after":"no goals","tactic":"abel","premises":[]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ φ₁.τ₃ = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ + h₂₃.h₂) ≫ S₂.g + φ₃.τ₃","state_after":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ h₁₂.h₃ + h₁₂.h₂ ≫ S₂.g + (h₂₃.h₃ + h₂₃.h₂ ≫ S₂.g + φ₃.τ₃) = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ ≫ S₂.g + h₂₃.h₂ ≫ S₂.g) + φ₃.τ₃","tactic":"rw [h₁₂.comm₃, h₂₃.comm₃, add_comp]","premises":[{"full_name":"CategoryTheory.Preadditive.add_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[58,2],"def_end_pos":[58,10]},{"full_name":"CategoryTheory.ShortComplex.Homotopy.comm₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean","def_pos":[376,2],"def_end_pos":[376,7]}]},{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.111346, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ h₁₂.h₃ + h₁₂.h₂ ≫ S₂.g + (h₂₃.h₃ + h₂₃.h₂ ≫ S₂.g + φ₃.τ₃) = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ ≫ S₂.g + h₂₃.h₂ ≫ S₂.g) + φ₃.τ₃","state_after":"no goals","tactic":"abel","premises":[]}]}
{"url":"Mathlib/Data/Bool/Basic.lean","commit":"","full_name":"Bool.true_eq_false_eq_False","start":[29,0],"end":[29,59],"file_path":"Mathlib/Data/Bool/Basic.lean","tactics":[{"state_before":"⊢ ¬true = false","state_after":"no goals","tactic":"decide","premises":[]}]}
{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"IsLocalizedModule.mkOfAlgebra","start":[1127,0],"end":[1151,12],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R✝ : Type u_1\ninst✝³¹ : CommSemiring R✝\nS✝ : Submonoid R✝\nM✝ : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³⁰ : AddCommMonoid M✝\ninst✝²⁹ : AddCommMonoid M'\ninst✝²⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝²⁷ : CommSemiring A\ninst✝²⁶ : Algebra R✝ A\ninst✝²⁵ : Module A M'\ninst✝²⁴ : IsLocalization S✝ A\ninst✝²³ : Module R✝ M✝\ninst✝²² : Module R✝ M'\ninst✝²¹ : Module R✝ M''\ninst✝²⁰ : IsScalarTower R✝ A M'\nf✝ : M✝ →ₗ[R✝] M'\ng : M✝ →ₗ[R✝] M''\nM₀ : Type ?u.1244734\nM₀' : Type ?u.1244737\ninst✝¹⁹ : AddCommGroup M₀\ninst✝¹⁸ : AddCommGroup M₀'\ninst✝¹⁷ : Module R✝ M₀\ninst✝¹⁶ : Module R✝ M₀'\nf₀ : M₀ →ₗ[R✝] M₀'\ninst✝¹⁵ : IsLocalizedModule S✝ f₀\nM₁ : Type ?u.1246805\nM₁' : Type ?u.1246808\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : AddCommGroup M₁'\ninst✝¹² : Module R✝ M₁\ninst✝¹¹ : Module R✝ M₁'\nf₁ : M₁ →ₗ[R✝] M₁'\ninst✝¹⁰ : IsLocalizedModule S✝ f₁\nM₂ : Type ?u.1248876\nM₂' : Type ?u.1248879\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : AddCommGroup M₂'\ninst✝⁷ : Module R✝ M₂\ninst✝⁶ : Module R✝ M₂'\nf₂ : M₂ →ₗ[R✝] M₂'\ninst✝⁵ : IsLocalizedModule S✝ f₂\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 → ∃ m, m • x = 0\n⊢ IsLocalizedModule M f.toLinearMap","state_after":"R✝ : Type u_1\ninst✝³¹ : CommSemiring R✝\nS✝ : Submonoid R✝\nM✝ : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³⁰ : AddCommMonoid M✝\ninst✝²⁹ : AddCommMonoid M'\ninst✝²⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝²⁷ : CommSemiring A\ninst✝²⁶ : Algebra R✝ A\ninst✝²⁵ : Module A M'\ninst✝²⁴ : IsLocalization S✝ A\ninst✝²³ : Module R✝ M✝\ninst✝²² : Module R✝ M'\ninst✝²¹ : Module R✝ M''\ninst✝²⁰ : IsScalarTower R✝ A M'\nf✝ : M✝ →ₗ[R✝] M'\ng : M✝ →ₗ[R✝] M''\nM₀ : Type ?u.1244734\nM₀' : Type ?u.1244737\ninst✝¹⁹ : AddCommGroup M₀\ninst✝¹⁸ : AddCommGroup M₀'\ninst✝¹⁷ : Module R✝ M₀\ninst✝¹⁶ : Module R✝ M₀'\nf₀ : M₀ →ₗ[R✝] M₀'\ninst✝¹⁵ : IsLocalizedModule S✝ f₀\nM₁ : Type ?u.1246805\nM₁' : Type ?u.1246808\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : AddCommGroup M₁'\ninst✝¹² : Module R✝ M₁\ninst✝¹¹ : Module R✝ M₁'\nf₁ : M₁ →ₗ[R✝] M₁'\ninst✝¹⁰ : IsLocalizedModule S✝ f₁\nM₂ : Type ?u.1248876\nM₂' : Type ?u.1248879\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : AddCommGroup M₂'\ninst✝⁷ : Module R✝ M₂\ninst✝⁶ : Module R✝ M₂'\nf₂ : M₂ →ₗ[R✝] M₂'\ninst✝⁵ : IsLocalizedModule S✝ f₂\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m • x = 0\n⊢ IsLocalizedModule M f.toLinearMap","tactic":"replace h₃ := fun x =>\n    Iff.intro (h₃ x) fun ⟨⟨m, hm⟩, e⟩ =>\n      (h₁ m hm).mul_left_cancel <| by\n        rw [← Algebra.smul_def]\n        simpa [Submonoid.smul_def] using f.congr_arg e","premises":[{"full_name":"AlgHom.congr_arg","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[186,18],"def_end_pos":[186,27]},{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"IsUnit.mul_left_cancel","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[696,18],"def_end_pos":[696,33]},{"full_name":"Submonoid.smul_def","def_path":"Mathlib/Algebra/Group/Submonoid/Operations.lean","def_pos":[1130,21],"def_end_pos":[1130,29]}]},{"state_before":"R✝ : Type u_1\ninst✝³¹ : CommSemiring R✝\nS✝ : Submonoid R✝\nM✝ : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³⁰ : AddCommMonoid M✝\ninst✝²⁹ : AddCommMonoid M'\ninst✝²⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝²⁷ : CommSemiring A\ninst✝²⁶ : Algebra R✝ A\ninst✝²⁵ : Module A M'\ninst✝²⁴ : IsLocalization S✝ A\ninst✝²³ : Module R✝ M✝\ninst✝²² : Module R✝ M'\ninst✝²¹ : Module R✝ M''\ninst✝²⁰ : IsScalarTower R✝ A M'\nf✝ : M✝ →ₗ[R✝] M'\ng : M✝ →ₗ[R✝] M''\nM₀ : Type ?u.1244734\nM₀' : Type ?u.1244737\ninst✝¹⁹ : AddCommGroup M₀\ninst✝¹⁸ : AddCommGroup M₀'\ninst✝¹⁷ : Module R✝ M₀\ninst✝¹⁶ : Module R✝ M₀'\nf₀ : M₀ →ₗ[R✝] M₀'\ninst✝¹⁵ : IsLocalizedModule S✝ f₀\nM₁ : Type ?u.1246805\nM₁' : Type ?u.1246808\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : AddCommGroup M₁'\ninst✝¹² : Module R✝ M₁\ninst✝¹¹ : Module R✝ M₁'\nf₁ : M₁ →ₗ[R✝] M₁'\ninst✝¹⁰ : IsLocalizedModule S✝ f₁\nM₂ : Type ?u.1248876\nM₂' : Type ?u.1248879\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : AddCommGroup M₂'\ninst✝⁷ : Module R✝ M₂\ninst✝⁶ : Module R✝ M₂'\nf₂ : M₂ →ₗ[R✝] M₂'\ninst✝⁵ : IsLocalizedModule S✝ f₂\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m • x = 0\n⊢ IsLocalizedModule M f.toLinearMap","state_after":"case map_units\nR✝ : Type u_1\ninst✝³¹ : CommSemiring R✝\nS✝ : Submonoid R✝\nM✝ : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³⁰ : AddCommMonoid M✝\ninst✝²⁹ : AddCommMonoid M'\ninst✝²⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝²⁷ : CommSemiring A\ninst✝²⁶ : Algebra R✝ A\ninst✝²⁵ : Module A M'\ninst✝²⁴ : IsLocalization S✝ A\ninst✝²³ : Module R✝ M✝\ninst✝²² : Module R✝ M'\ninst✝²¹ : Module R✝ M''\ninst✝²⁰ : IsScalarTower R✝ A M'\nf✝ : M✝ →ₗ[R✝] M'\ng : M✝ →ₗ[R✝] M''\nM₀ : Type ?u.1244734\nM₀' : Type ?u.1244737\ninst✝¹⁹ : AddCommGroup M₀\ninst✝¹⁸ : AddCommGroup M₀'\ninst✝¹⁷ : Module R✝ M₀\ninst✝¹⁶ : Module R✝ M₀'\nf₀ : M₀ →ₗ[R✝] M₀'\ninst✝¹⁵ : IsLocalizedModule S✝ f₀\nM₁ : Type ?u.1246805\nM₁' : Type ?u.1246808\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : AddCommGroup M₁'\ninst✝¹² : Module R✝ M₁\ninst✝¹¹ : Module R✝ M₁'\nf₁ : M₁ →ₗ[R✝] M₁'\ninst✝¹⁰ : IsLocalizedModule S✝ f₁\nM₂ : Type ?u.1248876\nM₂' : Type ?u.1248879\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : AddCommGroup M₂'\ninst✝⁷ : Module R✝ M₂\ninst✝⁶ : Module R✝ M₂'\nf₂ : M₂ →ₗ[R✝] M₂'\ninst✝⁵ : IsLocalizedModule S✝ f₂\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m • x = 0\n⊢ ∀ (x : ↥M), IsUnit ((algebraMap R (Module.End R S')) ↑x)\n\ncase surj'\nR✝ : Type u_1\ninst✝³¹ : CommSemiring R✝\nS✝ : Submonoid R✝\nM✝ : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³⁰ : AddCommMonoid M✝\ninst✝²⁹ : AddCommMonoid M'\ninst✝²⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝²⁷ : CommSemiring A\ninst✝²⁶ : Algebra R✝ A\ninst✝²⁵ : Module A M'\ninst✝²⁴ : IsLocalization S✝ A\ninst✝²³ : Module R✝ M✝\ninst✝²² : Module R✝ M'\ninst✝²¹ : Module R✝ M''\ninst✝²⁰ : IsScalarTower R✝ A M'\nf✝ : M✝ →ₗ[R✝] M'\ng : M✝ →ₗ[R✝] M''\nM₀ : Type ?u.1244734\nM₀' : Type ?u.1244737\ninst✝¹⁹ : AddCommGroup M₀\ninst✝¹⁸ : AddCommGroup M₀'\ninst✝¹⁷ : Module R✝ M₀\ninst✝¹⁶ : Module R✝ M₀'\nf₀ : M₀ →ₗ[R✝] M₀'\ninst✝¹⁵ : IsLocalizedModule S✝ f₀\nM₁ : Type ?u.1246805\nM₁' : Type ?u.1246808\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : AddCommGroup M₁'\ninst✝¹² : Module R✝ M₁\ninst✝¹¹ : Module R✝ M₁'\nf₁ : M₁ →ₗ[R✝] M₁'\ninst✝¹⁰ : IsLocalizedModule S✝ f₁\nM₂ : Type ?u.1248876\nM₂' : Type ?u.1248879\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : AddCommGroup M₂'\ninst✝⁷ : Module R✝ M₂\ninst✝⁶ : Module R✝ M₂'\nf₂ : M₂ →ₗ[R✝] M₂'\ninst✝⁵ : IsLocalizedModule S✝ f₂\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m • x = 0\n⊢ ∀ (y : S'), ∃ x, x.2 • y = f.toLinearMap x.1\n\ncase exists_of_eq\nR✝ : Type u_1\ninst✝³¹ : CommSemiring R✝\nS✝ : Submonoid R✝\nM✝ : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³⁰ : AddCommMonoid M✝\ninst✝²⁹ : AddCommMonoid M'\ninst✝²⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝²⁷ : CommSemiring A\ninst✝²⁶ : Algebra R✝ A\ninst✝²⁵ : Module A M'\ninst✝²⁴ : IsLocalization S✝ A\ninst✝²³ : Module R✝ M✝\ninst✝²² : Module R✝ M'\ninst✝²¹ : Module R✝ M''\ninst✝²⁰ : IsScalarTower R✝ A M'\nf✝ : M✝ →ₗ[R✝] M'\ng : M✝ →ₗ[R✝] M''\nM₀ : Type ?u.1244734\nM₀' : Type ?u.1244737\ninst✝¹⁹ : AddCommGroup M₀\ninst✝¹⁸ : AddCommGroup M₀'\ninst✝¹⁷ : Module R✝ M₀\ninst✝¹⁶ : Module R✝ M₀'\nf₀ : M₀ →ₗ[R✝] M₀'\ninst✝¹⁵ : IsLocalizedModule S✝ f₀\nM₁ : Type ?u.1246805\nM₁' : Type ?u.1246808\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : AddCommGroup M₁'\ninst✝¹² : Module R✝ M₁\ninst✝¹¹ : Module R✝ M₁'\nf₁ : M₁ →ₗ[R✝] M₁'\ninst✝¹⁰ : IsLocalizedModule S✝ f₁\nM₂ : Type ?u.1248876\nM₂' : Type ?u.1248879\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : AddCommGroup M₂'\ninst✝⁷ : Module R✝ M₂\ninst✝⁶ : Module R✝ M₂'\nf₂ : M₂ →ₗ[R✝] M₂'\ninst✝⁵ : IsLocalizedModule S✝ f₂\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m • x = 0\n⊢ ∀ {x₁ x₂ : S}, f.toLinearMap x₁ = f.toLinearMap x₂ → ∃ c, c • x₁ = c • x₂","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/HomologySequence.lean","commit":"","full_name":"HomologicalComplex.HomologySequence.composableArrows₃_exact","start":[124,0],"end":[148,37],"file_path":"Mathlib/Algebra/Homology/HomologySequence.lean","tactics":[{"state_before":"C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\n⊢ (composableArrows₃ K i j).Exact","tactic":"let S := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) (by simp)","premises":[{"full_name":"HomologicalComplex.homologyι","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[188,18],"def_end_pos":[188,27]},{"full_name":"HomologicalComplex.opcyclesToCycles","def_path":"Mathlib/Algebra/Homology/HomologySequence.lean","def_pos":[43,18],"def_end_pos":[43,34]}]},{"state_before":"C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\n⊢ (composableArrows₃ K i j).Exact","tactic":"let S' := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) (by simp)","premises":[{"full_name":"HomologicalComplex.fromOpcycles","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[213,18],"def_end_pos":[213,30]},{"full_name":"HomologicalComplex.homologyι","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[188,18],"def_end_pos":[188,27]}]},{"state_before":"C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\n⊢ (composableArrows₃ K i j).Exact","tactic":"let ι : S ⟶ S' :=\n    { τ₁ := 𝟙 _\n      τ₂ := 𝟙 _\n      τ₃ := K.iCycles j }","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"HomologicalComplex.iCycles","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[84,18],"def_end_pos":[84,25]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\n⊢ (composableArrows₃ K i j).Exact","tactic":"have hS : S.Exact := by\n    rw [ShortComplex.exact_iff_of_epi_of_isIso_of_mono ι]\n    exact S'.exact_of_f_is_kernel (K.homologyIsKernel i j (c.next_eq' hij))","premises":[{"full_name":"CategoryTheory.ShortComplex.Exact","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[41,10],"def_end_pos":[41,15]},{"full_name":"CategoryTheory.ShortComplex.exact_iff_of_epi_of_isIso_of_mono","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[135,6],"def_end_pos":[135,39]},{"full_name":"CategoryTheory.ShortComplex.exact_of_f_is_kernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[405,6],"def_end_pos":[405,26]},{"full_name":"ComplexShape.next_eq'","def_path":"Mathlib/Algebra/Homology/ComplexShape.lean","def_pos":[141,8],"def_end_pos":[141,16]},{"full_name":"HomologicalComplex.homologyIsKernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[278,18],"def_end_pos":[278,34]}]},{"state_before":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\n⊢ (composableArrows₃ K i j).Exact","tactic":"let T := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) (by simp)","premises":[{"full_name":"HomologicalComplex.homologyπ","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[87,18],"def_end_pos":[87,27]},{"full_name":"HomologicalComplex.opcyclesToCycles","def_path":"Mathlib/Algebra/Homology/HomologySequence.lean","def_pos":[43,18],"def_end_pos":[43,34]}]},{"state_before":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\n⊢ (composableArrows₃ K i j).Exact","tactic":"let T' := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) (by simp)","premises":[{"full_name":"HomologicalComplex.homologyπ","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[87,18],"def_end_pos":[87,27]},{"full_name":"HomologicalComplex.toCycles","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[112,18],"def_end_pos":[112,26]}]},{"state_before":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\n⊢ (composableArrows₃ K i j).Exact","tactic":"let π : T' ⟶ T :=\n    { τ₁ := K.pOpcycles i\n      τ₂ := 𝟙 _\n      τ₃ := 𝟙 _ }","premises":[{"full_name":"CategoryTheory.CategoryStruct.id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[87,2],"def_end_pos":[87,4]},{"full_name":"HomologicalComplex.pOpcycles","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[185,18],"def_end_pos":[185,27]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\n⊢ (composableArrows₃ K i j).Exact","state_after":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhT : T.Exact\n⊢ (composableArrows₃ K i j).Exact","tactic":"have hT : T.Exact := by\n    rw [← ShortComplex.exact_iff_of_epi_of_isIso_of_mono π]\n    exact T'.exact_of_g_is_cokernel (K.homologyIsCokernel i j (c.prev_eq' hij))","premises":[{"full_name":"CategoryTheory.ShortComplex.Exact","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[41,10],"def_end_pos":[41,15]},{"full_name":"CategoryTheory.ShortComplex.exact_iff_of_epi_of_isIso_of_mono","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[135,6],"def_end_pos":[135,39]},{"full_name":"CategoryTheory.ShortComplex.exact_of_g_is_cokernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/Exact.lean","def_pos":[415,6],"def_end_pos":[415,28]},{"full_name":"ComplexShape.prev_eq'","def_path":"Mathlib/Algebra/Homology/ComplexShape.lean","def_pos":[147,8],"def_end_pos":[147,16]},{"full_name":"HomologicalComplex.homologyIsCokernel","def_path":"Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean","def_pos":[176,18],"def_end_pos":[176,36]}]},{"state_before":"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhT : T.Exact\n⊢ (composableArrows₃ K i j).Exact","state_after":"case h\nC : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhT : T.Exact\n⊢ (ComposableArrows.mk₂ ((composableArrows₃ K i j).map' 0 1 ⋯ ⋯) ((composableArrows₃ K i j).map' 1 2 ⋯ ⋯)).Exact\n\ncase h₀\nC : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhT : T.Exact\n⊢ (composableArrows₃ K i j).δ₀.Exact","tactic":"apply ComposableArrows.exact_of_δ₀","premises":[{"full_name":"CategoryTheory.ComposableArrows.exact_of_δ₀","def_path":"Mathlib/Algebra/Homology/ExactSequence.lean","def_pos":[245,6],"def_end_pos":[245,17]}]}]}
{"url":"Mathlib/Analysis/Convex/Cone/Pointed.lean","commit":"","full_name":"PointedCone.toConvexCone_pointed","start":[50,0],"end":[52,41],"file_path":"Mathlib/Analysis/Convex/Cone/Pointed.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\nS : PointedCone 𝕜 E\n⊢ (↑S).Pointed","state_after":"no goals","tactic":"simp [toConvexCone, ConvexCone.Pointed]","premises":[{"full_name":"ConvexCone.Pointed","def_path":"Mathlib/Analysis/Convex/Cone/Basic.lean","def_pos":[297,4],"def_end_pos":[297,11]},{"full_name":"PointedCone.toConvexCone","def_path":"Mathlib/Analysis/Convex/Cone/Pointed.lean","def_pos":[39,4],"def_end_pos":[39,16]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean","commit":"","full_name":"SimpleGraph.Walk.finite_neighborSet_toSubgraph","start":[169,0],"end":[179,22],"file_path":"Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v w : V\np : G.Walk u v\n⊢ (p.toSubgraph.neighborSet w).Finite","state_after":"no goals","tactic":"induction p with\n  | nil =>\n    rw [Walk.toSubgraph, neighborSet_singletonSubgraph]\n    apply Set.toFinite\n  | cons ha _ ih =>\n    rw [Walk.toSubgraph, Subgraph.neighborSet_sup]\n    refine Set.Finite.union ?_ ih\n    refine Set.Finite.subset ?_ (neighborSet_subgraphOfAdj_subset ha)\n    apply Set.toFinite","premises":[{"full_name":"Set.Finite.subset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[633,8],"def_end_pos":[633,21]},{"full_name":"Set.Finite.union","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[637,8],"def_end_pos":[637,20]},{"full_name":"Set.toFinite","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[71,8],"def_end_pos":[71,16]},{"full_name":"SimpleGraph.Subgraph.neighborSet_sup","def_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","def_pos":[435,8],"def_end_pos":[435,23]},{"full_name":"SimpleGraph.Walk.cons","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[55,4],"def_end_pos":[55,8]},{"full_name":"SimpleGraph.Walk.nil","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[54,4],"def_end_pos":[54,7]},{"full_name":"SimpleGraph.Walk.toSubgraph","def_path":"Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean","def_pos":[115,14],"def_end_pos":[115,24]},{"full_name":"SimpleGraph.neighborSet_singletonSubgraph","def_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","def_pos":[758,8],"def_end_pos":[758,37]},{"full_name":"SimpleGraph.neighborSet_subgraphOfAdj_subset","def_path":"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean","def_pos":[824,8],"def_end_pos":[824,40]}]}]}
{"url":"Mathlib/Data/Holor.lean","commit":"","full_name":"Holor.slice_sum","start":[201,0],"end":[207,78],"file_path":"Mathlib/Data/Holor.lean","tactics":[{"state_before":"α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : AddCommMonoid α\nβ : Type\ni : ℕ\nhid : i < d\ns : Finset β\nf : β → Holor α (d :: ds)\n⊢ ∑ x ∈ s, (f x).slice i hid = (∑ x ∈ s, f x).slice i hid","state_after":"α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : AddCommMonoid α\nβ : Type\ni : ℕ\nhid : i < d\ns : Finset β\nf : β → Holor α (d :: ds)\nthis : DecidableEq β := Classical.decEq β\n⊢ ∑ x ∈ s, (f x).slice i hid = (∑ x ∈ s, f x).slice i hid","tactic":"letI := Classical.decEq β","premises":[{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]}]},{"state_before":"α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : AddCommMonoid α\nβ : Type\ni : ℕ\nhid : i < d\ns : Finset β\nf : β → Holor α (d :: ds)\nthis : DecidableEq β := Classical.decEq β\n⊢ ∑ x ∈ s, (f x).slice i hid = (∑ x ∈ s, f x).slice i hid","state_after":"case refine_1\nα : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : AddCommMonoid α\nβ : Type\ni : ℕ\nhid : i < d\ns : Finset β\nf : β → Holor α (d :: ds)\nthis : DecidableEq β := Classical.decEq β\n⊢ ∑ x ∈ ∅, (f x).slice i hid = (∑ x ∈ ∅, f x).slice i hid\n\ncase refine_2\nα : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : AddCommMonoid α\nβ : Type\ni : ℕ\nhid : i < d\ns : Finset β\nf : β → Holor α (d :: ds)\nthis : DecidableEq β := Classical.decEq β\n⊢ ∀ ⦃a : β⦄ {s : Finset β},\n    a ∉ s →\n      ∑ x ∈ s, (f x).slice i hid = (∑ x ∈ s, f x).slice i hid →\n        ∑ x ∈ insert a s, (f x).slice i hid = (∑ x ∈ insert a s, f x).slice i hid","tactic":"refine Finset.induction_on s ?_ ?_","premises":[{"full_name":"Finset.induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1076,18],"def_end_pos":[1076,30]}]}]}
{"url":"Mathlib/CategoryTheory/GradedObject.lean","commit":"","full_name":"CategoryTheory.Iso.map_hom_inv_id_eval_app","start":[131,0],"end":[135,36],"file_path":"Mathlib/CategoryTheory/GradedObject.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝² : Category.{u_5, u_1} C\ninst✝¹ : Category.{u_7, u_2} D\ninst✝ : Category.{u_6, u_3} E\nX Y✝ : GradedObject J C\ne : X ≅ Y✝\nF : C ⥤ D ⥤ E\nj : J\nY : D\n⊢ (F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)","state_after":"no goals","tactic":"rw [← NatTrans.comp_app, ← F.map_comp, hom_inv_id_eval,\n    Functor.map_id, NatTrans.id_app]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Iso.hom_inv_id_eval","def_path":"Mathlib/CategoryTheory/GradedObject.lean","def_pos":[108,6],"def_end_pos":[108,21]},{"full_name":"CategoryTheory.NatTrans.comp_app","def_path":"Mathlib/CategoryTheory/Functor/Category.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"CategoryTheory.NatTrans.id_app","def_path":"Mathlib/CategoryTheory/Functor/Category.lean","def_pos":[66,8],"def_end_pos":[66,14]}]}]}
{"url":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","commit":"","full_name":"MultilinearMap.curryFinFinset_apply_const","start":[1722,0],"end":[1728,35],"file_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","tactics":[{"state_before":"R : Type uR\nS : Type uS\nι : Type uι\nn✝ : ℕ\nM : Fin n✝.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n✝.succ) → Module R (M i)\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nι' : Type u_1\nk l n : ℕ\ns : Finset (Fin n)\nhk : s.card = k\nhl : sᶜ.card = l\nf : MultilinearMap R (fun x => M') M₂\nx y : M'\n⊢ ((((curryFinFinset R M₂ M' hk hl) f) fun x_1 => x) fun x => y) = f (s.piecewise (fun x_1 => x) fun x => y)","state_after":"R : Type uR\nS : Type uS\nι : Type uι\nn✝ : ℕ\nM : Fin n✝.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n✝.succ) → Module R (M i)\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nι' : Type u_1\nk l n : ℕ\ns : Finset (Fin n)\nhk : s.card = k\nhl : sᶜ.card = l\nf : MultilinearMap R (fun x => M') M₂\nx y : M'\n⊢ ((curryFinFinset R M₂ M' hk hl).symm ((curryFinFinset R M₂ M' hk hl) f)) (s.piecewise (fun x_1 => x) fun x => y) =\n    f (s.piecewise (fun x_1 => x) fun x => y)","tactic":"refine (curryFinFinset_symm_apply_piecewise_const hk hl _ _ _).symm.trans ?_","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"MultilinearMap.curryFinFinset_symm_apply_piecewise_const","def_path":"Mathlib/LinearAlgebra/Multilinear/Basic.lean","def_pos":[1688,8],"def_end_pos":[1688,49]}]},{"state_before":"R : Type uR\nS : Type uS\nι : Type uι\nn✝ : ℕ\nM : Fin n✝.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n✝.succ) → Module R (M i)\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nι' : Type u_1\nk l n : ℕ\ns : Finset (Fin n)\nhk : s.card = k\nhl : sᶜ.card = l\nf : MultilinearMap R (fun x => M') M₂\nx y : M'\n⊢ ((curryFinFinset R M₂ M' hk hl).symm ((curryFinFinset R M₂ M' hk hl) f)) (s.piecewise (fun x_1 => x) fun x => y) =\n    f (s.piecewise (fun x_1 => x) fun x => y)","state_after":"no goals","tactic":"rw [LinearEquiv.symm_apply_apply]","premises":[{"full_name":"LinearEquiv.symm_apply_apply","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[346,8],"def_end_pos":[346,24]}]}]}
{"url":"Mathlib/Algebra/Star/SelfAdjoint.lean","commit":"","full_name":"IsSelfAdjoint.conjugate","start":[151,0],"end":[153,75],"file_path":"Mathlib/Algebra/Star/SelfAdjoint.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝¹ : Semigroup R\ninst✝ : StarMul R\nx : R\nhx : IsSelfAdjoint x\nz : R\n⊢ IsSelfAdjoint (z * x * star z)","state_after":"no goals","tactic":"simp only [isSelfAdjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq]","premises":[{"full_name":"IsSelfAdjoint.star_eq","def_path":"Mathlib/Algebra/Star/SelfAdjoint.lean","def_pos":[68,8],"def_end_pos":[68,15]},{"full_name":"StarMul.star_mul","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[126,2],"def_end_pos":[126,10]},{"full_name":"isSelfAdjoint_iff","def_path":"Mathlib/Algebra/Star/SelfAdjoint.lean","def_pos":[71,8],"def_end_pos":[71,32]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"star_star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[88,8],"def_end_pos":[88,17]}]}]}
{"url":"Mathlib/MeasureTheory/PiSystem.lean","commit":"","full_name":"isPiSystem_piiUnionInter","start":[381,0],"end":[418,38],"file_path":"Mathlib/MeasureTheory/PiSystem.lean","tactics":[{"state_before":"α : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\n⊢ IsPiSystem (piiUnionInter π S)","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\n⊢ t1 ∩ t2 ∈ piiUnionInter π S","tactic":"rintro t1 ⟨p1, hp1S, f1, hf1m, ht1_eq⟩ t2 ⟨p2, hp2S, f2, hf2m, ht2_eq⟩ h_nonempty","premises":[]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\n⊢ t1 ∩ t2 ∈ piiUnionInter π S","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\n⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x","tactic":"simp_rw [piiUnionInter, Set.mem_setOf_eq]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"piiUnionInter","def_path":"Mathlib/MeasureTheory/PiSystem.lean","def_pos":[320,4],"def_end_pos":[320,17]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\n⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\n⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x","tactic":"let g n := ite (n ∈ p1) (f1 n) Set.univ ∩ ite (n ∈ p2) (f2 n) Set.univ","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\n⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x","state_after":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\n⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x","tactic":"have hp_union_ss : ↑(p1 ∪ p2) ⊆ S := by\n    simp only [hp1S, hp2S, Finset.coe_union, union_subset_iff, and_self_iff]","premises":[{"full_name":"Finset.coe_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1211,8],"def_end_pos":[1211,17]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Set.union_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[666,8],"def_end_pos":[666,24]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\n⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x","state_after":"case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\n⊢ ∃ (_ : ∀ x ∈ p1 ∪ p2, g x ∈ π x), t1 ∩ t2 = ⋂ x ∈ p1 ∪ p2, g x","tactic":"use p1 ∪ p2, hp_union_ss, g","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\n⊢ ∃ (_ : ∀ x ∈ p1 ∪ p2, g x ∈ π x), t1 ∩ t2 = ⋂ x ∈ p1 ∪ p2, g x","state_after":"case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\n⊢ g n ∈ π n","tactic":"refine ⟨fun n hn => ?_, h_inter_eq⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\n⊢ g n ∈ π n","state_after":"case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\n⊢ ((if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ) ∈ π n","tactic":"simp only [g]","premises":[]},{"state_before":"case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\n⊢ ((if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ) ∈ π n","state_after":"case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\nhn1 : n ∈ p1\nhn2 : n ∈ p2\n⊢ f1 n ∩ f2 n ∈ π n\n\ncase neg\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\nhn1 : n ∈ p1\nhn2 : n ∉ p2\n⊢ f1 n ∩ univ ∈ π n\n\ncase pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\nhn1 : n ∉ p1\nh : n ∈ p2\n⊢ univ ∩ f2 n ∈ π n\n\ncase neg\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\nhpi : ∀ (x : ι), IsPiSystem (π x)\nS : Set ι\nt1 : Set α\np1 : Finset ι\nhp1S : ↑p1 ⊆ S\nf1 : ι → Set α\nhf1m : ∀ x ∈ p1, f1 x ∈ π x\nht1_eq : t1 = ⋂ x ∈ p1, f1 x\nt2 : Set α\np2 : Finset ι\nhp2S : ↑p2 ⊆ S\nf2 : ι → Set α\nhf2m : ∀ x ∈ p2, f2 x ∈ π x\nht2_eq : t2 = ⋂ x ∈ p2, f2 x\nh_nonempty : (t1 ∩ t2).Nonempty\ng : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ\nhp_union_ss : ↑(p1 ∪ p2) ⊆ S\nh_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i\nn : ι\nhn : n ∈ p1 ∪ p2\nhn1 : n ∉ p1\nh : n ∉ p2\n⊢ univ ∩ univ ∈ π n","tactic":"split_ifs with hn1 hn2 h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Order/GaloisConnection.lean","commit":"","full_name":"GaloisConnection.l_iSup","start":[237,0],"end":[241,76],"file_path":"Mathlib/Order/GaloisConnection.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort u_1\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (range (l ∘ f)) (l (iSup f))","state_after":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort u_1\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (l '' range f) (l (sSup (range f)))","tactic":"rw [range_comp, ← sSup_range]","premises":[{"full_name":"Set.range_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[621,8],"def_end_pos":[621,18]},{"full_name":"sSup_range","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[500,8],"def_end_pos":[500,18]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort u_1\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (l '' range f) (l (sSup (range f)))","state_after":"no goals","tactic":"exact gc.isLUB_l_image (isLUB_sSup _)","premises":[{"full_name":"GaloisConnection.isLUB_l_image","def_path":"Mathlib/Order/GaloisConnection.lean","def_pos":[115,8],"def_end_pos":[115,21]},{"full_name":"isLUB_sSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[83,8],"def_end_pos":[83,18]}]}]}
{"url":"Mathlib/Topology/Constructions.lean","commit":"","full_name":"prod_eq_generateFrom","start":[632,0],"end":[640,73],"file_path":"Mathlib/Topology/Constructions.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nt : Set X\nht : t ∈ inst✝⁵.1\n⊢ IsOpen t ∧ IsOpen univ ∧ Prod.fst ⁻¹' t = t ×ˢ univ","state_after":"no goals","tactic":"simpa [Set.prod_eq] using ht","premises":[{"full_name":"Set.prod_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[177,8],"def_end_pos":[177,15]}]},{"state_before":"X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nt : Set Y\nht : t ∈ inst✝⁴.1\n⊢ IsOpen univ ∧ IsOpen t ∧ Prod.snd ⁻¹' t = univ ×ˢ t","state_after":"no goals","tactic":"simpa [Set.prod_eq] using ht","premises":[{"full_name":"Set.prod_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[177,8],"def_end_pos":[177,15]}]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Basic.lean","commit":"","full_name":"hasSum_geometric_of_lt_one","start":[251,0],"end":[257,51],"file_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nr : ℝ\nh₁ : 0 ≤ r\nh₂ : r < 1\nthis✝ : r ≠ 1\nthis : Tendsto (fun n => (r ^ n - 1) * (r - 1)⁻¹) atTop (𝓝 ((0 - 1) * (r - 1)⁻¹))\n⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, r ^ i) atTop (𝓝 (1 - r)⁻¹)","state_after":"no goals","tactic":"simp_all [neg_inv, geom_sum_eq, div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"geom_sum_eq","def_path":"Mathlib/Algebra/GeomSum.lean","def_pos":[242,8],"def_end_pos":[242,19]},{"full_name":"neg_inv","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[108,8],"def_end_pos":[108,15]}]}]}
{"url":"Mathlib/NumberTheory/PellMatiyasevic.lean","commit":"","full_name":"Pell.eq_of_xn_modEq_lem2","start":[581,0],"end":[589,44],"file_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","tactics":[{"state_before":"a : ℕ\na1 : 1 < a\nn : ℕ\nh : 2 * xn a1 n = xn a1 (n + 1)\n⊢ a = 2 ∧ n = 0","state_after":"a : ℕ\na1 : 1 < a\nn : ℕ\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\n⊢ a = 2 ∧ n = 0","tactic":"rw [xn_succ, mul_comm] at h","premises":[{"full_name":"Pell.xn_succ","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[125,8],"def_end_pos":[125,15]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"a : ℕ\na1 : 1 < a\nn : ℕ\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\n⊢ a = 2 ∧ n = 0","state_after":"a : ℕ\na1 : 1 < a\nn : ℕ\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\nthis : n = 0\n⊢ a = 2 ∧ n = 0","tactic":"have : n = 0 :=\n    n.eq_zero_or_pos.resolve_right fun np =>\n      _root_.ne_of_lt\n        (lt_of_le_of_lt (Nat.mul_le_mul_left _ a1)\n          (Nat.lt_add_of_pos_right <| mul_pos (d_pos a1) (strictMono_y a1 np)))\n        h","premises":[{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]},{"full_name":"Nat.lt_add_of_pos_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[510,18],"def_end_pos":[510,37]},{"full_name":"Nat.mul_le_mul_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[713,8],"def_end_pos":[713,23]},{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]},{"full_name":"Pell.d_pos","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[90,8],"def_end_pos":[90,13]},{"full_name":"Pell.strictMono_y","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[353,8],"def_end_pos":[353,20]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]},{"full_name":"ne_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[83,8],"def_end_pos":[83,16]}]},{"state_before":"a : ℕ\na1 : 1 < a\nn : ℕ\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\nthis : n = 0\n⊢ a = 2 ∧ n = 0","state_after":"case refl\na : ℕ\na1 : 1 < a\nh : xn a1 0 * 2 = xn a1 0 * a + Pell.d a1 * yn a1 0\n⊢ a = 2 ∧ 0 = 0","tactic":"cases this","premises":[]},{"state_before":"case refl\na : ℕ\na1 : 1 < a\nh : xn a1 0 * 2 = xn a1 0 * a + Pell.d a1 * yn a1 0\n⊢ a = 2 ∧ 0 = 0","state_after":"case refl\na : ℕ\na1 : 1 < a\nh : 2 = a\n⊢ a = 2 ∧ 0 = 0","tactic":"simp at h","premises":[]},{"state_before":"case refl\na : ℕ\na1 : 1 < a\nh : 2 = a\n⊢ a = 2 ∧ 0 = 0","state_after":"no goals","tactic":"exact ⟨h.symm, rfl⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","commit":"","full_name":"HurwitzZeta.hasSum_nat_cosZeta","start":[759,0],"end":[767,89],"file_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","tactics":[{"state_before":"a : ℝ\ns : ℂ\nhs : 1 < s.re\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)","state_after":"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n  HasSum (fun n => cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 + cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|(-↑n)| ^ s / 2)\n    (cosZeta (↑a) s + cexp (2 * ↑π * I * ↑a * ↑0) / ↑|0| ^ s / 2)\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)","tactic":"have := (hasSum_int_cosZeta a hs).nat_add_neg","premises":[{"full_name":"HasSum.nat_add_neg","def_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","def_pos":[399,2],"def_end_pos":[399,13]},{"full_name":"HurwitzZeta.hasSum_int_cosZeta","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean","def_pos":[752,6],"def_end_pos":[752,24]}]},{"state_before":"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n  HasSum (fun n => cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 + cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|(-↑n)| ^ s / 2)\n    (cosZeta (↑a) s + cexp (2 * ↑π * I * ↑a * ↑0) / ↑|0| ^ s / 2)\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)","state_after":"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)","tactic":"simp_rw [abs_neg, Int.cast_neg, Nat.abs_cast, Int.cast_natCast, mul_neg, abs_zero, Int.cast_zero,\n    zero_cpow (ne_zero_of_one_lt_re hs), div_zero, zero_div, add_zero, ← add_div,\n    div_right_comm _ _ (2 : ℂ)] at this","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.ne_zero_of_one_lt_re","def_path":"Mathlib/Data/Complex/Abs.lean","def_pos":[308,6],"def_end_pos":[308,26]},{"full_name":"Complex.zero_cpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean","def_pos":[47,8],"def_end_pos":[47,17]},{"full_name":"Int.cast_natCast","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Int.cast_neg","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[75,8],"def_end_pos":[75,16]},{"full_name":"Int.cast_zero","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[53,8],"def_end_pos":[53,17]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Nat.abs_cast","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[84,8],"def_end_pos":[84,16]},{"full_name":"abs_neg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[70,2],"def_end_pos":[70,13]},{"full_name":"abs_zero","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[91,2],"def_end_pos":[91,13]},{"full_name":"add_div","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[27,8],"def_end_pos":[27,15]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"div_right_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[540,8],"def_end_pos":[540,22]},{"full_name":"div_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[301,8],"def_end_pos":[301,16]},{"full_name":"mul_neg","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[272,8],"def_end_pos":[272,15]},{"full_name":"zero_div","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[298,8],"def_end_pos":[298,16]}]},{"state_before":"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)","state_after":"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n => (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n * I))) / 2 / ↑n ^ s) (cosZeta (↑a) s)","tactic":"simp_rw [push_cast, Complex.cos, neg_mul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex.cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[60,4],"def_end_pos":[60,7]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"neg_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[268,8],"def_end_pos":[268,15]}]},{"state_before":"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n => (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n * I))) / 2 / ↑n ^ s) (cosZeta (↑a) s)","state_after":"no goals","tactic":"exact this.congr_fun fun n ↦ by rw [show 2 * π * a * n * I = 2 * π * I * a * n by ring]","premises":[{"full_name":"Complex.I","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[231,4],"def_end_pos":[231,5]},{"full_name":"HasSum.congr_fun","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[57,2],"def_end_pos":[57,13]},{"full_name":"Real.pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[119,28],"def_end_pos":[119,30]}]}]}
{"url":"Mathlib/RingTheory/Multiplicity.lean","commit":"","full_name":"multiplicity.eq_top_iff","start":[146,0],"end":[156,23],"file_path":"Mathlib/RingTheory/Multiplicity.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), ¬¬a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b","tactic":"simp only [Classical.not_not]","premises":[{"full_name":"Classical.not_not","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[135,16],"def_end_pos":[135,23]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b","state_after":"no goals","tactic":"exact\n      ⟨fun h n =>\n        Nat.casesOn n\n          (by\n            rw [_root_.pow_zero]\n            exact one_dvd _)\n          fun n => h _,\n        fun h n => h _⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"one_dvd","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[130,8],"def_end_pos":[130,15]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Nat.lean","commit":"","full_name":"Nat.Iio_eq_range","start":[54,0],"end":[56,25],"file_path":"Mathlib/Order/Interval/Finset/Nat.lean","tactics":[{"state_before":"a b c : ℕ\n⊢ Iio = range","state_after":"case h.a\na b✝ c b x : ℕ\n⊢ x ∈ Iio b ↔ x ∈ range b","tactic":"ext b x","premises":[]},{"state_before":"case h.a\na b✝ c b x : ℕ\n⊢ x ∈ Iio b ↔ x ∈ range b","state_after":"no goals","tactic":"rw [mem_Iio, mem_range]","premises":[{"full_name":"Finset.mem_Iio","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[375,8],"def_end_pos":[375,15]},{"full_name":"Finset.mem_range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2450,8],"def_end_pos":[2450,17]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Polynomial.lean","commit":"","full_name":"Polynomial.hasStrictDerivAt","start":[56,0],"end":[61,84],"file_path":"Mathlib/Analysis/Calculus/Deriv/Polynomial.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\np : 𝕜[X]\nq : R[X]\nx : 𝕜\n⊢ HasStrictDerivAt (fun x => eval x p) (eval x (derivative p)) x","state_after":"no goals","tactic":"induction p using Polynomial.induction_on' with\n  | h_add p q hp hq => simpa using hp.add hq\n  | h_monomial n a => simpa [mul_assoc] using (hasStrictDerivAt_pow n x).const_mul a","premises":[{"full_name":"HasStrictDerivAt.add","def_path":"Mathlib/Analysis/Calculus/Deriv/Add.lean","def_pos":[48,15],"def_end_pos":[48,35]},{"full_name":"HasStrictDerivAt.const_mul","def_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","def_pos":[268,8],"def_end_pos":[268,34]},{"full_name":"Polynomial.induction_on'","def_path":"Mathlib/Algebra/Polynomial/Induction.lean","def_pos":[60,18],"def_end_pos":[60,31]},{"full_name":"hasStrictDerivAt_pow","def_path":"Mathlib/Analysis/Calculus/Deriv/Pow.lean","def_pos":[43,8],"def_end_pos":[43,28]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","commit":"","full_name":"Matrix.det_one_add_col_mul_row","start":[410,0],"end":[414,29],"file_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nι : Type u_5\ninst✝ : Unique ι\nu v : m → α\n⊢ (1 + col ι u * row ι v).det = 1 + v ⬝ᵥ u","state_after":"no goals","tactic":"rw [det_one_add_mul_comm, det_unique, Pi.add_apply, Pi.add_apply, Matrix.one_apply_eq,\n    Matrix.row_mul_col_apply]","premises":[{"full_name":"Matrix.det_one_add_mul_comm","def_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","def_pos":[395,8],"def_end_pos":[395,28]},{"full_name":"Matrix.det_unique","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[100,8],"def_end_pos":[100,18]},{"full_name":"Matrix.one_apply_eq","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[510,8],"def_end_pos":[510,20]},{"full_name":"Matrix.row_mul_col_apply","def_path":"Mathlib/Data/Matrix/RowCol.lean","def_pos":[140,8],"def_end_pos":[140,25]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]}]}]}
{"url":"Mathlib/Probability/CondCount.lean","commit":"","full_name":"ProbabilityTheory.condCount_inter_self","start":[91,0],"end":[92,52],"file_path":"Mathlib/Probability/CondCount.lean","tactics":[{"state_before":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\n⊢ (condCount s) (s ∩ t) = (condCount s) t","state_after":"no goals","tactic":"rw [condCount, cond_inter_self _ hs.measurableSet]","premises":[{"full_name":"ProbabilityTheory.condCount","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[52,4],"def_end_pos":[52,13]},{"full_name":"ProbabilityTheory.cond_inter_self","def_path":"Mathlib/Probability/ConditionalProbability.lean","def_pos":[143,8],"def_end_pos":[143,23]},{"full_name":"Set.Finite.measurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[269,8],"def_end_pos":[269,32]}]}]}
{"url":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","commit":"","full_name":"HurwitzZeta.oddKernel_undef","start":[121,0],"end":[125,27],"file_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","tactics":[{"state_before":"a : UnitAddCircle\nx : ℝ\nhx : x ≤ 0\n⊢ oddKernel a x = 0","state_after":"case H\nx : ℝ\nhx : x ≤ 0\na' : ℝ\n⊢ oddKernel (↑a') x = 0","tactic":"induction' a using QuotientAddGroup.induction_on' with a'","premises":[{"full_name":"QuotientAddGroup.induction_on'","def_path":"Mathlib/GroupTheory/Coset.lean","def_pos":[374,2],"def_end_pos":[374,13]}]},{"state_before":"case H\nx : ℝ\nhx : x ≤ 0\na' : ℝ\n⊢ oddKernel (↑a') x = 0","state_after":"no goals","tactic":"rw [← ofReal_eq_zero, oddKernel_def', jacobiTheta₂_undef, jacobiTheta₂'_undef, zero_div, zero_add,\n    mul_zero, mul_zero] <;>\n  rwa [I_mul_im, ofReal_re]","premises":[{"full_name":"Complex.I_mul_im","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[264,8],"def_end_pos":[264,16]},{"full_name":"Complex.ofReal_eq_zero","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[136,8],"def_end_pos":[136,22]},{"full_name":"Complex.ofReal_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"HurwitzZeta.oddKernel_def'","def_path":"Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean","def_pos":[116,6],"def_end_pos":[116,20]},{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"jacobiTheta₂'_undef","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[282,6],"def_end_pos":[282,25]},{"full_name":"jacobiTheta₂_undef","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[272,6],"def_end_pos":[272,24]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]},{"full_name":"zero_div","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[298,8],"def_end_pos":[298,16]}]}]}
{"url":"Mathlib/Data/Nat/Factors.lean","commit":"","full_name":"Nat.primeFactorsList_sublist_right","start":[205,0],"end":[211,41],"file_path":"Mathlib/Data/Nat/Factors.lean","tactics":[{"state_before":"n k : ℕ\nh : k ≠ 0\n⊢ n.primeFactorsList <+ (n * k).primeFactorsList","state_after":"case zero\nk : ℕ\nh : k ≠ 0\n⊢ primeFactorsList 0 <+ (0 * k).primeFactorsList\n\ncase succ\nk : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+ ((hn + 1) * k).primeFactorsList","tactic":"cases' n with hn","premises":[]},{"state_before":"case succ\nk : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+ ((hn + 1) * k).primeFactorsList","state_after":"k : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+~ ((hn + 1) * k).primeFactorsList","tactic":"apply sublist_of_subperm_of_sorted _ (primeFactorsList_sorted _) (primeFactorsList_sorted _)","premises":[{"full_name":"List.sublist_of_subperm_of_sorted","def_path":"Mathlib/Data/List/Sort.lean","def_pos":[113,8],"def_end_pos":[113,36]},{"full_name":"Nat.primeFactorsList_sorted","def_path":"Mathlib/Data/Nat/Factors.lean","def_pos":[105,8],"def_end_pos":[105,31]}]},{"state_before":"k : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+~ ((hn + 1) * k).primeFactorsList","state_after":"k : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+~ hn.succ.primeFactorsList ++ k.primeFactorsList","tactic":"simp only [(perm_primeFactorsList_mul (Nat.succ_ne_zero _) h).subperm_left]","premises":[{"full_name":"List.Perm.subperm_left","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[216,8],"def_end_pos":[216,25]},{"full_name":"Nat.perm_primeFactorsList_mul","def_path":"Mathlib/Data/Nat/Factors.lean","def_pos":[188,8],"def_end_pos":[188,33]},{"full_name":"Nat.succ_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[708,16],"def_end_pos":[708,28]}]},{"state_before":"k : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+~ hn.succ.primeFactorsList ++ k.primeFactorsList","state_after":"no goals","tactic":"exact (sublist_append_left _ _).subperm","premises":[{"full_name":"List.Sublist.subperm","def_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","def_pos":[225,8],"def_end_pos":[225,23]},{"full_name":"List.sublist_append_left","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[109,16],"def_end_pos":[109,35]}]}]}
{"url":"Mathlib/Analysis/Fourier/AddCircle.lean","commit":"","full_name":"fourier_neg'","start":[151,0],"end":[153,53],"file_path":"Mathlib/Analysis/Fourier/AddCircle.lean","tactics":[{"state_before":"T : ℝ\nn : ℤ\nx : AddCircle T\n⊢ ↑(-(n • x)).toCircle = (starRingEnd ℂ) ((fourier n) x)","state_after":"T : ℝ\nn : ℤ\nx : AddCircle T\n⊢ (fourier (-n)) x = (starRingEnd ℂ) ((fourier n) x)","tactic":"rw [← neg_smul, ← fourier_apply]","premises":[{"full_name":"fourier_apply","def_path":"Mathlib/Analysis/Fourier/AddCircle.lean","def_pos":[107,8],"def_end_pos":[107,21]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]}]},{"state_before":"T : ℝ\nn : ℤ\nx : AddCircle T\n⊢ (fourier (-n)) x = (starRingEnd ℂ) ((fourier n) x)","state_after":"no goals","tactic":"exact fourier_neg","premises":[{"full_name":"fourier_neg","def_path":"Mathlib/Analysis/Fourier/AddCircle.lean","def_pos":[144,8],"def_end_pos":[144,19]}]}]}
{"url":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","commit":"","full_name":"MeasurableEmbedding.measurable_extend","start":[99,0],"end":[105,41],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng✝ : β → γ\nhf : MeasurableEmbedding f\ng : α → γ\ng' : β → γ\nhg : Measurable g\nhg' : Measurable g'\n⊢ Measurable (extend f g g')","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng✝ : β → γ\nhf : MeasurableEmbedding f\ng : α → γ\ng' : β → γ\nhg : Measurable g\nhg' : Measurable g'\n⊢ Measurable ((range f).restrict (extend f g g'))\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng✝ : β → γ\nhf : MeasurableEmbedding f\ng : α → γ\ng' : β → γ\nhg : Measurable g\nhg' : Measurable g'\n⊢ Measurable ((range f)ᶜ.restrict (extend f g g'))","tactic":"refine measurable_of_restrict_of_restrict_compl hf.measurableSet_range ?_ ?_","premises":[{"full_name":"MeasurableEmbedding.measurableSet_range","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[85,8],"def_end_pos":[85,27]},{"full_name":"measurable_of_restrict_of_restrict_compl","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[534,8],"def_end_pos":[534,48]}]}]}
{"url":"Mathlib/NumberTheory/Divisors.lean","commit":"","full_name":"Nat.eq_properDivisors_of_subset_of_sum_eq_sum","start":[352,0],"end":[370,13],"file_path":"Mathlib/NumberTheory/Divisors.lean","tactics":[{"state_before":"n : ℕ\ns : Finset ℕ\nhsub : s ⊆ n.properDivisors\n⊢ ∑ x ∈ s, x = ∑ x ∈ n.properDivisors, x → s = n.properDivisors","state_after":"case zero\ns : Finset ℕ\nhsub : s ⊆ properDivisors 0\n⊢ ∑ x ∈ s, x = ∑ x ∈ properDivisors 0, x → s = properDivisors 0\n\ncase succ\ns : Finset ℕ\nn✝ : ℕ\nhsub : s ⊆ (n✝ + 1).properDivisors\n⊢ ∑ x ∈ s, x = ∑ x ∈ (n✝ + 1).properDivisors, x → s = (n✝ + 1).properDivisors","tactic":"cases n","premises":[]},{"state_before":"case succ\ns : Finset ℕ\nn✝ : ℕ\nhsub : s ⊆ (n✝ + 1).properDivisors\n⊢ ∑ x ∈ s, x = ∑ x ∈ (n✝ + 1).properDivisors, x → s = (n✝ + 1).properDivisors","state_after":"no goals","tactic":"classical\n    rw [← sum_sdiff hsub]\n    intro h\n    apply Subset.antisymm hsub\n    rw [← sdiff_eq_empty_iff_subset]\n    contrapose h\n    rw [← Ne, ← nonempty_iff_ne_empty] at h\n    apply ne_of_lt\n    rw [← zero_add (∑ x ∈ s, x), ← add_assoc, add_zero]\n    apply add_lt_add_right\n    have hlt :=\n      sum_lt_sum_of_nonempty h fun x hx => pos_of_mem_properDivisors (sdiff_subset hx)\n    simp only [sum_const_zero] at hlt\n    apply hlt","premises":[{"full_name":"Finset.Subset.antisymm","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[329,8],"def_end_pos":[329,23]},{"full_name":"Finset.nonempty_iff_ne_empty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[521,8],"def_end_pos":[521,29]},{"full_name":"Finset.sdiff_eq_empty_iff_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1850,8],"def_end_pos":[1850,33]},{"full_name":"Finset.sdiff_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1890,16],"def_end_pos":[1890,28]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_const_zero","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[362,2],"def_end_pos":[362,13]},{"full_name":"Finset.sum_lt_sum_of_nonempty","def_path":"Mathlib/Algebra/Order/BigOperators/Group/Finset.lean","def_pos":[388,14],"def_end_pos":[388,36]},{"full_name":"Finset.sum_sdiff","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[509,2],"def_end_pos":[509,13]},{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]},{"full_name":"Nat.ne_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[382,8],"def_end_pos":[382,16]},{"full_name":"Nat.pos_of_mem_properDivisors","def_path":"Mathlib/NumberTheory/Divisors.lean","def_pos":[192,8],"def_end_pos":[192,33]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_lt_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[127,31],"def_end_pos":[127,47]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","commit":"","full_name":"integral_cos_pow","start":[676,0],"end":[683,6],"file_path":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","tactics":[{"state_before":"a b : ℝ\nn : ℕ\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a) / (↑n + 2) + (↑n + 1) / (↑n + 2) * ∫ (x : ℝ) in a..b, cos x ^ n","state_after":"a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) * (↑n + 2) =\n    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n","tactic":"field_simp","premises":[]},{"state_before":"a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) * (↑n + 2) =\n    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n","state_after":"case h.e'_2\na b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) * (↑n + 2) =\n    (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","tactic":"convert eq_sub_iff_add_eq.mp (integral_cos_pow_aux n) using 1","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"eq_sub_iff_add_eq","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[753,14],"def_end_pos":[753,31]},{"full_name":"integral_cos_pow_aux","def_path":"Mathlib/Analysis/SpecialFunctions/Integrals.lean","def_pos":[653,8],"def_end_pos":[653,28]}]},{"state_before":"case h.e'_2\na b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) * (↑n + 2) =\n    (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)","state_after":"no goals","tactic":"ring","premises":[]}]}
{"url":"Mathlib/Topology/Homotopy/Basic.lean","commit":"","full_name":"ContinuousMap.HomotopicRel.equivalence","start":[642,0],"end":[643,39],"file_path":"Mathlib/Topology/Homotopy/Basic.lean","tactics":[{"state_before":"F : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\nι : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nS : Set X\n⊢ ∀ {x y : C(X, Y)}, x.HomotopicRel y S → y.HomotopicRel x S","state_after":"no goals","tactic":"apply symm","premises":[{"full_name":"ContinuousMap.HomotopicRel.symm","def_path":"Mathlib/Topology/Homotopy/Basic.lean","def_pos":[634,8],"def_end_pos":[634,12]}]},{"state_before":"F : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\nι : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nS : Set X\n⊢ ∀ {x y z : C(X, Y)}, x.HomotopicRel y S → y.HomotopicRel z S → x.HomotopicRel z S","state_after":"no goals","tactic":"apply trans","premises":[{"full_name":"ContinuousMap.HomotopicRel.trans","def_path":"Mathlib/Topology/Homotopy/Basic.lean","def_pos":[638,8],"def_end_pos":[638,13]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"ite_zero_add","start":[128,0],"end":[131,29],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM✝ : Type u_4\nM : Type u\ninst✝¹ : MulOneClass M\nP : Prop\ninst✝ : Decidable P\na b : M\n⊢ (if P then 1 else a * b) = (if P then 1 else a) * if P then 1 else b","state_after":"no goals","tactic":"by_cases h : P <;> simp [h]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Sum/Basic.lean","commit":"","full_name":"Sum.map_surjective","start":[203,0],"end":[215,29],"file_path":"Mathlib/Data/Sum/Basic.lean","tactics":[{"state_before":"α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → γ\ng : β → δ\nh : Surjective (Sum.map f g)\nc : γ\n⊢ ∃ a, f a = c","state_after":"case intro.inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nc : γ\na : α\nh : Sum.map f g (inl a) = inl c\n⊢ ∃ a, f a = c\n\ncase intro.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nc : γ\nb : β\nh : Sum.map f g (inr b) = inl c\n⊢ ∃ a, f a = c","tactic":"obtain ⟨a | b, h⟩ := h (inl c)","premises":[{"full_name":"Sum.inl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[132,4],"def_end_pos":[132,7]}]},{"state_before":"α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → γ\ng : β → δ\nh : Surjective (Sum.map f g)\nd : δ\n⊢ ∃ a, g a = d","state_after":"case intro.inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nd : δ\na : α\nh : Sum.map f g (inl a) = inr d\n⊢ ∃ a, g a = d\n\ncase intro.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → γ\ng : β → δ\nh✝ : Surjective (Sum.map f g)\nd : δ\nb : β\nh : Sum.map f g (inr b) = inr d\n⊢ ∃ a, g a = d","tactic":"obtain ⟨a | b, h⟩ := h (inr d)","premises":[{"full_name":"Sum.inr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[134,4],"def_end_pos":[134,7]}]}]}
{"url":"Mathlib/Analysis/Convex/EGauge.lean","commit":"","full_name":"egauge_zero_right","start":[105,0],"end":[108,40],"file_path":"Mathlib/Analysis/Convex/EGauge.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nα : Type u_2\nE : Type u_3\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns t : Set E\nx y : E\nr : ℝ≥0∞\nhs : s.Nonempty\n⊢ egauge 𝕜 s 0 = 0","state_after":"𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nα : Type u_2\nE : Type u_3\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns t : Set E\nx y : E\nr : ℝ≥0∞\nhs : s.Nonempty\nthis : 0 ∈ 0 • s\n⊢ egauge 𝕜 s 0 = 0","tactic":"have : 0 ∈ (0 : 𝕜) • s := by simp [zero_smul_set hs]","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.zero_smul_set","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[650,16],"def_end_pos":[650,29]}]},{"state_before":"𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nα : Type u_2\nE : Type u_3\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns t : Set E\nx y : E\nr : ℝ≥0∞\nhs : s.Nonempty\nthis : 0 ∈ 0 • s\n⊢ egauge 𝕜 s 0 = 0","state_after":"no goals","tactic":"simpa using egauge_le_of_mem_smul this","premises":[{"full_name":"egauge_le_of_mem_smul","def_path":"Mathlib/Analysis/Convex/EGauge.lean","def_pos":[55,6],"def_end_pos":[55,27]}]}]}
{"url":"Mathlib/Data/Set/Subsingleton.lean","commit":"","full_name":"Set.not_nontrivial_singleton","start":[207,0],"end":[211,36],"file_path":"Mathlib/Data/Set/Subsingleton.lean","tactics":[{"state_before":"α : Type u\na : α\ns t : Set α\nx : α\nH : {x}.Nontrivial\n⊢ False","state_after":"α : Type u\na : α\ns t : Set α\nx : α\nH : ∃ y ∈ {x}, y ≠ x\n⊢ False","tactic":"rw [nontrivial_iff_exists_ne (mem_singleton x)] at H","premises":[{"full_name":"Set.mem_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1015,8],"def_end_pos":[1015,21]},{"full_name":"Set.nontrivial_iff_exists_ne","def_path":"Mathlib/Data/Set/Subsingleton.lean","def_pos":[174,8],"def_end_pos":[174,32]}]},{"state_before":"α : Type u\na : α\ns t : Set α\nx : α\nH : ∃ y ∈ {x}, y ≠ x\n⊢ False","state_after":"α : Type u\na : α\ns t : Set α\nx : α\nH : ∃ y ∈ {x}, y ≠ x\ny : α\nhy : y ∈ {x}\nhya : y ≠ x\n⊢ False","tactic":"let ⟨y, hy, hya⟩ := H","premises":[]},{"state_before":"α : Type u\na : α\ns t : Set α\nx : α\nH : ∃ y ∈ {x}, y ≠ x\ny : α\nhy : y ∈ {x}\nhya : y ≠ x\n⊢ False","state_after":"no goals","tactic":"exact hya (mem_singleton_iff.1 hy)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.mem_singleton_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[999,8],"def_end_pos":[999,25]}]}]}
{"url":"Mathlib/LinearAlgebra/PerfectPairing.lean","commit":"","full_name":"PerfectPairing.toDualRight_symm_comp_toDualLeft","start":[102,0],"end":[105,39],"file_path":"Mathlib/LinearAlgebra/PerfectPairing.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : PerfectPairing R M N\n⊢ ↑p.toDualRight.symm.dualMap ∘ₗ ↑p.toDualLeft = Dual.eval R M","state_after":"case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : PerfectPairing R M N\nx : M\n⊢ (↑p.toDualRight.symm.dualMap ∘ₗ ↑p.toDualLeft) x = (Dual.eval R M) x","tactic":"ext1 x","premises":[]},{"state_before":"case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : PerfectPairing R M N\nx : M\n⊢ (↑p.toDualRight.symm.dualMap ∘ₗ ↑p.toDualLeft) x = (Dual.eval R M) x","state_after":"no goals","tactic":"exact p.toDualRight_symm_toDualLeft x","premises":[{"full_name":"PerfectPairing.toDualRight_symm_toDualLeft","def_path":"Mathlib/LinearAlgebra/PerfectPairing.lean","def_pos":[96,8],"def_end_pos":[96,35]}]}]}
{"url":"Mathlib/Order/CompleteBooleanAlgebra.lean","commit":"","full_name":"inf_iSup₂_eq","start":[370,0],"end":[372,25],"file_path":"Mathlib/Order/CompleteBooleanAlgebra.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\ninst✝ : Frame α\ns t : Set α\na✝ b : α\nf : (i : ι) → κ i → α\na : α\n⊢ a ⊓ ⨆ i, ⨆ j, f i j = ⨆ i, ⨆ j, a ⊓ f i j","state_after":"no goals","tactic":"simp only [inf_iSup_eq]","premises":[{"full_name":"inf_iSup_eq","def_path":"Mathlib/Order/CompleteBooleanAlgebra.lean","def_pos":[363,8],"def_end_pos":[363,19]}]}]}
{"url":"Mathlib/Data/DFinsupp/NeLocus.lean","commit":"","full_name":"DFinsupp.mem_neLocus","start":[37,0],"end":[40,46],"file_path":"Mathlib/Data/DFinsupp/NeLocus.lean","tactics":[{"state_before":"α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf✝ g✝ f g : Π₀ (a : α), N a\na : α\n⊢ a ∈ f.neLocus g ↔ f a ≠ g a","state_after":"no goals","tactic":"simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,\n    and_iff_right_iff_imp] using Ne.ne_or_ne _","premises":[{"full_name":"DFinsupp.mem_support_iff","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[1010,8],"def_end_pos":[1010,23]},{"full_name":"DFinsupp.neLocus","def_path":"Mathlib/Data/DFinsupp/NeLocus.lean","def_pos":[34,4],"def_end_pos":[34,11]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"Finset.mem_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1191,8],"def_end_pos":[1191,17]},{"full_name":"Ne.ne_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[527,8],"def_end_pos":[527,19]},{"full_name":"and_iff_right_iff_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[165,16],"def_end_pos":[165,37]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","commit":"","full_name":"ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat","start":[205,0],"end":[216,69],"file_path":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = ((κ a) (s ×ˢ Iic x)).toReal","state_after":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ENNReal.ofReal (∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a) = (κ a) (s ×ˢ Iic x)\n\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ (κ a) (s ×ˢ Iic x) ≠ ⊤\n\ncase hp\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a\n\ncase hq\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ ((κ a) (s ×ˢ Iic x)).toReal","tactic":"rw [← ENNReal.ofReal_eq_ofReal_iff, ENNReal.ofReal_toReal]","premises":[{"full_name":"ENNReal.ofReal_eq_ofReal_iff","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[178,8],"def_end_pos":[178,28]},{"full_name":"ENNReal.ofReal_toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[198,8],"def_end_pos":[198,21]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ENNReal.ofReal (∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a) = (κ a) (s ×ˢ Iic x)\n\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ (κ a) (s ×ˢ Iic x) ≠ ⊤\n\ncase hp\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a\n\ncase hq\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ ((κ a) (s ×ˢ Iic x)).toReal","state_after":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ (κ a) (s ×ˢ Iic x) ≠ ⊤\n\ncase hp\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a\n\ncase hq\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ ((κ a) (s ×ˢ Iic x)).toReal\n\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ENNReal.ofReal (∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a) = (κ a) (s ×ˢ Iic x)","tactic":"rotate_left","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ENNReal.ofReal (∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a) = (κ a) (s ×ˢ Iic x)","state_after":"case hfi\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ Integrable (fun b => ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ((ν a).restrict s)\n\ncase f_nn\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ᶠ[ae ((ν a).restrict s)] fun b => ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x","tactic":"rw [ofReal_integral_eq_lintegral_ofReal, setLIntegral_stieltjesOfMeasurableRat hf _ _ hs]","premises":[{"full_name":"MeasureTheory.ofReal_integral_eq_lintegral_ofReal","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1087,8],"def_end_pos":[1087,43]},{"full_name":"ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat","def_path":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","def_pos":[119,6],"def_end_pos":[119,43]}]}]}
{"url":"Mathlib/Order/Heyting/Basic.lean","commit":"","full_name":"disjoint_compl_compl_right_iff","start":[716,0],"end":[718,60],"file_path":"Mathlib/Order/Heyting/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ Disjoint a bᶜᶜ ↔ Disjoint a b","state_after":"no goals","tactic":"simp_rw [← le_compl_iff_disjoint_right, compl_compl_compl]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"compl_compl_compl","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[709,8],"def_end_pos":[709,25]},{"full_name":"le_compl_iff_disjoint_right","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[625,8],"def_end_pos":[625,35]}]}]}
{"url":"Mathlib/GroupTheory/GroupAction/Blocks.lean","commit":"","full_name":"MulAction.IsBlock.isBlockSystem","start":[327,0],"end":[347,36],"file_path":"Mathlib/GroupTheory/GroupAction/Blocks.lean","tactics":[{"state_before":"G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ IsBlockSystem G (Set.range fun g => g • B)","state_after":"case nonempty\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∅ ∉ Set.range fun g => g • B\n\ncase cover\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ (a : X), ∃! b, (b ∈ Set.range fun g => g • B) ∧ a ∈ b\n\ncase mem_blocks\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b","tactic":"refine ⟨⟨?nonempty, ?cover⟩, ?mem_blocks⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]},{"state_before":"case nonempty\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∅ ∉ Set.range fun g => g • B\n\ncase cover\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ (a : X), ∃! b, (b ∈ Set.range fun g => g • B) ∧ a ∈ b\n\ncase mem_blocks\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b","state_after":"case nonempty\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∅ ∉ Set.range fun g => g • B\n\ncase cover\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ (a : X), ∃! b, (b ∈ Set.range fun g => g • B) ∧ a ∈ b","tactic":"case mem_blocks => rintro B' ⟨g, rfl⟩; exact hB.translate g","premises":[{"full_name":"MulAction.IsBlock.translate","def_path":"Mathlib/GroupTheory/GroupAction/Blocks.lean","def_pos":[314,8],"def_end_pos":[314,25]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean","commit":"","full_name":"CategoryTheory.PreservesImage.inv_comp_image_ι_map","start":[56,0],"end":[58,68],"file_path":"Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean","tactics":[{"state_before":"A : Type u₁\nB : Type u₂\ninst✝⁷ : Category.{v₁, u₁} A\ninst✝⁶ : Category.{v₂, u₂} B\ninst✝⁵ : HasEqualizers A\ninst✝⁴ : HasImages A\ninst✝³ : StrongEpiCategory B\ninst✝² : HasImages B\nL : A ⥤ B\ninst✝¹ : {X Y Z : A} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PreservesLimit (cospan f g) L\ninst✝ : {X Y Z : A} → (f : X ⟶ Y) → (g : X ⟶ Z) → PreservesColimit (span f g) L\nX Y : A\nf : X ⟶ Y\n⊢ (iso L f).inv ≫ image.ι (L.map f) = L.map (image.ι f)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Nat/Digits.lean","commit":"","full_name":"Nat.NormDigits.digits_one","start":[804,0],"end":[809,74],"file_path":"Mathlib/Data/Nat/Digits.lean","tactics":[{"state_before":"n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\n⊢ b.digits n = [n] ∧ 1 < b ∧ 0 < n","state_after":"n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ b.digits n = [n] ∧ 1 < b ∧ 0 < n","tactic":"have b2 : 1 < b :=\n    lt_iff_add_one_le.mpr (le_trans (add_le_add_right (lt_iff_add_one_le.mp n0) 1) nb)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.lt_iff_add_one_le","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[125,6],"def_end_pos":[125,23]},{"full_name":"add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[74,31],"def_end_pos":[74,47]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]},{"state_before":"n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ b.digits n = [n] ∧ 1 < b ∧ 0 < n","state_after":"n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ b.digits n = [n]","tactic":"refine ⟨?_, b2, n0⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]},{"state_before":"n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ b.digits n = [n]","state_after":"no goals","tactic":"rw [Nat.digits_def' b2 n0, Nat.mod_eq_of_lt nb,\n    (Nat.div_eq_zero_iff ((zero_le n).trans_lt nb)).2 nb, Nat.digits_zero]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.digits_def'","def_path":"Mathlib/Data/Nat/Digits.lean","def_pos":[114,8],"def_end_pos":[114,19]},{"full_name":"Nat.digits_zero","def_path":"Mathlib/Data/Nat/Digits.lean","def_pos":[82,8],"def_end_pos":[82,19]},{"full_name":"Nat.div_eq_zero_iff","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[980,16],"def_end_pos":[980,31]},{"full_name":"Nat.mod_eq_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[131,8],"def_end_pos":[131,20]},{"full_name":"Nat.zero_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1663,8],"def_end_pos":[1663,19]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Circumcenter.lean","commit":"","full_name":"Affine.Simplex.circumcenter_eq_centroid","start":[348,0],"end":[365,35],"file_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\n⊢ s.circumcenter = centroid ℝ univ s.points","state_after":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nhr :\n  Set.univ.Pairwise fun i j =>\n    dist (s.points i) (centroid ℝ univ s.points) = dist (s.points j) (centroid ℝ univ s.points)\n⊢ s.circumcenter = centroid ℝ univ s.points","tactic":"have hr :\n    Set.Pairwise Set.univ fun i j : Fin 2 =>\n      dist (s.points i) (Finset.univ.centroid ℝ s.points) =\n        dist (s.points j) (Finset.univ.centroid ℝ s.points) := by\n    intro i hi j hj hij\n    rw [Finset.centroid_pair_fin, dist_eq_norm_vsub V (s.points i),\n      dist_eq_norm_vsub V (s.points j), vsub_vadd_eq_vsub_sub, vsub_vadd_eq_vsub_sub, ←\n      one_smul ℝ (s.points i -ᵥ s.points 0), ← one_smul ℝ (s.points j -ᵥ s.points 0)]\n    fin_cases i <;> fin_cases j <;> simp [-one_smul, ← sub_smul] <;> norm_num","premises":[{"full_name":"Affine.Simplex.points","def_path":"Mathlib/LinearAlgebra/AffineSpace/Independent.lean","def_pos":[763,2],"def_end_pos":[763,8]},{"full_name":"Dist.dist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[80,2],"def_end_pos":[80,6]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Finset.centroid","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[748,4],"def_end_pos":[748,12]},{"full_name":"Finset.centroid_pair_fin","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[783,8],"def_end_pos":[783,25]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.Pairwise","def_path":"Mathlib/Logic/Pairwise.lean","def_pos":[62,14],"def_end_pos":[62,22]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"VSub.vsub","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[94,2],"def_end_pos":[94,6]},{"full_name":"dist_eq_norm_vsub","def_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","def_pos":[68,8],"def_end_pos":[68,25]},{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]},{"full_name":"sub_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[245,8],"def_end_pos":[245,16]},{"full_name":"vsub_vadd_eq_vsub_sub","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[147,8],"def_end_pos":[147,29]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nhr :\n  Set.univ.Pairwise fun i j =>\n    dist (s.points i) (centroid ℝ univ s.points) = dist (s.points j) (centroid ℝ univ s.points)\n⊢ s.circumcenter = centroid ℝ univ s.points","state_after":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nhr : ∃ z, ∀ x ∈ Set.univ, dist (s.points x) (centroid ℝ univ s.points) = z\n⊢ s.circumcenter = centroid ℝ univ s.points","tactic":"rw [Set.pairwise_eq_iff_exists_eq] at hr","premises":[{"full_name":"Set.pairwise_eq_iff_exists_eq","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[113,8],"def_end_pos":[113,33]}]},{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nhr : ∃ z, ∀ x ∈ Set.univ, dist (s.points x) (centroid ℝ univ s.points) = z\n⊢ s.circumcenter = centroid ℝ univ s.points","state_after":"case intro\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nr : ℝ\nhr : ∀ x ∈ Set.univ, dist (s.points x) (centroid ℝ univ s.points) = r\n⊢ s.circumcenter = centroid ℝ univ s.points","tactic":"cases' hr with r hr","premises":[]},{"state_before":"case intro\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nr : ℝ\nhr : ∀ x ∈ Set.univ, dist (s.points x) (centroid ℝ univ s.points) = r\n⊢ s.circumcenter = centroid ℝ univ s.points","state_after":"no goals","tactic":"exact\n    (s.eq_circumcenter_of_dist_eq\n        (centroid_mem_affineSpan_of_card_eq_add_one ℝ _ (Finset.card_fin 2)) fun i =>\n        hr i (Set.mem_univ _)).symm","premises":[{"full_name":"Affine.Simplex.eq_circumcenter_of_dist_eq","def_path":"Mathlib/Geometry/Euclidean/Circumcenter.lean","def_pos":[299,8],"def_end_pos":[299,34]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset.card_fin","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[291,8],"def_end_pos":[291,23]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"centroid_mem_affineSpan_of_card_eq_add_one","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[1114,8],"def_end_pos":[1114,50]}]}]}
{"url":"Mathlib/Algebra/Module/Submodule/Map.lean","commit":"","full_name":"Submodule.map_comap_subtype","start":[370,0],"end":[372,97],"file_path":"Mathlib/Algebra/Module/Submodule/Map.lean","tactics":[{"state_before":"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nx✝ : M\nF : Type u_9\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F σ₁₂ M M₂\nx : M\n⊢ x ∈ map p.subtype (comap p.subtype p') → x ∈ p ⊓ p'","state_after":"case intro.mk.intro\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nx : M\nF : Type u_9\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F σ₁₂ M M₂\nval✝ : M\nh₁ : val✝ ∈ p\nh₂ : ⟨val✝, h₁⟩ ∈ ↑(comap p.subtype p')\n⊢ p.subtype ⟨val✝, h₁⟩ ∈ p ⊓ p'","tactic":"rintro ⟨⟨_, h₁⟩, h₂, rfl⟩","premises":[]},{"state_before":"case intro.mk.intro\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nx : M\nF : Type u_9\ninst✝¹ : FunLike F M M₂\ninst✝ : SemilinearMapClass F σ₁₂ M M₂\nval✝ : M\nh₁ : val✝ ∈ p\nh₂ : ⟨val✝, h₁⟩ ∈ ↑(comap p.subtype p')\n⊢ p.subtype ⟨val✝, h₁⟩ ∈ p ⊓ p'","state_after":"no goals","tactic":"exact ⟨h₁, h₂⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]}]}
{"url":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","commit":"","full_name":"MvQPF.Fix.rec_eq","start":[189,0],"end":[203,85],"file_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","tactics":[{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)","state_after":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)","tactic":"have : recF g ∘ fixToW = Fix.rec g := by\n    apply funext\n    apply Quotient.ind\n    intro x\n    apply recF_eq_of_wEquiv\n    apply wrepr_equiv","premises":[{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"MvQPF.Fix.rec","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[174,4],"def_end_pos":[174,11]},{"full_name":"MvQPF.fixToW","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"MvQPF.recF","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[57,4],"def_end_pos":[57,8]},{"full_name":"MvQPF.recF_eq_of_wEquiv","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[82,8],"def_end_pos":[82,25]},{"full_name":"MvQPF.wrepr_equiv","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[125,8],"def_end_pos":[125,19]},{"full_name":"Quotient.ind","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1617,18],"def_end_pos":[1617,21]},{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)","state_after":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ recF g ((P F).wMk' ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x)","tactic":"conv =>\n    lhs\n    rw [Fix.rec, Fix.mk]\n    dsimp","premises":[{"full_name":"MvQPF.Fix.mk","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[182,4],"def_end_pos":[182,10]},{"full_name":"MvQPF.Fix.rec","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[174,4],"def_end_pos":[174,11]}]},{"state_before":"n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ recF g ((P F).wMk' ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x)","state_after":"case mk\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : (P F).B a ⟹ α ::: Fix F α\nh : repr x = ⟨a, f⟩\n⊢ recF g ((P F).wMk' ((TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)","tactic":"cases' h : repr x with a f","premises":[{"full_name":"MvQPF.repr","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]}]},{"state_before":"case mk\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : (P F).B a ⟹ α ::: Fix F α\nh : repr x = ⟨a, f⟩\n⊢ recF g ((P F).wMk' ((TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)","state_after":"case mk\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : (P F).B a ⟹ α ::: Fix F α\nh : repr x = ⟨a, f⟩\n⊢ g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)","tactic":"rw [MvPFunctor.map_eq, recF_eq', ← MvPFunctor.map_eq, MvPFunctor.wDest'_wMk']","premises":[{"full_name":"MvPFunctor.map_eq","def_path":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","def_pos":[59,8],"def_end_pos":[59,14]},{"full_name":"MvPFunctor.wDest'_wMk'","def_path":"Mathlib/Data/PFunctor/Multivariate/W.lean","def_pos":[252,8],"def_end_pos":[252,19]},{"full_name":"MvQPF.recF_eq'","def_path":"Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean","def_pos":[65,8],"def_end_pos":[65,16]}]},{"state_before":"case mk\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : (P F).B a ⟹ α ::: Fix F α\nh : repr x = ⟨a, f⟩\n⊢ g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)","state_after":"no goals","tactic":"rw [← MvPFunctor.comp_map, abs_map, ← h, abs_repr, ← appendFun_comp, id_comp, this]","premises":[{"full_name":"MvPFunctor.comp_map","def_path":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","def_pos":[66,8],"def_end_pos":[66,16]},{"full_name":"MvQPF.abs_map","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[91,2],"def_end_pos":[91,9]},{"full_name":"MvQPF.abs_repr","def_path":"Mathlib/Data/QPF/Multivariate/Basic.lean","def_pos":[90,2],"def_end_pos":[90,10]},{"full_name":"TypeVec.appendFun_comp","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[213,8],"def_end_pos":[213,22]},{"full_name":"TypeVec.id_comp","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[72,8],"def_end_pos":[72,15]}]}]}
{"url":"Mathlib/Probability/Independence/Conditional.lean","commit":"","full_name":"ProbabilityTheory.iCondIndepSets_iff","start":[154,0],"end":[191,61],"file_path":"Mathlib/Probability/Independence/Conditional.lean","tactics":[{"state_before":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ iCondIndepSets m' hm' π μ ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","state_after":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","tactic":"simp only [iCondIndepSets, Kernel.iIndepSets]","premises":[{"full_name":"ProbabilityTheory.Kernel.iIndepSets","def_path":"Mathlib/Probability/Independence/Kernel.lean","def_pos":[62,4],"def_end_pos":[62,14]},{"full_name":"ProbabilityTheory.iCondIndepSets","def_path":"Mathlib/Probability/Independence/Conditional.lean","def_pos":[74,4],"def_end_pos":[74,18]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","state_after":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","tactic":"have h_eq' : ∀ (s : Finset ι) (f : ι → Set Ω) (_H : ∀ i, i ∈ s → f i ∈ π i) i (_hi : i ∈ s),\n      (fun ω ↦ ENNReal.toReal (condexpKernel μ m' ω (f i))) =ᵐ[μ] μ⟦f i | m'⟧ :=\n    fun s f H i hi ↦ condexpKernel_ae_eq_condexp hm' (hπ i (f i) (H i hi))","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[90,30],"def_end_pos":[90,37]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ProbabilityTheory.condexpKernel","def_path":"Mathlib/Probability/Kernel/Condexp.lean","def_pos":[67,30],"def_end_pos":[67,43]},{"full_name":"ProbabilityTheory.condexpKernel_ae_eq_condexp","def_path":"Mathlib/Probability/Kernel/Condexp.lean","def_pos":[162,6],"def_end_pos":[162,33]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","state_after":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","tactic":"have h_eq : ∀ (s : Finset ι) (f : ι → Set Ω) (_H : ∀ i, i ∈ s → f i ∈ π i), ∀ᵐ ω ∂μ,\n      ∀ i ∈ s, ENNReal.toReal (condexpKernel μ m' ω (f i)) = (μ⟦f i | m'⟧) ω := by\n    intros s f H\n    simp_rw [← Finset.mem_coe]\n    rw [ae_ball_iff (Finset.countable_toSet s)]\n    exact h_eq' s f H","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.countable_toSet","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[300,8],"def_end_pos":[300,30]},{"full_name":"Finset.mem_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[195,8],"def_end_pos":[195,15]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.ae_ball_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[105,8],"def_end_pos":[105,19]},{"full_name":"MeasureTheory.condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[90,30],"def_end_pos":[90,37]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ProbabilityTheory.condexpKernel","def_path":"Mathlib/Probability/Kernel/Condexp.lean","def_pos":[67,30],"def_end_pos":[67,43]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","state_after":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\nh_inter_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      (fun ω => (((condexpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᶠ[ae μ] μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m']\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","tactic":"have h_inter_eq : ∀ (s : Finset ι) (f : ι → Set Ω) (_H : ∀ i, i ∈ s → f i ∈ π i),\n      (fun ω ↦ ENNReal.toReal (condexpKernel μ m' ω (⋂ i ∈ s, f i)))\n        =ᵐ[μ] μ⟦⋂ i ∈ s, f i | m'⟧ := by\n    refine fun s f H ↦ condexpKernel_ae_eq_condexp hm' ?_\n    exact MeasurableSet.biInter (Finset.countable_toSet _) (fun i hi ↦ hπ i _ (H i hi))","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.countable_toSet","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[300,8],"def_end_pos":[300,30]},{"full_name":"MeasurableSet.biInter","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[142,8],"def_end_pos":[142,29]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[90,30],"def_end_pos":[90,37]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ProbabilityTheory.condexpKernel","def_path":"Mathlib/Probability/Kernel/Condexp.lean","def_pos":[67,30],"def_end_pos":[67,43]},{"full_name":"ProbabilityTheory.condexpKernel_ae_eq_condexp","def_path":"Mathlib/Probability/Kernel/Condexp.lean","def_pos":[162,6],"def_end_pos":[162,33]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[182,4],"def_end_pos":[182,10]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\nh_inter_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      (fun ω => (((condexpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᶠ[ae μ] μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m']\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']","state_after":"case refine_1\nΩ : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\nh_inter_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      (fun ω => (((condexpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᶠ[ae μ] μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m']\ns : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ s, f i ∈ π i\nh : ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)\n⊢ μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']\n\ncase refine_2\nΩ : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\nh_inter_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      (fun ω => (((condexpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᶠ[ae μ] μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m']\ns : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ s, f i ∈ π i\nh : μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']\n⊢ ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)","tactic":"refine ⟨fun h s f hf ↦ ?_, fun h s f hf ↦ ?_⟩ <;> specialize h s hf","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Order/Filter/Basic.lean","commit":"","full_name":"Filter.range_mem_map","start":[1674,0],"end":[1676,30],"file_path":"Mathlib/Order/Filter/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f : Filter α\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ range m ∈ map m f","state_after":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f : Filter α\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ m '' univ ∈ map m f","tactic":"rw [← image_univ]","premises":[{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f : Filter α\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ m '' univ ∈ map m f","state_after":"no goals","tactic":"exact image_mem_map univ_mem","premises":[{"full_name":"Filter.image_mem_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1664,8],"def_end_pos":[1664,21]},{"full_name":"Filter.univ_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[136,8],"def_end_pos":[136,16]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Splits.lean","commit":"","full_name":"Polynomial.splits_id_iff_splits","start":[138,0],"end":[139,38],"file_path":"Mathlib/Algebra/Polynomial/Splits.lean","tactics":[{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\n⊢ Splits (RingHom.id L) (map i f) ↔ Splits i f","state_after":"no goals","tactic":"rw [splits_map_iff, RingHom.id_comp]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Polynomial.splits_map_iff","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[109,8],"def_end_pos":[109,22]},{"full_name":"RingHom.id_comp","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[584,8],"def_end_pos":[584,15]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","commit":"","full_name":"CategoryTheory.NonPreadditiveAbelian.add_neg_self","start":[367,0],"end":[367,84],"file_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a + -a = 0","state_after":"no goals","tactic":"rw [add_neg, sub_self]","premises":[{"full_name":"CategoryTheory.NonPreadditiveAbelian.add_neg","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[365,8],"def_end_pos":[365,15]},{"full_name":"CategoryTheory.NonPreadditiveAbelian.sub_self","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[333,8],"def_end_pos":[333,16]}]}]}
{"url":"Mathlib/Algebra/Order/Group/MinMax.lean","commit":"","full_name":"abs_min_sub_min_le_max","start":[84,0],"end":[86,46],"file_path":"Mathlib/Algebra/Order/Group/MinMax.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c✝ a b c d : α\n⊢ |min a b - min c d| ≤ max |a - c| |b - d|","state_after":"no goals","tactic":"simpa only [max_neg_neg, neg_sub_neg, abs_sub_comm] using\n    abs_max_sub_max_le_max (-a) (-b) (-c) (-d)","premises":[{"full_name":"abs_max_sub_max_le_max","def_path":"Mathlib/Algebra/Order/Group/MinMax.lean","def_pos":[78,8],"def_end_pos":[78,30]},{"full_name":"abs_sub_comm","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[72,2],"def_end_pos":[72,13]},{"full_name":"max_neg_neg","def_path":"Mathlib/Algebra/Order/Group/MinMax.lean","def_pos":[41,14],"def_end_pos":[41,25]},{"full_name":"neg_sub_neg","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[530,2],"def_end_pos":[530,13]}]}]}
{"url":"Mathlib/Topology/Instances/EReal.lean","commit":"","full_name":"_private.Mathlib.Topology.Instances.EReal.0.EReal.limsup_add_le_of_lt","start":[161,0],"end":[172,78],"file_path":"Mathlib/Topology/Instances/EReal.lean","tactics":[{"state_before":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\n⊢ limsup (u + v) f ≤ a + b","state_after":"case inl\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nu v : α → EReal\na b : EReal\nha : limsup u ⊥ < a\nhb : limsup v ⊥ < b\n⊢ limsup (u + v) ⊥ ≤ a + b\n\ncase inr\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ limsup (u + v) f ≤ a + b","tactic":"rcases eq_or_neBot f with (rfl | _)","premises":[{"full_name":"Filter.eq_or_neBot","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[484,8],"def_end_pos":[484,19]}]},{"state_before":"case inr\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ limsup (u + v) f ≤ a + b","state_after":"case inr\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ limsup (u + v) f ≤ limsup (fun x => a + b) f","tactic":"rw [← @limsup_const EReal α _ f _ (a + b)]","premises":[{"full_name":"EReal","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[55,4],"def_end_pos":[55,9]},{"full_name":"Filter.limsup_const","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[646,8],"def_end_pos":[646,20]}]},{"state_before":"case inr\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ limsup (u + v) f ≤ limsup (fun x => a + b) f","state_after":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ ∀ (x : α), u x < a ∧ v x < b → (u + v) x ≤ a + b","tactic":"apply limsup_le_limsup (Eventually.mp (Eventually.and (eventually_lt_of_limsup_lt ha)\n    (eventually_lt_of_limsup_lt hb)) (eventually_of_forall _))","premises":[{"full_name":"EReal.limsup_le_limsup","def_path":"Mathlib/Topology/Instances/EReal.lean","def_pos":[158,6],"def_end_pos":[158,22]},{"full_name":"Filter.Eventually.and","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[973,18],"def_end_pos":[973,32]},{"full_name":"Filter.Eventually.mp","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[998,8],"def_end_pos":[998,21]},{"full_name":"Filter.eventually_lt_of_limsup_lt","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[1162,8],"def_end_pos":[1162,34]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]}]},{"state_before":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ ∀ (x : α), u x < a ∧ v x < b → (u + v) x ≤ a + b","state_after":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ ∀ (x : α), u x < a → v x < b → u x + v x ≤ a + b","tactic":"simp only [Pi.add_apply, and_imp]","premises":[{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]}]},{"state_before":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ ∀ (x : α), u x < a → v x < b → u x + v x ≤ a + b","state_after":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\nx : α\n⊢ u x < a → v x < b → u x + v x ≤ a + b","tactic":"intro x","premises":[]},{"state_before":"α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\nx : α\n⊢ u x < a → v x < b → u x + v x ≤ a + b","state_after":"no goals","tactic":"exact fun ux_lt_a vx_lt_b ↦ add_le_add (le_of_lt ux_lt_a) (le_of_lt vx_lt_b)","premises":[{"full_name":"add_le_add","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[182,31],"def_end_pos":[182,41]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]}]}]}
{"url":"Mathlib/Algebra/Group/UniqueProds.lean","commit":"","full_name":"UniqueSums.of_addHom","start":[300,0],"end":[308,62],"file_path":"Mathlib/Algebra/Group/UniqueProds.lean","tactics":[{"state_before":"G : Type u\nH : Type v\ninst✝² : Mul G\ninst✝¹ : Mul H\nf : H →ₙ* G\nhf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d\ninst✝ : UniqueProds G\nA B : Finset H\nA0 : A.Nonempty\nB0 : B.Nonempty\n⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0","state_after":"no goals","tactic":"classical\n    obtain ⟨a0, ha0, b0, hb0, h⟩ := uniqueMul_of_nonempty (A0.image f) (B0.image f)\n    obtain ⟨a', ha', rfl⟩ := mem_image.mp ha0\n    obtain ⟨b', hb', rfl⟩ := mem_image.mp hb0\n    exact ⟨a', ha', b', hb', UniqueMul.of_mulHom_image f hf h⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.Nonempty.image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[361,18],"def_end_pos":[361,32]},{"full_name":"Finset.mem_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[303,8],"def_end_pos":[303,17]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"UniqueMul.of_mulHom_image","def_path":"Mathlib/Algebra/Group/UniqueProds.lean","def_pos":[141,23],"def_end_pos":[141,38]},{"full_name":"UniqueProds.uniqueMul_of_nonempty","def_path":"Mathlib/Algebra/Group/UniqueProds.lean","def_pos":[224,2],"def_end_pos":[224,23]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Ordinal.lean","commit":"","full_name":"Cardinal.mk_multiset_of_isEmpty","start":[1113,0],"end":[1114,56],"file_path":"Mathlib/SetTheory/Cardinal/Ordinal.lean","tactics":[{"state_before":"α : Type u\ninst✝ : IsEmpty α\n⊢ #(α →₀ ℕ) = 1","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Grothendieck.lean","commit":"","full_name":"CategoryTheory.Grothendieck.grothendieckTypeToCat_counitIso_inv_app_coe","start":[265,0],"end":[302,7],"file_path":"Mathlib/CategoryTheory/Grothendieck.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nX : Grothendieck (G ⋙ typeToCat)\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).obj X ≅ (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).obj X","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base✝, fiber := { as := as✝ } } ≅\n    (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).obj { base := base✝, fiber := { as := as✝ } }","tactic":"rcases X with ⟨_, ⟨⟩⟩","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base✝, fiber := { as := as✝ } } ≅\n    (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).obj { base := base✝, fiber := { as := as✝ } }","state_after":"no goals","tactic":"exact Iso.refl _","premises":[{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ {X Y : Grothendieck (G ⋙ typeToCat)} (f : X ⟶ Y),\n    (𝟭 (Grothendieck (G ⋙ typeToCat))).map f ≫\n        ((fun X =>\n              Grothendieck.casesOn X fun base fiber =>\n                Discrete.casesOn fiber fun as =>\n                  Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n            Y).hom =\n      ((fun X =>\n              Grothendieck.casesOn X fun base fiber =>\n                Discrete.casesOn fiber fun as =>\n                  Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n            X).hom ≫\n        (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).map f","state_after":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).map { base := base, fiber := { down := { down := f } } } ≫\n      ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝, fiber := { as := as✝ } }).hom =\n    ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝¹, fiber := { as := as✝¹ } }).hom ≫\n      (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).map\n        { base := base, fiber := { down := { down := f } } }","tactic":"rintro ⟨_, ⟨⟩⟩ ⟨_, ⟨⟩⟩ ⟨base, ⟨⟨f⟩⟩⟩","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).map { base := base, fiber := { down := { down := f } } } ≫\n      ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝, fiber := { as := as✝ } }).hom =\n    ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝¹, fiber := { as := as✝¹ } }).hom ≫\n      (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).map\n        { base := base, fiber := { down := { down := f } } }","state_after":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } ≫ 𝟙 { base := base✝, fiber := { as := as✝ } } =\n    𝟙 { base := base✝¹, fiber := { as := as✝¹ } } ≫\n      (grothendieckTypeToCatInverse G).map\n        ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","tactic":"dsimp at *","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } ≫ 𝟙 { base := base✝, fiber := { as := as✝ } } =\n    𝟙 { base := base✝¹, fiber := { as := as✝¹ } } ≫\n      (grothendieckTypeToCatInverse G).map\n        ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","state_after":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } =\n    (grothendieckTypeToCatInverse G).map\n      ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","tactic":"simp","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } =\n    (grothendieckTypeToCatInverse G).map\n      ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nX : G.Elements\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X","state_after":"case mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝ : C\nsnd✝ : G.obj fst✝\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst✝, snd✝⟩ ≅ (𝟭 G.Elements).obj ⟨fst✝, snd✝⟩","tactic":"cases X","premises":[]},{"state_before":"case mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝ : C\nsnd✝ : G.obj fst✝\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst✝, snd✝⟩ ≅ (𝟭 G.Elements).obj ⟨fst✝, snd✝⟩","state_after":"no goals","tactic":"exact Iso.refl _","premises":[{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ {X Y : G.Elements} (f : X ⟶ Y),\n    (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map f ≫\n        ((fun X =>\n              Sigma.casesOn (motive := fun t =>\n                X = t →\n                  ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                X\n                (fun fst snd h =>\n                  ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                ⋯)\n            Y).hom =\n      ((fun X =>\n              Sigma.casesOn (motive := fun t =>\n                X = t →\n                  ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                X\n                (fun fst snd h =>\n                  ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                ⋯)\n            X).hom ≫\n        (𝟭 G.Elements).map f","state_after":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map ⟨f, e⟩ ≫\n      ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝, snd✝⟩).hom =\n    ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝¹, snd✝¹⟩).hom ≫\n      (𝟭 G.Elements).map ⟨f, e⟩","tactic":"rintro ⟨⟩ ⟨⟩ ⟨f, e⟩","premises":[]},{"state_before":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map ⟨f, e⟩ ≫\n      ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝, snd✝⟩).hom =\n    ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝¹, snd✝¹⟩).hom ≫\n      (𝟭 G.Elements).map ⟨f, e⟩","state_after":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) ≫\n      𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝, snd✝⟩)) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝¹, snd✝¹⟩)) ≫ ⟨f, e⟩","tactic":"dsimp at *","premises":[]},{"state_before":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) ≫\n      𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝, snd✝⟩)) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝¹, snd✝¹⟩)) ≫ ⟨f, e⟩","state_after":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) = ⟨f, e⟩","tactic":"simp","premises":[]},{"state_before":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) = ⟨f, e⟩","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ (X : Grothendieck (G ⋙ typeToCat)),\n    (grothendieckTypeToCatFunctor G).map\n          ((NatIso.ofComponents\n                  (fun X =>\n                    Grothendieck.casesOn X fun base fiber =>\n                      Discrete.casesOn fiber fun as =>\n                        Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n                  ⋯).hom.app\n            X) ≫\n        (NatIso.ofComponents\n                (fun X =>\n                  Sigma.casesOn (motive := fun t =>\n                    X = t →\n                      ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                    X\n                    (fun fst snd h =>\n                      ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                    ⋯)\n                ⋯).hom.app\n          ((grothendieckTypeToCatFunctor G).obj X) =\n      𝟙 ((grothendieckTypeToCatFunctor G).obj X)","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map\n        ((NatIso.ofComponents\n                (fun X =>\n                  Grothendieck.casesOn X fun base fiber =>\n                    Discrete.casesOn fiber fun as =>\n                      Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n                ⋯).hom.app\n          { base := base✝, fiber := { as := as✝ } }) ≫\n      (NatIso.ofComponents\n              (fun X =>\n                Sigma.casesOn (motive := fun t =>\n                  X = t →\n                    ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                  X\n                  (fun fst snd h =>\n                    ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                  ⋯)\n              ⋯).hom.app\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","tactic":"rintro ⟨_, ⟨⟩⟩","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map\n        ((NatIso.ofComponents\n                (fun X =>\n                  Grothendieck.casesOn X fun base fiber =>\n                    Discrete.casesOn fiber fun as =>\n                      Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n                ⋯).hom.app\n          { base := base✝, fiber := { as := as✝ } }) ≫\n      (NatIso.ofComponents\n              (fun X =>\n                Sigma.casesOn (motive := fun t =>\n                  X = t →\n                    ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                  X\n                  (fun fst snd h =>\n                    ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                  ⋯)\n              ⋯).hom.app\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map (𝟙 { base := base✝, fiber := { as := as✝ } }) ≫\n      (Sigma.rec (motive := fun t =>\n          (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n            ((grothendieckTypeToCatFunctor G).obj\n                ((grothendieckTypeToCatInverse G).obj\n                  ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n              (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n          (fun fst snd h =>\n            ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n          ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","tactic":"dsimp","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map (𝟙 { base := base✝, fiber := { as := as✝ } }) ≫\n      (Sigma.rec (motive := fun t =>\n          (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n            ((grothendieckTypeToCatFunctor G).obj\n                ((grothendieckTypeToCatInverse G).obj\n                  ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n              (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n          (fun fst snd h =>\n            ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n          ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (Sigma.rec (motive := fun t =>\n        (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n          ((grothendieckTypeToCatFunctor G).obj\n              ((grothendieckTypeToCatInverse G).obj\n                ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n            (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n        (fun fst snd h =>\n          ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","tactic":"simp","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (Sigma.rec (motive := fun t =>\n        (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n          ((grothendieckTypeToCatFunctor G).obj\n              ((grothendieckTypeToCatInverse G).obj\n                ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n            (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n        (fun fst snd h =>\n          ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Real/Cardinality.lean","commit":"","full_name":"Cardinal.mk_real","start":[186,0],"end":[197,14],"file_path":"Mathlib/Data/Real/Cardinality.lean","tactics":[{"state_before":"c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ #ℝ = 𝔠","state_after":"case a\nc : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ #ℝ ≤ 𝔠\n\ncase a\nc : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ 𝔠 ≤ #ℝ","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
{"url":"Mathlib/Topology/Semicontinuous.lean","commit":"","full_name":"lowerSemicontinuousWithinAt_univ_iff","start":[139,0],"end":[141,76],"file_path":"Mathlib/Topology/Semicontinuous.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type u_2\ninst✝ : Preorder β\nf g : α → β\nx : α\ns t : Set α\ny z : β\n⊢ LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x","state_after":"no goals","tactic":"simp [LowerSemicontinuousWithinAt, LowerSemicontinuousAt, nhdsWithin_univ]","premises":[{"full_name":"LowerSemicontinuousAt","def_path":"Mathlib/Topology/Semicontinuous.lean","def_pos":[94,4],"def_end_pos":[94,25]},{"full_name":"LowerSemicontinuousWithinAt","def_path":"Mathlib/Topology/Semicontinuous.lean","def_pos":[82,4],"def_end_pos":[82,31]},{"full_name":"nhdsWithin_univ","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[66,14],"def_end_pos":[66,29]}]}]}
{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.le_add_pred_of_pos","start":[208,0],"end":[208,75],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a✝ b c d m n k : ℕ\np q : ℕ → Prop\na : ℕ\nhb : b ≠ 0\n⊢ a ≤ b + (a - 1)","state_after":"no goals","tactic":"omega","premises":[]}]}
{"url":"Mathlib/RingTheory/Jacobson.lean","commit":"","full_name":"Ideal.Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson","start":[525,0],"end":[548,82],"file_path":"Mathlib/RingTheory/Jacobson.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\n⊢ ((Quotient.mk P).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\n⊢ ((Quotient.mk P).comp C).IsIntegral","tactic":"let P' : Ideal R := P.comap C","premises":[{"full_name":"Ideal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[39,7],"def_end_pos":[39,12]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\n⊢ ((Quotient.mk P).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\n⊢ ((Quotient.mk P).comp C).IsIntegral","tactic":"haveI : P'.IsPrime := comap_isPrime C P","premises":[{"full_name":"Ideal.IsPrime","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[203,6],"def_end_pos":[203,13]},{"full_name":"Ideal.comap_isPrime","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[193,8],"def_end_pos":[193,21]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\n⊢ ((Quotient.mk P).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\n⊢ ((Quotient.mk P).comp C).IsIntegral","tactic":"let f : R[X] →+* Polynomial (R ⧸ P') := Polynomial.mapRingHom (Quotient.mk P')","premises":[{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[641,4],"def_end_pos":[641,14]},{"full_name":"RingHom","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[297,10],"def_end_pos":[297,17]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\n⊢ ((Quotient.mk P).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\n⊢ ((Quotient.mk P).comp C).IsIntegral","tactic":"have hf : Function.Surjective ↑f := map_surjective (Quotient.mk P') Quotient.mk_surjective","premises":[{"full_name":"Function.Surjective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[109,4],"def_end_pos":[109,14]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Ideal.Quotient.mk_surjective","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[141,8],"def_end_pos":[141,21]},{"full_name":"Polynomial.map_surjective","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[711,8],"def_end_pos":[711,22]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\n⊢ ((Quotient.mk P).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ ((Quotient.mk P).comp C).IsIntegral","tactic":"have hPJ : P = (P.map f).comap f := by\n    rw [comap_map_of_surjective _ hf]\n    refine le_antisymm (le_sup_of_le_left le_rfl) (sup_le le_rfl ?_)\n    refine fun p hp =>\n      polynomial_mem_ideal_of_coeff_mem_ideal P p fun n => Quotient.eq_zero_iff_mem.mp ?_\n    simpa only [f, coeff_map, coe_mapRingHom] using (Polynomial.ext_iff.mp hp) n","premises":[{"full_name":"Ideal.Quotient.eq_zero_iff_mem","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[112,8],"def_end_pos":[112,23]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.comap_map_of_surjective","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[350,8],"def_end_pos":[350,31]},{"full_name":"Ideal.map","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[34,4],"def_end_pos":[34,7]},{"full_name":"Ideal.polynomial_mem_ideal_of_coeff_mem_ideal","def_path":"Mathlib/RingTheory/Polynomial/Basic.lean","def_pos":[541,8],"def_end_pos":[541,47]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Polynomial.coe_mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[649,8],"def_end_pos":[649,22]},{"full_name":"Polynomial.coeff_map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[671,8],"def_end_pos":[671,17]},{"full_name":"Polynomial.ext_iff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[687,8],"def_end_pos":[687,15]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"le_sup_of_le_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[117,8],"def_end_pos":[117,25]},{"full_name":"sup_le","def_path":"Mathlib/Order/Lattice.lean","def_pos":[129,8],"def_end_pos":[129,14]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ ((Quotient.mk P).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ ((quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P))) ⋯).comp\n      ((Quotient.mk P).comp C)).IsIntegral","tactic":"refine RingHom.IsIntegral.tower_bot _ _ (injective_quotient_le_comap_map P) ?_","premises":[{"full_name":"Ideal.injective_quotient_le_comap_map","def_path":"Mathlib/RingTheory/Ideal/Over.lean","def_pos":[72,8],"def_end_pos":[72,39]},{"full_name":"RingHom.IsIntegral.tower_bot","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[616,15],"def_end_pos":[616,43]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ ((quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P))) ⋯).comp\n      ((Quotient.mk P).comp C)).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ (((Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)).comp C).comp (Quotient.mk (comap C P))).IsIntegral","tactic":"rw [← quotient_mk_maps_eq]","premises":[{"full_name":"Ideal.quotient_mk_maps_eq","def_path":"Mathlib/RingTheory/Ideal/Over.lean","def_pos":[95,8],"def_end_pos":[95,27]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ (((Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)).comp C).comp (Quotient.mk (comap C P))).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ ((Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)).comp C).IsIntegral","tactic":"refine ((Quotient.mk P').isIntegral_of_surjective Quotient.mk_surjective).trans _ _ ?_","premises":[{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Ideal.Quotient.mk_surjective","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[141,8],"def_end_pos":[141,21]},{"full_name":"RingHom.IsIntegral.trans","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[584,18],"def_end_pos":[584,42]},{"full_name":"RingHom.isIntegral_of_surjective","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[603,8],"def_end_pos":[603,40]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\n⊢ ((Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\n⊢ ((Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)).comp C).IsIntegral","tactic":"have : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P) :=\n    Or.recOn (map_eq_top_or_isMaximal_of_surjective f hf hP)\n      (fun h => absurd (_root_.trans (h ▸ hPJ : P = comap f ⊤) comap_top : P = ⊤) hP.ne_top) id","premises":[{"full_name":"Ideal.IsMaximal","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[246,6],"def_end_pos":[246,15]},{"full_name":"Ideal.IsMaximal.ne_top","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[254,8],"def_end_pos":[254,24]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Ideal.comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[38,4],"def_end_pos":[38,9]},{"full_name":"Ideal.comap_top","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[148,8],"def_end_pos":[148,17]},{"full_name":"Ideal.map","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[34,4],"def_end_pos":[34,7]},{"full_name":"Ideal.map_eq_top_or_isMaximal_of_surjective","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[378,8],"def_end_pos":[378,45]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[641,4],"def_end_pos":[641,14]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"absurd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[246,20],"def_end_pos":[246,26]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"trans","def_path":"Mathlib/Init/Algebra/Classes.lean","def_pos":[261,8],"def_end_pos":[261,13]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\n⊢ ((Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)).comp C).IsIntegral","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\n⊢ IsJacobson (R ⧸ P')\n\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\nx : R ⧸ P'\nhx : C x ∈ map (mapRingHom (Quotient.mk (comap C P))) P\n⊢ x = 0","tactic":"apply quotient_mk_comp_C_isIntegral_of_jacobson' _ ?_ (fun x hx => ?_)","premises":[{"full_name":"_private.Mathlib.RingTheory.Jacobson.0.Ideal.Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson'","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[491,16],"def_end_pos":[491,58]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\n⊢ IsJacobson (R ⧸ P')\n\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\nx : R ⧸ P'\nhx : C x ∈ map (mapRingHom (Quotient.mk (comap C P))) P\n⊢ x = 0","state_after":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\nx : R ⧸ P'\nhx : C x ∈ map (mapRingHom (Quotient.mk (comap C P))) P\n⊢ x = 0","tactic":"any_goals exact Ideal.isJacobson_quotient","premises":[{"full_name":"Ideal.isJacobson_quotient","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[107,27],"def_end_pos":[107,46]}]},{"state_before":"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\nx : R ⧸ P'\nhx : C x ∈ map (mapRingHom (Quotient.mk (comap C P))) P\n⊢ x = 0","state_after":"case intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\nz : R\nhx : C ((Quotient.mk P') z) ∈ map (mapRingHom (Quotient.mk (comap C P))) P\n⊢ (Quotient.mk P') z = 0","tactic":"obtain ⟨z, rfl⟩ := Quotient.mk_surjective x","premises":[{"full_name":"Ideal.Quotient.mk_surjective","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[141,8],"def_end_pos":[141,21]}]},{"state_before":"case intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (map f P)\nthis : (map (mapRingHom (Quotient.mk (comap C P))) P).IsMaximal\nz : R\nhx : C ((Quotient.mk P') z) ∈ map (mapRingHom (Quotient.mk (comap C P))) P\n⊢ (Quotient.mk P') z = 0","state_after":"no goals","tactic":"rwa [Quotient.eq_zero_iff_mem, mem_comap, hPJ, mem_comap, coe_mapRingHom, map_C]","premises":[{"full_name":"Ideal.Quotient.eq_zero_iff_mem","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[112,8],"def_end_pos":[112,23]},{"full_name":"Ideal.mem_comap","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[66,8],"def_end_pos":[66,17]},{"full_name":"Polynomial.coe_mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[649,8],"def_end_pos":[649,22]},{"full_name":"Polynomial.map_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[601,8],"def_end_pos":[601,13]}]}]}
{"url":"Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean","commit":"","full_name":"MeasureTheory.Measure.rnDeriv_withDensity_left_of_absolutelyContinuous","start":[116,0],"end":[130,57],"file_path":"Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ✝ ν✝¹ μ ν✝ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\n⊢ (μ.withDensity f).rnDeriv ν =ᶠ[ae ν] fun x => f x * μ.rnDeriv ν x","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ✝ ν✝¹ μ ν✝ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\n⊢ AEMeasurable (fun x => f x * μ.rnDeriv ν x) ν\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ✝ ν✝¹ μ ν✝ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\n⊢ μ.withDensity f = 0 + ν.withDensity fun x => f x * μ.rnDeriv ν x","tactic":"refine (Measure.eq_rnDeriv₀ ?_ Measure.MutuallySingular.zero_left ?_).symm","premises":[{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"MeasureTheory.Measure.MutuallySingular.zero_left","def_path":"Mathlib/MeasureTheory/Measure/MutuallySingular.lean","def_pos":[85,8],"def_end_pos":[85,17]},{"full_name":"MeasureTheory.Measure.eq_rnDeriv₀","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[518,8],"def_end_pos":[518,19]}]}]}
{"url":"Mathlib/RingTheory/OreLocalization/Basic.lean","commit":"","full_name":"OreLocalization.smul_one_smul","start":[548,0],"end":[551,47],"file_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : R\nx : OreLocalization S X\n⊢ (r • 1) • x = r • x","state_after":"case c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : R\nr' : X\ns : ↥S\n⊢ (r • 1) • (r' /ₒ s) = r • (r' /ₒ s)","tactic":"induction' x using OreLocalization.ind with r' s","premises":[{"full_name":"OreLocalization.ind","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[103,18],"def_end_pos":[103,21]}]},{"state_before":"case c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : R\nr' : X\ns : ↥S\n⊢ (r • 1) • (r' /ₒ s) = r • (r' /ₒ s)","state_after":"no goals","tactic":"simp only [smul_oreDiv, smul_eq_mul, mul_one]","premises":[{"full_name":"OreLocalization.smul_oreDiv","def_path":"Mathlib/RingTheory/OreLocalization/Basic.lean","def_pos":[540,8],"def_end_pos":[540,19]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
{"url":"Mathlib/GroupTheory/GroupAction/FixedPoints.lean","commit":"","full_name":"MulAction.smul_mem_fixedBy_iff_mem_fixedBy","start":[64,0],"end":[68,5],"file_path":"Mathlib/GroupTheory/GroupAction/FixedPoints.lean","tactics":[{"state_before":"α : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nM : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\na : α\ng : G\n⊢ g • a ∈ fixedBy α g ↔ a ∈ fixedBy α g","state_after":"α : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nM : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\na : α\ng : G\n⊢ g • a = a ↔ a ∈ fixedBy α g","tactic":"rw [mem_fixedBy, smul_left_cancel_iff]","premises":[{"full_name":"MulAction.mem_fixedBy","def_path":"Mathlib/GroupTheory/GroupAction/Basic.lean","def_pos":[148,8],"def_end_pos":[148,19]},{"full_name":"smul_left_cancel_iff","def_path":"Mathlib/Algebra/Group/Action/Basic.lean","def_pos":[91,6],"def_end_pos":[91,26]}]},{"state_before":"α : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nM : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\na : α\ng : G\n⊢ g • a = a ↔ a ∈ fixedBy α g","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Topology/Order/LowerUpperTopology.lean","commit":"","full_name":"Topology.IsLower.isClosed_upperClosure","start":[217,0],"end":[220,50],"file_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : ?m.5263\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\ns : Set α\nh : s.Finite\n⊢ IsClosed ↑(upperClosure s)","state_after":"α : Type u_1\nβ : ?m.5263\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\ns : Set α\nh : s.Finite\n⊢ IsClosed (⋃ i ∈ s, ↑(UpperSet.Ici i))","tactic":"simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]","premises":[{"full_name":"UpperSet.coe_iInf","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[589,8],"def_end_pos":[589,16]},{"full_name":"UpperSet.iInf_Ici","def_path":"Mathlib/Order/UpperLower/Basic.lean","def_pos":[1310,8],"def_end_pos":[1310,25]}]},{"state_before":"α : Type u_1\nβ : ?m.5263\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\ns : Set α\nh : s.Finite\n⊢ IsClosed (⋃ i ∈ s, ↑(UpperSet.Ici i))","state_after":"no goals","tactic":"exact h.isClosed_biUnion fun _ _ => isClosed_Ici","premises":[{"full_name":"Set.Finite.isClosed_biUnion","def_path":"Mathlib/Topology/Basic.lean","def_pos":[189,8],"def_end_pos":[189,35]},{"full_name":"isClosed_Ici","def_path":"Mathlib/Topology/Order/OrderClosed.lean","def_pos":[331,8],"def_end_pos":[331,20]}]}]}
{"url":"Mathlib/Data/Multiset/Fold.lean","commit":"","full_name":"Multiset.fold_union_inter","start":[91,0],"end":[93,50],"file_path":"Mathlib/Data/Multiset/Fold.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nop : α → α → α\nhc : Std.Commutative op\nha : Std.Associative op\ninst✝ : DecidableEq α\ns₁ s₂ : Multiset α\nb₁ b₂ : α\n⊢ op (fold op b₁ (s₁ ∪ s₂)) (fold op b₂ (s₁ ∩ s₂)) = op (fold op b₁ s₁) (fold op b₂ s₂)","state_after":"no goals","tactic":"rw [← fold_add op, union_add_inter, fold_add op]","premises":[{"full_name":"Multiset.fold_add","def_path":"Mathlib/Data/Multiset/Fold.lean","def_pos":[63,8],"def_end_pos":[63,16]},{"full_name":"Multiset.union_add_inter","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1639,8],"def_end_pos":[1639,23]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/LocallyInjective.lean","commit":"","full_name":"CategoryTheory.Presheaf.isLocallyInjective_iff_of_fac","start":[146,0],"end":[153,18],"file_path":"Mathlib/CategoryTheory/Sites/LocallyInjective.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : ConcreteCategory D\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nφψ : F₁ ⟶ F₃\nfac : φ ≫ ψ = φψ\ninst✝ : IsLocallyInjective J ψ\n⊢ IsLocallyInjective J φψ ↔ IsLocallyInjective J φ","state_after":"case mp\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : ConcreteCategory D\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nφψ : F₁ ⟶ F₃\nfac : φ ≫ ψ = φψ\ninst✝ : IsLocallyInjective J ψ\n⊢ IsLocallyInjective J φψ → IsLocallyInjective J φ\n\ncase mpr\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : ConcreteCategory D\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nφψ : F₁ ⟶ F₃\nfac : φ ≫ ψ = φψ\ninst✝ : IsLocallyInjective J ψ\n⊢ IsLocallyInjective J φ → IsLocallyInjective J φψ","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Inv.lean","commit":"","full_name":"deriv_inv","start":[79,0],"end":[82,36],"file_path":"Mathlib/Analysis/Calculus/Deriv/Inv.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\n⊢ deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹","state_after":"case inl\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\ns t : Set 𝕜\nL : Filter 𝕜\n⊢ deriv (fun x => x⁻¹) 0 = -(0 ^ 2)⁻¹\n\ncase inr\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhne : x ≠ 0\n⊢ deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹","tactic":"rcases eq_or_ne x 0 with (rfl | hne)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","commit":"","full_name":"affineSpan_insert_eq_affineSpan","start":[1254,0],"end":[1259,77],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : P\nps : Set P\nh : p ∈ affineSpan k ps\n⊢ affineSpan k (insert p ps) = affineSpan k ps","state_after":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : P\nps : Set P\nh : p ∈ ↑(affineSpan k ps)\n⊢ affineSpan k (insert p ps) = affineSpan k ps","tactic":"rw [← mem_coe] at h","premises":[{"full_name":"AffineSubspace.mem_coe","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[166,8],"def_end_pos":[166,15]}]},{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : P\nps : Set P\nh : p ∈ ↑(affineSpan k ps)\n⊢ affineSpan k (insert p ps) = affineSpan k ps","state_after":"no goals","tactic":"rw [← affineSpan_insert_affineSpan, Set.insert_eq_of_mem h, affineSpan_coe]","premises":[{"full_name":"AffineSubspace.affineSpan_coe","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[913,8],"def_end_pos":[913,22]},{"full_name":"Set.insert_eq_of_mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[899,8],"def_end_pos":[899,24]},{"full_name":"affineSpan_insert_affineSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[1250,8],"def_end_pos":[1250,36]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Basic.lean","commit":"","full_name":"EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection","start":[398,0],"end":[406,74],"file_path":"Mathlib/Geometry/Euclidean/Basic.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np : P\n⊢ (_root_.orthogonalProjection s.direction) (p -ᵥ ↑((orthogonalProjection s) p)) = 0","state_after":"case hv\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np : P\n⊢ p -ᵥ ↑((orthogonalProjection s) p) ∈ s.directionᗮ","tactic":"apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero","premises":[{"full_name":"orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero","def_path":"Mathlib/Analysis/InnerProductSpace/Projection.lean","def_pos":[811,8],"def_end_pos":[811,70]}]},{"state_before":"case hv\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np : P\n⊢ p -ᵥ ↑((orthogonalProjection s) p) ∈ s.directionᗮ","state_after":"case hv\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np : P\nc : V\nhc : c ∈ s.direction\n⊢ ⟪c, p -ᵥ ↑((orthogonalProjection s) p)⟫_ℝ = 0","tactic":"intro c hc","premises":[]},{"state_before":"case hv\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np : P\nc : V\nhc : c ∈ s.direction\n⊢ ⟪c, p -ᵥ ↑((orthogonalProjection s) p)⟫_ℝ = 0","state_after":"no goals","tactic":"rw [← neg_vsub_eq_vsub_rev, inner_neg_right,\n    orthogonalProjection_vsub_mem_direction_orthogonal s p c hc, neg_zero]","premises":[{"full_name":"EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal","def_path":"Mathlib/Geometry/Euclidean/Basic.lean","def_pos":[386,8],"def_end_pos":[386,58]},{"full_name":"inner_neg_right","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[539,8],"def_end_pos":[539,23]},{"full_name":"neg_vsub_eq_vsub_rev","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[138,8],"def_end_pos":[138,28]},{"full_name":"neg_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[950,2],"def_end_pos":[950,13]}]}]}
{"url":"Mathlib/Data/Finset/NoncommProd.lean","commit":"","full_name":"Finset.noncommSum_lemma","start":[232,0],"end":[239,26],"file_path":"Mathlib/Data/Finset/NoncommProd.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf : α → β\ncomm : (↑s).Pairwise fun a b => Commute (f a) (f b)\n⊢ {x | x ∈ Multiset.map f s.val}.Pairwise Commute","state_after":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf : α → β\ncomm : (↑s).Pairwise fun a b => Commute (f a) (f b)\n⊢ {x | ∃ a ∈ s.val, f a = x}.Pairwise Commute","tactic":"simp_rw [Multiset.mem_map]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]}]},{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf : α → β\ncomm : (↑s).Pairwise fun a b => Commute (f a) (f b)\n⊢ {x | ∃ a ∈ s.val, f a = x}.Pairwise Commute","state_after":"case intro.intro.intro.intro\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf : α → β\ncomm : (↑s).Pairwise fun a b => Commute (f a) (f b)\na : α\nha : a ∈ s.val\nb : α\nhb : b ∈ s.val\na✝ : f a ≠ f b\n⊢ Commute (f a) (f b)","tactic":"rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ _","premises":[]},{"state_before":"case intro.intro.intro.intro\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf : α → β\ncomm : (↑s).Pairwise fun a b => Commute (f a) (f b)\na : α\nha : a ∈ s.val\nb : α\nhb : b ∈ s.val\na✝ : f a ≠ f b\n⊢ Commute (f a) (f b)","state_after":"no goals","tactic":"exact comm.of_refl ha hb","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","commit":"","full_name":"Real.logb_zero","start":[44,0],"end":[45,50],"file_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","tactics":[{"state_before":"b x y : ℝ\n⊢ logb b 0 = 0","state_after":"no goals","tactic":"simp [logb]","premises":[{"full_name":"Real.logb","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Base.lean","def_pos":[38,18],"def_end_pos":[38,22]}]}]}
{"url":"Mathlib/Data/Nat/Squarefree.lean","commit":"","full_name":"Nat.primeFactors_prod","start":[378,0],"end":[384,48],"file_path":"Mathlib/Data/Nat/Squarefree.lean","tactics":[{"state_before":"s : Finset ℕ\nm n p : ℕ\nhs : ∀ p ∈ s, Prime p\n⊢ (∏ p ∈ s, p).primeFactors = s","state_after":"s : Finset ℕ\nm n p : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\n⊢ (∏ p ∈ s, p).primeFactors = s","tactic":"have hn : ∏ p ∈ s, p ≠ 0 := prod_ne_zero_iff.2 fun p hp ↦ (hs _ hp).ne_zero","premises":[{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Finset.prod_ne_zero_iff","def_path":"Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean","def_pos":[49,6],"def_end_pos":[49,22]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.Prime.ne_zero","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[48,8],"def_end_pos":[48,21]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"s : Finset ℕ\nm n p : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\n⊢ (∏ p ∈ s, p).primeFactors = s","state_after":"case a\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ p ∈ (∏ p ∈ s, p).primeFactors ↔ p ∈ s","tactic":"ext p","premises":[]},{"state_before":"case a\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ p ∈ (∏ p ∈ s, p).primeFactors ↔ p ∈ s","state_after":"case a\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ (Prime p ∧ ∃ a ∈ s, p ∣ a) ↔ p ∈ s","tactic":"rw [mem_primeFactors_of_ne_zero hn, and_congr_right (fun hp ↦ hp.prime.dvd_finset_prod_iff _)]","premises":[{"full_name":"Nat.mem_primeFactors_of_ne_zero","def_path":"Mathlib/Data/Nat/PrimeFin.lean","def_pos":[38,6],"def_end_pos":[38,33]},{"full_name":"Prime.dvd_finset_prod_iff","def_path":"Mathlib/Algebra/BigOperators/Associated.lean","def_pos":[176,8],"def_end_pos":[176,33]},{"full_name":"and_congr_right","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[142,8],"def_end_pos":[142,23]}]},{"state_before":"case a\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ (Prime p ∧ ∃ a ∈ s, p ∣ a) ↔ p ∈ s","state_after":"case a\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ (Prime p ∧ ∃ a ∈ s, p ∣ a) → p ∈ s","tactic":"refine ⟨?_, fun hp ↦ ⟨hs _ hp, _, hp, dvd_rfl⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"dvd_rfl","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[125,8],"def_end_pos":[125,15]}]},{"state_before":"case a\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ (Prime p ∧ ∃ a ∈ s, p ∣ a) → p ∈ s","state_after":"case a.intro.intro.intro\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\nhp : Prime p\nq : ℕ\nhq : q ∈ s\nhpq : p ∣ q\n⊢ p ∈ s","tactic":"rintro ⟨hp, q, hq, hpq⟩","premises":[]},{"state_before":"case a.intro.intro.intro\ns : Finset ℕ\nm n p✝ : ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\nhp : Prime p\nq : ℕ\nhq : q ∈ s\nhpq : p ∣ q\n⊢ p ∈ s","state_after":"no goals","tactic":"rwa [← ((hs _ hq).dvd_iff_eq hp.ne_one).1 hpq]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.Prime.dvd_iff_eq","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[173,8],"def_end_pos":[173,24]},{"full_name":"Nat.Prime.ne_one","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[67,8],"def_end_pos":[67,20]}]}]}
{"url":"Mathlib/Algebra/Module/LocalizedModule.lean","commit":"","full_name":"isLocalizedModule_id","start":[558,0],"end":[563,38],"file_path":"Mathlib/Algebra/Module/LocalizedModule.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M'\ninst✝¹³ : AddCommMonoid M''\nA : Type u_5\ninst✝¹² : CommSemiring A\ninst✝¹¹ : Algebra R A\ninst✝¹⁰ : Module A M'\ninst✝⁹ : IsLocalization S A\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M'\ninst✝⁶ : Module R M''\ninst✝⁵ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\nR' : Type u_6\ninst✝⁴ : CommSemiring R'\ninst✝³ : Algebra R R'\ninst✝² : IsLocalization S R'\ninst✝¹ : Module R' M\ninst✝ : IsScalarTower R R' M\ns : ↥S\n⊢ IsUnit ((algebraMap R (Module.End R M)) ↑s)","state_after":"R : Type u_1\ninst✝¹⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M'\ninst✝¹³ : AddCommMonoid M''\nA : Type u_5\ninst✝¹² : CommSemiring A\ninst✝¹¹ : Algebra R A\ninst✝¹⁰ : Module A M'\ninst✝⁹ : IsLocalization S A\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M'\ninst✝⁶ : Module R M''\ninst✝⁵ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\nR' : Type u_6\ninst✝⁴ : CommSemiring R'\ninst✝³ : Algebra R R'\ninst✝² : IsLocalization S R'\ninst✝¹ : Module R' M\ninst✝ : IsScalarTower R R' M\ns : ↥S\n⊢ IsUnit ((Algebra.lsmul R R M) ((algebraMap R R') ↑s))","tactic":"rw [← (Algebra.lsmul R (A := R') R M).commutes]","premises":[{"full_name":"AlgHom.commutes","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[201,8],"def_end_pos":[201,16]},{"full_name":"Algebra.lsmul","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[63,4],"def_end_pos":[63,9]}]},{"state_before":"R : Type u_1\ninst✝¹⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M'\ninst✝¹³ : AddCommMonoid M''\nA : Type u_5\ninst✝¹² : CommSemiring A\ninst✝¹¹ : Algebra R A\ninst✝¹⁰ : Module A M'\ninst✝⁹ : IsLocalization S A\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M'\ninst✝⁶ : Module R M''\ninst✝⁵ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\nR' : Type u_6\ninst✝⁴ : CommSemiring R'\ninst✝³ : Algebra R R'\ninst✝² : IsLocalization S R'\ninst✝¹ : Module R' M\ninst✝ : IsScalarTower R R' M\ns : ↥S\n⊢ IsUnit ((Algebra.lsmul R R M) ((algebraMap R R') ↑s))","state_after":"no goals","tactic":"exact (IsLocalization.map_units R' s).map _","premises":[{"full_name":"IsLocalization.map_units","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"IsUnit.map","def_path":"Mathlib/Algebra/Group/Units/Hom.lean","def_pos":[162,8],"def_end_pos":[162,11]}]}]}
{"url":"Mathlib/Data/Matroid/Dual.lean","commit":"","full_name":"Matroid.setOf_dual_base_eq","start":[151,0],"end":[156,52],"file_path":"Mathlib/Data/Matroid/Dual.lean","tactics":[{"state_before":"α : Type u_1\nM : Matroid α\nI B X : Set α\n⊢ {B | M✶.Base B} = (fun X => M.E \\ X) '' {B | M.Base B}","state_after":"case h\nα : Type u_1\nM : Matroid α\nI B✝ X B : Set α\n⊢ B ∈ {B | M✶.Base B} ↔ B ∈ (fun X => M.E \\ X) '' {B | M.Base B}","tactic":"ext B","premises":[]},{"state_before":"case h\nα : Type u_1\nM : Matroid α\nI B✝ X B : Set α\n⊢ B ∈ {B | M✶.Base B} ↔ B ∈ (fun X => M.E \\ X) '' {B | M.Base B}","state_after":"case h\nα : Type u_1\nM : Matroid α\nI B✝ X B : Set α\n⊢ M.Base (M.E \\ B) ∧ B ⊆ M.E ↔ ∃ x, M.Base x ∧ M.E \\ x = B","tactic":"simp only [mem_setOf_eq, mem_image, dual_base_iff']","premises":[{"full_name":"Matroid.dual_base_iff'","def_path":"Mathlib/Data/Matroid/Dual.lean","def_pos":[147,8],"def_end_pos":[147,22]},{"full_name":"Set.mem_image","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[118,8],"def_end_pos":[118,17]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα : Type u_1\nM : Matroid α\nI B✝ X B : Set α\n⊢ M.Base (M.E \\ B) ∧ B ⊆ M.E ↔ ∃ x, M.Base x ∧ M.E \\ x = B","state_after":"case h\nα : Type u_1\nM : Matroid α\nI B✝ X B : Set α\nx✝ : ∃ x, M.Base x ∧ M.E \\ x = B\nB' : Set α\nhB' : M.Base B'\nh : M.E \\ B' = B\n⊢ M.Base (M.E \\ B)","tactic":"refine ⟨fun h ↦ ⟨_, h.1, diff_diff_cancel_left h.2⟩,\n    fun ⟨B', hB', h⟩ ↦ ⟨?_,h.symm.trans_subset diff_subset⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Set.diff_diff_cancel_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1636,8],"def_end_pos":[1636,29]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]}]},{"state_before":"case h\nα : Type u_1\nM : Matroid α\nI B✝ X B : Set α\nx✝ : ∃ x, M.Base x ∧ M.E \\ x = B\nB' : Set α\nhB' : M.Base B'\nh : M.E \\ B' = B\n⊢ M.Base (M.E \\ B)","state_after":"no goals","tactic":"rwa [← h, diff_diff_cancel_left hB'.subset_ground]","premises":[{"full_name":"Matroid.Base.subset_ground","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[364,8],"def_end_pos":[364,26]},{"full_name":"Set.diff_diff_cancel_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1636,8],"def_end_pos":[1636,29]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","commit":"","full_name":"CategoryTheory.Limits.isoZeroOfMonoZero_inv","start":[379,0],"end":[384,54],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nh : Mono 0\n⊢ (0 ≫ 0) ≫ 0 = 𝟙 X ≫ 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/RingTheory/HahnSeries/Basic.lean","commit":"","full_name":"HahnSeries.support_embDomain_subset","start":[396,0],"end":[400,71],"file_path":"Mathlib/RingTheory/HahnSeries/Basic.lean","tactics":[{"state_before":"Γ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_3\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx : HahnSeries Γ R\n⊢ (embDomain f x).support ⊆ ⇑f '' x.support","state_after":"Γ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_3\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx : HahnSeries Γ R\ng : Γ'\nhg : g ∈ (embDomain f x).support\n⊢ g ∈ ⇑f '' x.support","tactic":"intro g hg","premises":[]},{"state_before":"Γ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_3\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx : HahnSeries Γ R\ng : Γ'\nhg : g ∈ (embDomain f x).support\n⊢ g ∈ ⇑f '' x.support","state_after":"Γ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_3\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx : HahnSeries Γ R\ng : Γ'\nhg : g ∉ ⇑f '' x.support\n⊢ g ∉ (embDomain f x).support","tactic":"contrapose! hg","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"Γ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_3\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx : HahnSeries Γ R\ng : Γ'\nhg : g ∉ ⇑f '' x.support\n⊢ g ∉ (embDomain f x).support","state_after":"no goals","tactic":"rw [mem_support, embDomain_notin_image_support hg, Classical.not_not]","premises":[{"full_name":"Classical.not_not","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[135,16],"def_end_pos":[135,23]},{"full_name":"HahnSeries.embDomain_notin_image_support","def_path":"Mathlib/RingTheory/HahnSeries/Basic.lean","def_pos":[392,8],"def_end_pos":[392,37]},{"full_name":"HahnSeries.mem_support","def_path":"Mathlib/RingTheory/HahnSeries/Basic.lean","def_pos":[76,8],"def_end_pos":[76,19]}]}]}
{"url":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","commit":"","full_name":"Set.Sized.uvCompression","start":[278,0],"end":[283,33],"file_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v a : Finset α\nr : ℕ\nhuv : u.card = v.card\nh𝒜 : Set.Sized r ↑𝒜\n⊢ Set.Sized r ↑(𝓒 u v 𝒜)","state_after":"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v a : Finset α\nr : ℕ\nhuv : u.card = v.card\nh𝒜 : Set.Sized r ↑𝒜\n⊢ ∀ ⦃x : Finset α⦄, (x ∈ 𝒜 ∧ compress u v x ∈ 𝒜 ∨ x ∉ 𝒜 ∧ ∃ b ∈ 𝒜, compress u v b = x) → x.card = r","tactic":"simp_rw [Set.Sized, mem_coe, mem_compression]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.mem_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[195,8],"def_end_pos":[195,15]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.Sized","def_path":"Mathlib/Data/Finset/Slice.lean","def_pos":[42,4],"def_end_pos":[42,9]},{"full_name":"UV.mem_compression","def_path":"Mathlib/Combinatorics/SetFamily/Compression/UV.lean","def_pos":[144,8],"def_end_pos":[144,23]}]},{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v a : Finset α\nr : ℕ\nhuv : u.card = v.card\nh𝒜 : Set.Sized r ↑𝒜\n⊢ ∀ ⦃x : Finset α⦄, (x ∈ 𝒜 ∧ compress u v x ∈ 𝒜 ∨ x ∉ 𝒜 ∧ ∃ b ∈ 𝒜, compress u v b = x) → x.card = r","state_after":"case inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v a : Finset α\nr : ℕ\nhuv : u.card = v.card\nh𝒜 : Set.Sized r ↑𝒜\ns : Finset α\nhs : s ∈ 𝒜 ∧ compress u v s ∈ 𝒜\n⊢ s.card = r\n\ncase inr.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v a : Finset α\nr : ℕ\nhuv : u.card = v.card\nh𝒜 : Set.Sized r ↑𝒜\nt : Finset α\nht : t ∈ 𝒜\nhuvt : compress u v t ∉ 𝒜\n⊢ (compress u v t).card = r","tactic":"rintro s (hs | ⟨huvt, t, ht, rfl⟩)","premises":[]}]}
{"url":"Mathlib/NumberTheory/ClassNumber/Finite.lean","commit":"","full_name":"ClassGroup.exists_mem_finset_approx'","start":[237,0],"end":[255,34],"file_path":"Mathlib/NumberTheory/ClassNumber/Finite.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","state_after":"R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","tactic":"have inj : Function.Injective (algebraMap R L) := by\n    rw [IsScalarTower.algebraMap_eq R S L]\n    exact (IsIntegralClosure.algebraMap_injective S R L).comp bS.algebraMap_injective","premises":[{"full_name":"Basis.algebraMap_injective","def_path":"Mathlib/RingTheory/AlgebraTower.lean","def_pos":[171,8],"def_end_pos":[171,34]},{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]},{"full_name":"Function.Injective.comp","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[104,8],"def_end_pos":[104,22]},{"full_name":"IsIntegralClosure.algebraMap_injective","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean","def_pos":[438,8],"def_end_pos":[438,28]},{"full_name":"IsScalarTower.algebraMap_eq","def_path":"Mathlib/Algebra/Algebra/Tower.lean","def_pos":[118,8],"def_end_pos":[118,21]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","state_after":"case intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","tactic":"obtain ⟨a', b', hb', h⟩ := IsIntegralClosure.exists_smul_eq_mul inj a hb","premises":[{"full_name":"IsIntegralClosure.exists_smul_eq_mul","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[387,8],"def_end_pos":[387,44]}]},{"state_before":"case intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","state_after":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","tactic":"obtain ⟨q, r, hr, hqr⟩ := exists_mem_finsetApprox bS adm a' hb'","premises":[{"full_name":"ClassGroup.exists_mem_finsetApprox","def_path":"Mathlib/NumberTheory/ClassNumber/Finite.lean","def_pos":[185,8],"def_end_pos":[185,31]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","state_after":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","tactic":"refine ⟨q, r, hr, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)","state_after":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) <\n    abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) b)","tactic":"refine\n    lt_of_mul_lt_mul_left ?_ (show 0 ≤ abv (Algebra.norm R (algebraMap R S b')) from abv.nonneg _)","premises":[{"full_name":"AbsoluteValue.nonneg","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[88,18],"def_end_pos":[88,24]},{"full_name":"Algebra.norm","def_path":"Mathlib/RingTheory/Norm/Defs.lean","def_pos":[57,18],"def_end_pos":[57,22]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"lt_of_mul_lt_mul_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[205,8],"def_end_pos":[205,29]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) <\n    abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) b)","state_after":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) =\n    abv ((Algebra.norm R) (r • a' - b' • q)) * abv ((Algebra.norm R) b)","tactic":"refine\n    lt_of_le_of_lt (le_of_eq ?_)\n      (mul_lt_mul hqr le_rfl (abv.pos ((Algebra.norm_ne_zero_iff_of_basis bS).mpr hb))\n        (abv.nonneg _))","premises":[{"full_name":"AbsoluteValue.nonneg","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[88,18],"def_end_pos":[88,24]},{"full_name":"AbsoluteValue.pos","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[105,18],"def_end_pos":[105,21]},{"full_name":"Algebra.norm_ne_zero_iff_of_basis","def_path":"Mathlib/RingTheory/Norm/Basic.lean","def_pos":[124,8],"def_end_pos":[124,33]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]},{"full_name":"lt_of_le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[99,8],"def_end_pos":[99,22]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) =\n    abv ((Algebra.norm R) (r • a' - b' • q)) * abv ((Algebra.norm R) b)","state_after":"no goals","tactic":"rw [← abv.map_mul, ← MonoidHom.map_mul, ← abv.map_mul, ← MonoidHom.map_mul, ← Algebra.smul_def,\n    smul_sub b', sub_mul, smul_comm, h, mul_comm b a', Algebra.smul_mul_assoc r a' b,\n    Algebra.smul_mul_assoc b' q b]","premises":[{"full_name":"AbsoluteValue.map_mul","def_path":"Mathlib/Algebra/Order/AbsoluteValue.lean","def_pos":[99,18],"def_end_pos":[99,25]},{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"Algebra.smul_mul_assoc","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[320,18],"def_end_pos":[320,32]},{"full_name":"MonoidHom.map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[630,18],"def_end_pos":[630,35]},{"full_name":"SMulCommClass.smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[154,2],"def_end_pos":[154,11]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]}]}]}
{"url":"Mathlib/Algebra/Lie/IdealOperations.lean","commit":"","full_name":"LieSubmodule.lie_le_iff","start":[90,0],"end":[94,19],"file_path":"Mathlib/Algebra/Lie/IdealOperations.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'","state_after":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'","tactic":"rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]","premises":[{"full_name":"LieSubmodule.lieIdeal_oper_eq_span","def_path":"Mathlib/Algebra/Lie/IdealOperations.lean","def_pos":[52,8],"def_end_pos":[52,29]},{"full_name":"LieSubmodule.lieSpan_le","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[663,8],"def_end_pos":[663,18]}]},{"state_before":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'","state_after":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ (∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N') → {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N'","tactic":"refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ (∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N') → {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N'","state_after":"case intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nh : ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'\nx : L\nhx : x ∈ I\nm : M\nhm : m ∈ N\n⊢ ⁅↑⟨x, hx⟩, ↑⟨m, hm⟩⁆ ∈ ↑N'","tactic":"rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩","premises":[]},{"state_before":"case intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nh : ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'\nx : L\nhx : x ∈ I\nm : M\nhm : m ∈ N\n⊢ ⁅↑⟨x, hx⟩, ↑⟨m, hm⟩⁆ ∈ ↑N'","state_after":"no goals","tactic":"exact h x hx m hm","premises":[]}]}
{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Complex.exp_ne_zero","start":[210,0],"end":[212,71],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x y : ℂ\nh : cexp x = 0\n⊢ 0 = 1","state_after":"x y : ℂ\nh : cexp x = 0\n⊢ x + -x = 0 * cexp (-x)","tactic":"rw [← exp_zero, ← add_neg_self x, exp_add, h]","premises":[{"full_name":"Complex.exp_add","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[165,8],"def_end_pos":[165,15]},{"full_name":"Complex.exp_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[152,8],"def_end_pos":[152,16]},{"full_name":"add_neg_self","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1054,2],"def_end_pos":[1054,13]}]},{"state_before":"x y : ℂ\nh : cexp x = 0\n⊢ x + -x = 0 * cexp (-x)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/RingDivision.lean","commit":"","full_name":"Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le","start":[606,0],"end":[619,68],"file_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","tactics":[{"state_before":"R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdiv : p ∣ q\nhdeg : q.natDegree ≤ p.natDegree\n⊢ q = C q.leadingCoeff * p","state_after":"case intro\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\n⊢ q = C q.leadingCoeff * p","tactic":"obtain ⟨r, hr⟩ := hdiv","premises":[]},{"state_before":"case intro\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\n⊢ q = C q.leadingCoeff * p","state_after":"case intro.inl\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : natDegree 0 ≤ p.natDegree\nhr : 0 = p * r\n⊢ 0 = C (leadingCoeff 0) * p\n\ncase intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\nhq : q ≠ 0\n⊢ q = C q.leadingCoeff * p","tactic":"obtain rfl | hq := eq_or_ne q 0","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\nhq : q ≠ 0\n⊢ q = C q.leadingCoeff * p","state_after":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p","tactic":"have rzero : r ≠ 0 := fun h => by simp [h, hq] at hr","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p","state_after":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p\n\ncase intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : (p * r).natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ p.leadingCoeff * r.leadingCoeff ≠ 0","tactic":"rw [hr, natDegree_mul'] at hdeg","premises":[{"full_name":"Polynomial.natDegree_mul'","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[863,8],"def_end_pos":[863,22]}]},{"state_before":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p\n\ncase intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : (p * r).natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ p.leadingCoeff * r.leadingCoeff ≠ 0","state_after":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : (p * r).natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ p.leadingCoeff * r.leadingCoeff ≠ 0\n\ncase intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p","tactic":"swap","premises":[]},{"state_before":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p","state_after":"case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = C (r.coeff 0) * p\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p\n\ncase intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = r * p\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ r.natDegree = 0","tactic":"rw [mul_comm, @eq_C_of_natDegree_eq_zero _ _ r] at hr","premises":[{"full_name":"Polynomial.eq_C_of_natDegree_eq_zero","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[1032,8],"def_end_pos":[1032,33]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.encard_le_one_iff_eq","start":[294,0],"end":[296,18],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\n⊢ s.encard ≤ 1 ↔ s = ∅ ∨ ∃ x, s = {x}","state_after":"no goals","tactic":"rw [le_iff_lt_or_eq, lt_iff_not_le, ENat.one_le_iff_ne_zero, not_not, encard_eq_zero,\n    encard_eq_one]","premises":[{"full_name":"Classical.not_not","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[135,16],"def_end_pos":[135,23]},{"full_name":"ENat.one_le_iff_ne_zero","def_path":"Mathlib/Data/ENat/Basic.lean","def_pos":[239,8],"def_end_pos":[239,26]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.encard_eq_one","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[289,8],"def_end_pos":[289,21]},{"full_name":"Set.encard_eq_zero","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[89,16],"def_end_pos":[89,30]},{"full_name":"le_iff_lt_or_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[194,8],"def_end_pos":[194,23]},{"full_name":"lt_iff_not_le","def_path":"Mathlib/Order/Basic.lean","def_pos":[391,8],"def_end_pos":[391,21]}]}]}
{"url":"Mathlib/Data/List/Pi.lean","commit":"","full_name":"Multiset.pi_coe","start":[85,0],"end":[93,53],"file_path":"Mathlib/Data/List/Pi.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝ : DecidableEq ι\nα : ι → Type u_2\nl : List ι\nfs : (i : ι) → List (α i)\n⊢ ((↑l).pi fun x => ↑(fs x)) = ↑(l.pi fs)","state_after":"case nil\nι : Type u_1\ninst✝ : DecidableEq ι\nα : ι → Type u_2\nfs : (i : ι) → List (α i)\n⊢ ((↑[]).pi fun x => ↑(fs x)) = ↑([].pi fs)\n\ncase cons\nι : Type u_1\ninst✝ : DecidableEq ι\nα : ι → Type u_2\nfs : (i : ι) → List (α i)\ni : ι\nl : List ι\nih : ((↑l).pi fun x => ↑(fs x)) = ↑(l.pi fs)\n⊢ ((↑(i :: l)).pi fun x => ↑(fs x)) = ↑((i :: l).pi fs)","tactic":"induction' l with i l ih","premises":[]}]}
{"url":"Mathlib/Deprecated/Subgroup.lean","commit":"","full_name":"IsSubgroup.inv_mem_iff","start":[121,0],"end":[123,52],"file_path":"Mathlib/Deprecated/Subgroup.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\nA : Type u_3\na a₁ a₂ b c : G\ninst✝ : Group G\ns : Set G\nhs : IsSubgroup s\nh : a⁻¹ ∈ s\n⊢ a ∈ s","state_after":"no goals","tactic":"simpa using hs.inv_mem h","premises":[{"full_name":"IsSubgroup.inv_mem","def_path":"Mathlib/Deprecated/Subgroup.lean","def_pos":[50,2],"def_end_pos":[50,9]}]}]}
{"url":"Mathlib/CategoryTheory/Extensive.lean","commit":"","full_name":"CategoryTheory.FinitaryPreExtensive.sigma_desc_iso","start":[535,0],"end":[551,54],"file_path":"Mathlib/CategoryTheory/Extensive.lean","tactics":[{"state_before":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\n⊢ IsIso (Sigma.desc fun x => pullback.fst f (π x))","state_after":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\n⊢ IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))","tactic":"suffices IsColimit (Cofan.mk _ ((fun _ ↦ pullback.fst _ _) : (a : α) → pullback f (π a) ⟶ _)) by\n    change IsIso (this.coconePointUniqueUpToIso (getColimitCocone _).2).inv\n    infer_instance","premises":[{"full_name":"CategoryTheory.IsIso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[229,6],"def_end_pos":[229,11]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.Limits.Cofan.mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[64,4],"def_end_pos":[64,12]},{"full_name":"CategoryTheory.Limits.ColimitCocone.isColimit","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[590,2],"def_end_pos":[590,11]},{"full_name":"CategoryTheory.Limits.IsColimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[507,10],"def_end_pos":[507,19]},{"full_name":"CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[587,4],"def_end_pos":[587,28]},{"full_name":"CategoryTheory.Limits.getColimitCocone","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[601,4],"def_end_pos":[601,20]},{"full_name":"CategoryTheory.Limits.pullback","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[92,7],"def_end_pos":[92,15]},{"full_name":"CategoryTheory.Limits.pullback.fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[108,7],"def_end_pos":[108,19]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\n⊢ IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))","state_after":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\nthis : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso\n⊢ IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))","tactic":"let this : IsColimit (Cofan.mk X π) := by\n    refine @IsColimit.ofPointIso (t := Cofan.mk X π) (P := coproductIsCoproduct Z) ?_\n    convert hπ\n    simp [coproductIsCoproduct]","premises":[{"full_name":"CategoryTheory.Limits.Cofan.mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[64,4],"def_end_pos":[64,12]},{"full_name":"CategoryTheory.Limits.IsColimit","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[507,10],"def_end_pos":[507,19]},{"full_name":"CategoryTheory.Limits.IsColimit.ofPointIso","def_path":"Mathlib/CategoryTheory/Limits/IsLimit.lean","def_pos":[640,4],"def_end_pos":[640,14]},{"full_name":"CategoryTheory.Limits.coproductIsCoproduct","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[219,4],"def_end_pos":[219,24]}]},{"state_before":"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\nthis : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso\n⊢ IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))","state_after":"case refine_1\nJ : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\nthis : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso\n⊢ (Discrete.natTrans fun i => pullback.snd f (π i.as)) ≫ (Cofan.mk X π).ι =\n    (Cofan.mk Y fun x => pullback.fst f (π x)).ι ≫ (Functor.const (Discrete α)).map f\n\ncase refine_2\nJ : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\nthis : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso\n⊢ ∀ (j : Discrete α),\n    IsPullback ((Cofan.mk Y fun x => pullback.fst f (π x)).ι.app j)\n      ((Discrete.natTrans fun i => pullback.snd f (π i.as)).app j) f ((Cofan.mk X π).ι.app j)","tactic":"refine (FinitaryPreExtensive.isUniversal_finiteCoproducts this\n    (Cofan.mk _ ((fun _ ↦ pullback.fst _ _) : (a : α) → pullback f (π a) ⟶ _))\n    (Discrete.natTrans fun i ↦ pullback.snd _ _) f ?_\n    (NatTrans.equifibered_of_discrete _) ?_).some","premises":[{"full_name":"CategoryTheory.Discrete.natTrans","def_path":"Mathlib/CategoryTheory/DiscreteCategory.lean","def_pos":[198,4],"def_end_pos":[198,12]},{"full_name":"CategoryTheory.FinitaryPreExtensive.isUniversal_finiteCoproducts","def_path":"Mathlib/CategoryTheory/Extensive.lean","def_pos":[441,8],"def_end_pos":[441,57]},{"full_name":"CategoryTheory.Limits.Cofan.mk","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[64,4],"def_end_pos":[64,12]},{"full_name":"CategoryTheory.Limits.pullback","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[92,7],"def_end_pos":[92,15]},{"full_name":"CategoryTheory.Limits.pullback.fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[108,7],"def_end_pos":[108,19]},{"full_name":"CategoryTheory.Limits.pullback.snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[112,7],"def_end_pos":[112,19]},{"full_name":"CategoryTheory.NatTrans.equifibered_of_discrete","def_path":"Mathlib/CategoryTheory/Limits/VanKampen.lean","def_pos":[76,8],"def_end_pos":[76,40]},{"full_name":"Nonempty.some","def_path":"Mathlib/Logic/Nonempty.lean","def_pos":[81,31],"def_end_pos":[81,44]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","commit":"","full_name":"MeasureTheory.Measure.isHaarMeasure_eq_of_isProbabilityMeasure","start":[663,0],"end":[678,17],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","tactics":[{"state_before":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\n⊢ μ' = μ","state_after":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\n⊢ μ' = μ","tactic":"have : CompactSpace G := by\n    by_contra H\n    rw [not_compactSpace_iff] at H\n    simpa using measure_univ_of_isMulLeftInvariant μ","premises":[{"full_name":"Classical.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[129,8],"def_end_pos":[129,23]},{"full_name":"CompactSpace","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[260,6],"def_end_pos":[260,18]},{"full_name":"MeasureTheory.measure_univ_of_isMulLeftInvariant","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[609,8],"def_end_pos":[609,42]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"not_compactSpace_iff","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[795,8],"def_end_pos":[795,28]}]},{"state_before":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\n⊢ μ' = μ","state_after":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\n⊢ μ' = μ","tactic":"have A s : μ' s = haarScalarFactor μ' μ • μ s :=\n    measure_isMulInvariant_eq_smul_of_isCompact_closure _ _ isClosed_closure.isCompact","premises":[{"full_name":"IsClosed.isCompact","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[752,8],"def_end_pos":[752,26]},{"full_name":"MeasureTheory.Measure.haarScalarFactor","def_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","def_pos":[294,18],"def_end_pos":[294,34]},{"full_name":"MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure","def_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","def_pos":[615,8],"def_end_pos":[615,59]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]}]},{"state_before":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\n⊢ μ' = μ","state_after":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\nZ : μ' univ = μ'.haarScalarFactor μ • μ univ\n⊢ μ' = μ","tactic":"have Z := A univ","premises":[{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]}]},{"state_before":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\nZ : μ' univ = μ'.haarScalarFactor μ • μ univ\n⊢ μ' = μ","state_after":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\nZ : 1 = μ'.haarScalarFactor μ\n⊢ μ' = μ","tactic":"simp only [measure_univ, ENNReal.smul_def, smul_eq_mul, mul_one, ENNReal.one_eq_coe] at Z","premises":[{"full_name":"ENNReal.one_eq_coe","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[336,27],"def_end_pos":[336,37]},{"full_name":"ENNReal.smul_def","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[472,8],"def_end_pos":[472,16]},{"full_name":"MeasureTheory.IsProbabilityMeasure.measure_univ","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[208,2],"def_end_pos":[208,14]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]},{"state_before":"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\nZ : 1 = μ'.haarScalarFactor μ\n⊢ μ' = μ","state_after":"case h\nG : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\nZ : 1 = μ'.haarScalarFactor μ\ns : Set G\n_hs : MeasurableSet s\n⊢ μ' s = μ s","tactic":"ext s _hs","premises":[]},{"state_before":"case h\nG : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝³ : IsProbabilityMeasure μ\ninst✝² : IsProbabilityMeasure μ'\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nthis : CompactSpace G\nA : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s\nZ : 1 = μ'.haarScalarFactor μ\ns : Set G\n_hs : MeasurableSet s\n⊢ μ' s = μ s","state_after":"no goals","tactic":"simp [A s, ← Z]","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","commit":"","full_name":"AffineBasis.det_smul_coords_eq_cramer_coords","start":[155,0],"end":[161,94],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","tactics":[{"state_before":"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\n⊢ (b.toMatrix ⇑b₂).det • b₂.coords x = (b.toMatrix ⇑b₂)ᵀ.cramer (b.coords x)","state_after":"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\nhu : IsUnit (b.toMatrix ⇑b₂)\n⊢ (b.toMatrix ⇑b₂).det • b₂.coords x = (b.toMatrix ⇑b₂)ᵀ.cramer (b.coords x)","tactic":"have hu := b.isUnit_toMatrix b₂","premises":[{"full_name":"AffineBasis.isUnit_toMatrix","def_path":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","def_pos":[120,8],"def_end_pos":[120,23]}]},{"state_before":"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\nhu : IsUnit (b.toMatrix ⇑b₂)\n⊢ (b.toMatrix ⇑b₂).det • b₂.coords x = (b.toMatrix ⇑b₂)ᵀ.cramer (b.coords x)","state_after":"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\nhu : IsUnit (b.toMatrix ⇑b₂).det\n⊢ (b.toMatrix ⇑b₂).det • b₂.coords x = (b.toMatrix ⇑b₂)ᵀ.cramer (b.coords x)","tactic":"rw [Matrix.isUnit_iff_isUnit_det] at hu","premises":[{"full_name":"Matrix.isUnit_iff_isUnit_det","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[137,8],"def_end_pos":[137,29]}]},{"state_before":"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : CommRing k\ninst✝² : Module k V\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb b₂ : AffineBasis ι k P\nx : P\nhu : IsUnit (b.toMatrix ⇑b₂).det\n⊢ (b.toMatrix ⇑b₂).det • b₂.coords x = (b.toMatrix ⇑b₂)ᵀ.cramer (b.coords x)","state_after":"no goals","tactic":"rw [← b.toMatrix_inv_vecMul_toMatrix, Matrix.det_smul_inv_vecMul_eq_cramer_transpose _ _ hu]","premises":[{"full_name":"AffineBasis.toMatrix_inv_vecMul_toMatrix","def_path":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","def_pos":[148,8],"def_end_pos":[148,36]},{"full_name":"Matrix.det_smul_inv_vecMul_eq_cramer_transpose","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[634,8],"def_end_pos":[634,47]}]}]}
{"url":"Mathlib/Analysis/Convex/Function.lean","commit":"","full_name":"ConvexOn.sup","start":[576,0],"end":[584,79],"file_path":"Mathlib/Analysis/Convex/Function.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\n⊢ ConvexOn 𝕜 s (f ⊔ g)","state_after":"case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (a • x + b • y) ≤ a • (f ⊔ g) x + b • (f ⊔ g) y\n\ncase refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\nι : Type u_6\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhg : ConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ g (a • x + b • y) ≤ a • (f ⊔ g) x + b • (f ⊔ g) y","tactic":"refine ⟨hf.left, fun x hx y hy a b ha hb hab => sup_le ?_ ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"sup_le","def_path":"Mathlib/Order/Lattice.lean","def_pos":[129,8],"def_end_pos":[129,14]}]}]}
{"url":"Mathlib/Data/Nat/Cast/Basic.lean","commit":"","full_name":"MonoidHom.apply_mnat","start":[239,0],"end":[241,72],"file_path":"Mathlib/Data/Nat/Cast/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\ninst✝ : AddMonoid β\nf : Multiplicative ℕ →* α\nn : Multiplicative ℕ\n⊢ f n = f (ofAdd 1) ^ toAdd n","state_after":"no goals","tactic":"rw [← powersHom_symm_apply, ← powersHom_apply, Equiv.apply_symm_apply]","premises":[{"full_name":"Equiv.apply_symm_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[239,16],"def_end_pos":[239,32]},{"full_name":"powersHom_apply","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[230,6],"def_end_pos":[230,21]},{"full_name":"powersHom_symm_apply","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[236,6],"def_end_pos":[236,26]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesLeftHomology","start":[764,0],"end":[774,18],"file_path":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\n⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ","state_after":"C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\nγ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData\n⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ","tactic":"have γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData := default","premises":[{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[254,10],"def_end_pos":[254,29]},{"full_name":"CategoryTheory.ShortComplex.leftHomologyData","def_path":"Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean","def_pos":[220,18],"def_end_pos":[220,34]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\nγ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData\n⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ","state_after":"C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\nγ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData\n⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH","tactic":"rw [γ.quasiIso_iff, (γ.map F).quasiIso_iff, LeftHomologyMapData.map_φH]","premises":[{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData.map","def_path":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","def_pos":[136,4],"def_end_pos":[136,27]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData.map_φH","def_path":"Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean","def_pos":[135,2],"def_end_pos":[135,7]},{"full_name":"CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff","def_path":"Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean","def_pos":[97,6],"def_end_pos":[97,38]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\nγ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData\n⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH","state_after":"case mp\nC : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\nγ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData\n⊢ IsIso (F.map γ.φH) → IsIso γ.φH\n\ncase mpr\nC : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{u_3, u_1} C\ninst✝¹⁰ : Category.{u_4, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nS : ShortComplex C\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhl₁ : S₁.LeftHomologyData\nhr₁ : S₁.RightHomologyData\nhl₂ : S₂.LeftHomologyData\nhr₂ : S₂.RightHomologyData\nψl : LeftHomologyMapData φ hl₁ hl₂\nψr : RightHomologyMapData φ hr₁ hr₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology\ninst✝³ : (F.mapShortComplex.obj S₂).HasHomology\ninst✝² : F.PreservesLeftHomologyOf S₁\ninst✝¹ : F.PreservesLeftHomologyOf S₂\ninst✝ : F.ReflectsIsomorphisms\nγ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData\n⊢ IsIso γ.φH → IsIso (F.map γ.φH)","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Combination.lean","commit":"","full_name":"Finset.convexHull_eq","start":[350,0],"end":[368,96],"file_path":"Mathlib/Analysis/Convex/Combination.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Finset E\n⊢ (convexHull R) ↑s = {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}","state_after":"case refine_1\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Finset E\n⊢ ↑s ⊆ {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}\n\ncase refine_2\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Finset E\n⊢ Convex R {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}\n\ncase refine_3\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Finset E\n⊢ {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x} ⊆ (convexHull R) ↑s","tactic":"refine Set.Subset.antisymm (convexHull_min ?_ ?_) ?_","premises":[{"full_name":"Set.Subset.antisymm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[302,8],"def_end_pos":[302,23]},{"full_name":"convexHull_min","def_path":"Mathlib/Analysis/Convex/Hull.lean","def_pos":[59,8],"def_end_pos":[59,22]}]}]}
{"url":"Mathlib/MeasureTheory/Function/LpSpace.lean","commit":"","full_name":"MeasureTheory.norm_indicatorConstLp'","start":[785,0],"end":[790,45],"file_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : E\nhp_pos : p ≠ 0\nhμs_pos : μ s ≠ 0\n⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)","state_after":"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : E\nhp_pos : p ≠ 0\nhμs_pos : μ s ≠ 0\nhp_top : p = ⊤\n⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : E\nhp_pos : p ≠ 0\nhμs_pos : μ s ≠ 0\nhp_top : ¬p = ⊤\n⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)","tactic":"by_cases hp_top : p = ∞","premises":[{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"div_div_cancel_left","start":[914,0],"end":[915,66],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : CommGroup G\na✝ b✝ c d a b : G\n⊢ a / b / a = b⁻¹","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Order/SuccPred/LinearLocallyFinite.lean","commit":"","full_name":"toZ_iterate_pred_of_not_isMin","start":[240,0],"end":[257,64],"file_path":"Mathlib/Order/SuccPred/LinearLocallyFinite.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n] i0)\n⊢ toZ i0 (pred^[n] i0) = -↑n","state_after":"case zero\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhn : ¬IsMin (pred^[0] i0)\n⊢ toZ i0 (pred^[0] i0) = -↑0\n\ncase succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","tactic":"cases' n with n n","premises":[]},{"state_before":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","state_after":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","tactic":"have : pred^[n.succ] i0 < i0 := by\n    refine lt_of_le_of_ne (pred_iterate_le _ _) fun h_pred_iterate_eq ↦ hn ?_\n    have h_pred_eq_pred : pred^[n.succ] i0 = pred^[0] i0 := by\n      rwa [Function.iterate_zero, id]\n    exact isMin_iterate_pred_of_eq_of_ne h_pred_eq_pred (Nat.succ_ne_zero n)","premises":[{"full_name":"Function.iterate_zero","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[50,8],"def_end_pos":[50,20]},{"full_name":"Nat.iterate","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[36,4],"def_end_pos":[36,15]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Nat.succ_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[708,16],"def_end_pos":[708,28]},{"full_name":"Order.isMin_iterate_pred_of_eq_of_ne","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[604,8],"def_end_pos":[604,38]},{"full_name":"Order.pred","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[529,4],"def_end_pos":[529,8]},{"full_name":"Order.pred_iterate_le","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[596,8],"def_end_pos":[596,23]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"lt_of_le_of_ne","def_path":"Mathlib/Order/Defs.lean","def_pos":[164,8],"def_end_pos":[164,22]}]},{"state_before":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","state_after":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\nm : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","tactic":"let m := (-toZ i0 (pred^[n.succ] i0)).toNat","premises":[{"full_name":"Int.toNat","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[289,4],"def_end_pos":[289,9]},{"full_name":"Nat.iterate","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[36,4],"def_end_pos":[36,15]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Order.pred","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[529,4],"def_end_pos":[529,8]},{"full_name":"toZ","def_path":"Mathlib/Order/SuccPred/LinearLocallyFinite.lean","def_pos":[178,4],"def_end_pos":[178,7]}]},{"state_before":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\nm : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","state_after":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\nm : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat\nh_eq : pred^[m] i0 = pred^[n.succ] i0\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","tactic":"have h_eq : pred^[m] i0 = pred^[n.succ] i0 := iterate_pred_toZ _ this","premises":[{"full_name":"Nat.iterate","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[36,4],"def_end_pos":[36,15]},{"full_name":"Nat.succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1083,4],"def_end_pos":[1083,8]},{"full_name":"Order.pred","def_path":"Mathlib/Order/SuccPred/Basic.lean","def_pos":[529,4],"def_end_pos":[529,8]},{"full_name":"iterate_pred_toZ","def_path":"Mathlib/Order/SuccPred/LinearLocallyFinite.lean","def_pos":[199,8],"def_end_pos":[199,24]}]},{"state_before":"case succ\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\nm : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat\nh_eq : pred^[m] i0 = pred^[n.succ] i0\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","state_after":"case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\nm : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat\nh_eq : pred^[m] i0 = pred^[n.succ] i0\nhmn : m = n + 1\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)\n\ncase neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nn : ℕ\nhn : ¬IsMin (pred^[n + 1] i0)\nthis : pred^[n.succ] i0 < i0\nm : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat\nh_eq : pred^[m] i0 = pred^[n.succ] i0\nhmn : ¬m = n + 1\n⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)","tactic":"by_cases hmn : m = n + 1","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Functor.lean","commit":"","full_name":"CategoryTheory.Functor.mapTriangle_map_hom₁","start":[32,0],"end":[47,40],"file_path":"Mathlib/CategoryTheory/Triangulated/Functor.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) X✝).mor₁ ≫\n      F.map f.hom₂ =\n    F.map f.hom₁ ≫\n      ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) Y✝).mor₁","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₁ ≫ F.map f.hom₂ = F.map f.hom₁ ≫ F.map Y✝.mor₁","tactic":"dsimp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₁ ≫ F.map f.hom₂ = F.map f.hom₁ ≫ F.map Y✝.mor₁","state_after":"no goals","tactic":"simp only [← F.map_comp, f.comm₁]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Pretriangulated.TriangleMorphism.comm₁","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[109,2],"def_end_pos":[109,7]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) X✝).mor₂ ≫\n      F.map f.hom₃ =\n    F.map f.hom₂ ≫\n      ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) Y✝).mor₂","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₂ ≫ F.map f.hom₃ = F.map f.hom₂ ≫ F.map Y✝.mor₂","tactic":"dsimp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ F.map X✝.mor₂ ≫ F.map f.hom₃ = F.map f.hom₂ ≫ F.map Y✝.mor₂","state_after":"no goals","tactic":"simp only [← F.map_comp, f.comm₂]","premises":[{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Pretriangulated.TriangleMorphism.comm₂","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[111,2],"def_end_pos":[111,7]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) X✝).mor₃ ≫\n      (shiftFunctor D 1).map (F.map f.hom₁) =\n    F.map f.hom₃ ≫\n      ((fun T => Triangle.mk (F.map T.mor₁) (F.map T.mor₂) (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁)) Y✝).mor₃","state_after":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ (F.map X✝.mor₃ ≫ (F.commShiftIso 1).hom.app X✝.obj₁) ≫ (shiftFunctor D 1).map (F.map f.hom₁) =\n    F.map f.hom₃ ≫ F.map Y✝.mor₃ ≫ (F.commShiftIso 1).hom.app Y✝.obj₁","tactic":"dsimp [Functor.comp]","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{?u.284, u_1} C\ninst✝⁶ : Category.{?u.288, u_2} D\ninst✝⁵ : Category.{?u.292, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ⟶ Y✝\n⊢ (F.map X✝.mor₃ ≫ (F.commShiftIso 1).hom.app X✝.obj₁) ≫ (shiftFunctor D 1).map (F.map f.hom₁) =\n    F.map f.hom₃ ≫ F.map Y✝.mor₃ ≫ (F.commShiftIso 1).hom.app Y✝.obj₁","state_after":"no goals","tactic":"simp only [Category.assoc, ← NatTrans.naturality,\n          ← F.map_comp_assoc, f.comm₃]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.map_comp_assoc","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[57,6],"def_end_pos":[57,28]},{"full_name":"CategoryTheory.NatTrans.naturality","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[50,2],"def_end_pos":[50,12]},{"full_name":"CategoryTheory.Pretriangulated.TriangleMorphism.comm₃","def_path":"Mathlib/CategoryTheory/Triangulated/Basic.lean","def_pos":[113,2],"def_end_pos":[113,7]}]}]}
{"url":"Mathlib/Algebra/Order/Group/Defs.lean","commit":"","full_name":"exists_one_lt'","start":[125,0],"end":[130,16],"file_path":"Mathlib/Algebra/Order/Group/Defs.lean","tactics":[{"state_before":"α : Type u\ninst✝¹ : LinearOrderedCommGroup α\na b c : α\ninst✝ : Nontrivial α\n⊢ ∃ a, 1 < a","state_after":"case intro\nα : Type u\ninst✝¹ : LinearOrderedCommGroup α\na b c : α\ninst✝ : Nontrivial α\ny : α\nhy : y ≠ 1\n⊢ ∃ a, 1 < a","tactic":"obtain ⟨y, hy⟩ := Decidable.exists_ne (1 : α)","premises":[{"full_name":"Decidable.exists_ne","def_path":"Mathlib/Logic/Nontrivial/Defs.lean","def_pos":[40,18],"def_end_pos":[40,37]}]},{"state_before":"case intro\nα : Type u\ninst✝¹ : LinearOrderedCommGroup α\na b c : α\ninst✝ : Nontrivial α\ny : α\nhy : y ≠ 1\n⊢ ∃ a, 1 < a","state_after":"case intro.inl\nα : Type u\ninst✝¹ : LinearOrderedCommGroup α\na b c : α\ninst✝ : Nontrivial α\ny : α\nhy : y ≠ 1\nh : y < 1\n⊢ ∃ a, 1 < a\n\ncase intro.inr\nα : Type u\ninst✝¹ : LinearOrderedCommGroup α\na b c : α\ninst✝ : Nontrivial α\ny : α\nhy : y ≠ 1\nh : 1 < y\n⊢ ∃ a, 1 < a","tactic":"obtain h|h := hy.lt_or_lt","premises":[{"full_name":"Ne.lt_or_lt","def_path":"Mathlib/Order/Basic.lean","def_pos":[394,8],"def_end_pos":[394,19]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","commit":"","full_name":"MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure","start":[539,0],"end":[549,15],"file_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","tactics":[{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν ν₁ ν₂ : Measure α\ns t : Set α\ninst✝ : SFinite μ\nc : ℕ → ℝ≥0\nhc₀ : ∀ (i : ℕ), 0 < c i\nhc : ∑' (i : ℕ), (sFiniteSeq μ i) univ * ↑(c i) < ⊤\n⊢ ∃ ν, IsFiniteMeasure ν ∧ μ ≪ ν","state_after":"case intro.intro.refine_1\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν ν₁ ν₂ : Measure α\ns t : Set α\ninst✝ : SFinite μ\nc : ℕ → ℝ≥0\nhc₀ : ∀ (i : ℕ), 0 < c i\nhc : ∑' (i : ℕ), (sFiniteSeq μ i) univ * ↑(c i) < ⊤\n⊢ (sum fun n => c n • sFiniteSeq μ n) univ < ⊤\n\ncase intro.intro.refine_2\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν ν₁ ν₂ : Measure α\ns t : Set α\ninst✝ : SFinite μ\nc : ℕ → ℝ≥0\nhc₀ : ∀ (i : ℕ), 0 < c i\nhc : ∑' (i : ℕ), (sFiniteSeq μ i) univ * ↑(c i) < ⊤\n⊢ μ ≪ sum fun n => c n • sFiniteSeq μ n","tactic":"refine ⟨.sum fun n ↦ c n • sFiniteSeq μ n, ⟨?_⟩, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"MeasureTheory.Measure.sum","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1325,18],"def_end_pos":[1325,21]},{"full_name":"MeasureTheory.sFiniteSeq","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[494,4],"def_end_pos":[494,14]}]}]}
{"url":"Mathlib/Analysis/Convolution.lean","commit":"","full_name":"MeasureTheory.support_convolution_subset_swap","start":[501,0],"end":[512,40],"file_path":"Mathlib/Analysis/Convolution.lean","tactics":[{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\n⊢ support (f ⋆[L, μ] g) ⊆ support g + support f","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\n⊢ x ∈ support g + support f","tactic":"intro x h2x","premises":[]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\n⊢ x ∈ support g + support f","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : x ∉ support g + support f\n⊢ False","tactic":"by_contra hx","premises":[{"full_name":"Classical.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[129,8],"def_end_pos":[129,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : x ∉ support g + support f\n⊢ False","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : x ∉ support g + support f\n⊢ (f ⋆[L, μ] g) x = 0","tactic":"apply h2x","premises":[]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : x ∉ support g + support f\n⊢ (f ⋆[L, μ] g) x = 0","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ (f ⋆[L, μ] g) x = 0","tactic":"simp_rw [Set.mem_add, ← exists_and_left, not_exists, not_and_or, nmem_support] at hx","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.nmem_support","def_path":"Mathlib/Algebra/Group/Support.lean","def_pos":[35,2],"def_end_pos":[35,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_add","def_path":"Mathlib/Data/Set/Pointwise/Basic.lean","def_pos":[272,2],"def_end_pos":[272,13]},{"full_name":"exists_and_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[288,16],"def_end_pos":[288,31]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]},{"full_name":"not_exists","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[254,16],"def_end_pos":[254,26]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ (f ⋆[L, μ] g) x = 0","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ ∫ (t : G), (L (f t)) (g (x - t)) ∂μ = 0","tactic":"rw [convolution_def]","premises":[{"full_name":"MeasureTheory.convolution_def","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[421,8],"def_end_pos":[421,23]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ ∫ (t : G), (L (f t)) (g (x - t)) ∂μ = 0","state_after":"case h.e'_2.h.e'_7\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ (fun t => (L (f t)) (g (x - t))) = fun x => 0","tactic":"convert integral_zero G F using 2","premises":[{"full_name":"MeasureTheory.integral_zero","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[769,8],"def_end_pos":[769,21]}]},{"state_before":"case h.e'_2.h.e'_7\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ (fun t => (L (f t)) (g (x - t))) = fun x => 0","state_after":"case h.e'_2.h.e'_7.h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\nt : G\n⊢ (L (f t)) (g (x - t)) = 0","tactic":"ext t","premises":[]},{"state_before":"case h.e'_2.h.e'_7.h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\nt : G\n⊢ (L (f t)) (g (x - t)) = 0","state_after":"case h.e'_2.h.e'_7.h.inl\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\nt : G\nh : g (x - t) = 0\n⊢ (L (f t)) (g (x - t)) = 0\n\ncase h.e'_2.h.e'_7.h.inr.inl\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\nt : G\nh : f t = 0\n⊢ (L (f t)) (g (x - t)) = 0\n\ncase h.e'_2.h.e'_7.h.inr.inr\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\nt : G\nh : ¬x - t + t = x\n⊢ (L (f t)) (g (x - t)) = 0","tactic":"rcases hx (x - t) t with (h | h | h)","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","commit":"","full_name":"CategoryTheory.Limits.HasZeroObject.zeroIsoTerminal_hom","start":[289,0],"end":[290,92],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasTerminal C\n⊢ zeroIsoTerminal.hom = 0","state_after":"no goals","tactic":"ext","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/Opposite.lean","commit":"","full_name":"HomologicalComplex.unop_d","start":[88,0],"end":[94,67],"file_path":"Mathlib/Algebra/Homology/Opposite.lean","tactics":[{"state_before":"ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{?u.20862, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\ni j : ι\nhij : ¬c.symm.Rel i j\n⊢ (fun i j => (X.d j i).unop) i j = 0","state_after":"ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{?u.20862, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\ni j : ι\nhij : ¬c.symm.Rel i j\n⊢ (X.d j i).unop = 0","tactic":"simp only","premises":[]},{"state_before":"ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{?u.20862, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\ni j : ι\nhij : ¬c.symm.Rel i j\n⊢ (X.d j i).unop = 0","state_after":"no goals","tactic":"rw [X.shape j i hij, unop_zero]","premises":[{"full_name":"CategoryTheory.Limits.unop_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[124,14],"def_end_pos":[124,23]},{"full_name":"HomologicalComplex.shape","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[58,2],"def_end_pos":[58,7]}]},{"state_before":"ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{?u.20862, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\nx✝⁴ x✝³ x✝² : ι\nx✝¹ : c.symm.Rel x✝⁴ x✝³\nx✝ : c.symm.Rel x✝³ x✝²\n⊢ (fun i j => (X.d j i).unop) x✝⁴ x✝³ ≫ (fun i j => (X.d j i).unop) x✝³ x✝² = 0","state_after":"no goals","tactic":"rw [← unop_comp, X.d_comp_d, unop_zero]","premises":[{"full_name":"CategoryTheory.Limits.unop_zero","def_path":"Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean","def_pos":[124,14],"def_end_pos":[124,23]},{"full_name":"CategoryTheory.unop_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[80,8],"def_end_pos":[80,17]},{"full_name":"HomologicalComplex.d_comp_d","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[68,8],"def_end_pos":[68,16]}]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.tendsto_atBot_diagonal","start":[1345,0],"end":[1347,37],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeInf α\n⊢ Tendsto (fun a => (a, a)) atBot atBot","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeInf α\n⊢ Tendsto (fun a => (a, a)) atBot (atBot ×ˢ atBot)","tactic":"rw [← prod_atBot_atBot_eq]","premises":[{"full_name":"Filter.prod_atBot_atBot_eq","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[1326,8],"def_end_pos":[1326,27]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeInf α\n⊢ Tendsto (fun a => (a, a)) atBot (atBot ×ˢ atBot)","state_after":"no goals","tactic":"exact tendsto_id.prod_mk tendsto_id","premises":[{"full_name":"Filter.Tendsto.prod_mk","def_path":"Mathlib/Order/Filter/Prod.lean","def_pos":[144,8],"def_end_pos":[144,23]},{"full_name":"Filter.tendsto_id","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2649,8],"def_end_pos":[2649,18]}]}]}
{"url":"Mathlib/Topology/Bases.lean","commit":"","full_name":"TopologicalSpace.exists_countable_basis","start":[724,0],"end":[728,78],"file_path":"Mathlib/Topology/Bases.lean","tactics":[{"state_before":"α : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\n⊢ ∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b","state_after":"case intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b","tactic":"obtain ⟨b, hb₁, hb₂⟩ := @SecondCountableTopology.is_open_generated_countable α _ _","premises":[{"full_name":"SecondCountableTopology.is_open_generated_countable","def_path":"Mathlib/Topology/Bases.lean","def_pos":[706,2],"def_end_pos":[706,29]}]},{"state_before":"case intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b","state_after":"case intro.intro.refine_1\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ b} \\ {∅}).Countable\n\ncase intro.intro.refine_2\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ∅ ∈ {∅}","tactic":"refine ⟨_, ?_, not_mem_diff_of_mem ?_, (isTopologicalBasis_of_subbasis hb₂).diff_empty⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.not_mem_diff_of_mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1419,8],"def_end_pos":[1419,27]},{"full_name":"TopologicalSpace.IsTopologicalBasis.diff_empty","def_path":"Mathlib/Topology/Bases.lean","def_pos":[90,8],"def_end_pos":[90,37]},{"full_name":"TopologicalSpace.isTopologicalBasis_of_subbasis","def_path":"Mathlib/Topology/Bases.lean","def_pos":[104,8],"def_end_pos":[104,38]}]},{"state_before":"case intro.intro.refine_1\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ b} \\ {∅}).Countable\n\ncase intro.intro.refine_2\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ∅ ∈ {∅}","state_after":"no goals","tactic":"exacts [((countable_setOf_finite_subset hb₁).image _).mono diff_subset, rfl]","premises":[{"full_name":"Set.Countable.image","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[149,8],"def_end_pos":[149,23]},{"full_name":"Set.Countable.mono","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[109,8],"def_end_pos":[109,22]},{"full_name":"Set.countable_setOf_finite_subset","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[251,8],"def_end_pos":[251,37]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Order/Interval/Set/Basic.lean","commit":"","full_name":"Set.Icc_union_Icc'","start":[1382,0],"end":[1391,42],"file_path":"Mathlib/Order/Interval/Set/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\n⊢ Icc a b ∪ Icc c d = Icc (min a c) (max b d)","state_after":"case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\n⊢ x ∈ Icc a b ∪ Icc c d ↔ x ∈ Icc (min a c) (max b d)","tactic":"ext1 x","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\n⊢ x ∈ Icc a b ∪ Icc c d ↔ x ∈ Icc (min a c) (max b d)","state_after":"case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d)","tactic":"simp_rw [mem_union, mem_Icc, min_le_iff, le_max_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_Icc","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[110,8],"def_end_pos":[110,15]},{"full_name":"Set.mem_union","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[609,8],"def_end_pos":[609,17]},{"full_name":"le_max_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[35,8],"def_end_pos":[35,18]},{"full_name":"min_le_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[39,8],"def_end_pos":[39,18]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d)","state_after":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\nhc : c ≤ x\nhd : x ≤ d\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d)\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\nhc : c ≤ x\nhd : ¬x ≤ d\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d)\n\ncase pos\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\nhc : ¬c ≤ x\nhd : x ≤ d\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d)\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\nhc : ¬c ≤ x\nhd : ¬x ≤ d\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d)","tactic":"by_cases hc : c ≤ x <;> by_cases hd : x ≤ d","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Gluing.lean","commit":"","full_name":"Metric.le_glueDist_inr_inl","start":[102,0],"end":[104,47],"file_path":"Mathlib/Topology/MetricSpace/Gluing.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nZ : Type w\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nx : Y\ny : X\n⊢ ε ≤ glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y)","state_after":"X : Type u\nY : Type v\nZ : Type w\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nx : Y\ny : X\n⊢ ε ≤ glueDist Φ Ψ ε (Sum.inl y) (Sum.inr x)","tactic":"rw [glueDist_comm]","premises":[{"full_name":"_private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_comm","def_path":"Mathlib/Topology/MetricSpace/Gluing.lean","def_pos":[84,16],"def_end_pos":[84,29]}]},{"state_before":"X : Type u\nY : Type v\nZ : Type w\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nx : Y\ny : X\n⊢ ε ≤ glueDist Φ Ψ ε (Sum.inl y) (Sum.inr x)","state_after":"no goals","tactic":"apply le_glueDist_inl_inr","premises":[{"full_name":"Metric.le_glueDist_inl_inr","def_path":"Mathlib/Topology/MetricSpace/Gluing.lean","def_pos":[98,8],"def_end_pos":[98,27]}]}]}
{"url":"Mathlib/Logic/Equiv/Set.lean","commit":"","full_name":"dite_comp_equiv_update","start":[605,0],"end":[626,18],"file_path":"Mathlib/Logic/Equiv/Set.lean","tactics":[{"state_before":"α✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Sort u_2\nγ : Sort u_3\np : α → Prop\ne : β ≃ Subtype p\nv : β → γ\nw : α → γ\nj : β\nx : γ\ninst✝² : DecidableEq β\ninst✝¹ : DecidableEq α\ninst✝ : (j : α) → Decidable (p j)\n⊢ (fun i => if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =\n    update (fun i => if h : p i then v (e.symm ⟨i, h⟩) else w i) (↑(e j)) x","state_after":"case h\nα✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Sort u_2\nγ : Sort u_3\np : α → Prop\ne : β ≃ Subtype p\nv : β → γ\nw : α → γ\nj : β\nx : γ\ninst✝² : DecidableEq β\ninst✝¹ : DecidableEq α\ninst✝ : (j : α) → Decidable (p j)\ni : α\n⊢ (if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =\n    update (fun i => if h : p i then v (e.symm ⟨i, h⟩) else w i) (↑(e j)) x i","tactic":"ext i","premises":[]},{"state_before":"case h\nα✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Sort u_2\nγ : Sort u_3\np : α → Prop\ne : β ≃ Subtype p\nv : β → γ\nw : α → γ\nj : β\nx : γ\ninst✝² : DecidableEq β\ninst✝¹ : DecidableEq α\ninst✝ : (j : α) → Decidable (p j)\ni : α\n⊢ (if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =\n    update (fun i => if h : p i then v (e.symm ⟨i, h⟩) else w i) (↑(e j)) x i","state_after":"case pos\nα✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Sort u_2\nγ : Sort u_3\np : α → Prop\ne : β ≃ Subtype p\nv : β → γ\nw : α → γ\nj : β\nx : γ\ninst✝² : DecidableEq β\ninst✝¹ : DecidableEq α\ninst✝ : (j : α) → Decidable (p j)\ni : α\nh : p i\n⊢ (if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =\n    update (fun i => if h : p i then v (e.symm ⟨i, h⟩) else w i) (↑(e j)) x i\n\ncase neg\nα✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Sort u_2\nγ : Sort u_3\np : α → Prop\ne : β ≃ Subtype p\nv : β → γ\nw : α → γ\nj : β\nx : γ\ninst✝² : DecidableEq β\ninst✝¹ : DecidableEq α\ninst✝ : (j : α) → Decidable (p j)\ni : α\nh : ¬p i\n⊢ (if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =\n    update (fun i => if h : p i then v (e.symm ⟨i, h⟩) else w i) (↑(e j)) x i","tactic":"by_cases h : p i","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Fintype/Card.lean","commit":"","full_name":"Fintype.card_compl_eq_card_compl","start":[781,0],"end":[787,43],"file_path":"Mathlib/Data/Fintype/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : Finite α\np q : α → Prop\ninst✝³ : Fintype { x // p x }\ninst✝² : Fintype { x // ¬p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // ¬q x }\nh : card { x // p x } = card { x // q x }\n⊢ card { x // ¬p x } = card { x // ¬q x }","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : Finite α\np q : α → Prop\ninst✝³ : Fintype { x // p x }\ninst✝² : Fintype { x // ¬p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // ¬q x }\nh : card { x // p x } = card { x // q x }\nval✝ : Fintype α\n⊢ card { x // ¬p x } = card { x // ¬q x }","tactic":"cases nonempty_fintype α","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : Finite α\np q : α → Prop\ninst✝³ : Fintype { x // p x }\ninst✝² : Fintype { x // ¬p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // ¬q x }\nh : card { x // p x } = card { x // q x }\nval✝ : Fintype α\n⊢ card { x // ¬p x } = card { x // ¬q x }","state_after":"no goals","tactic":"simp only [Fintype.card_subtype_compl, h]","premises":[{"full_name":"Fintype.card_subtype_compl","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[768,8],"def_end_pos":[768,34]}]}]}
{"url":"Mathlib/Algebra/Group/Units.lean","commit":"","full_name":"AddUnits.neg_eq_of_add_eq_zero_right","start":[281,0],"end":[285,45],"file_path":"Mathlib/Algebra/Group/Units.lean","tactics":[{"state_before":"α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\nh : ↑u * a = 1\n⊢ ↑u⁻¹ = ↑u⁻¹ * 1","state_after":"no goals","tactic":"rw [mul_one]","premises":[{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\nh : ↑u * a = 1\n⊢ ↑u⁻¹ * 1 = a","state_after":"no goals","tactic":"rw [← h, inv_mul_cancel_left]","premises":[{"full_name":"Units.inv_mul_cancel_left","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[235,8],"def_end_pos":[235,27]}]}]}
{"url":"Mathlib/Topology/Bases.lean","commit":"","full_name":"TopologicalSpace.IsSeparable.prod","start":[492,0],"end":[499,8],"file_path":"Mathlib/Topology/Bases.lean","tactics":[{"state_before":"α : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\nhs : IsSeparable s\nht : IsSeparable t\n⊢ IsSeparable (s ×ˢ t)","state_after":"case intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\nht : IsSeparable t\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\n⊢ IsSeparable (s ×ˢ t)","tactic":"rcases hs with ⟨cs, cs_count, hcs⟩","premises":[]},{"state_before":"case intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\nht : IsSeparable t\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\n⊢ IsSeparable (s ×ˢ t)","state_after":"case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ IsSeparable (s ×ˢ t)","tactic":"rcases ht with ⟨ct, ct_count, hct⟩","premises":[]},{"state_before":"case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ IsSeparable (s ×ˢ t)","state_after":"case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ s ×ˢ t ⊆ closure (cs ×ˢ ct)","tactic":"refine ⟨cs ×ˢ ct, cs_count.prod ct_count, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Set.Countable.prod","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[268,18],"def_end_pos":[268,32]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ s ×ˢ t ⊆ closure (cs ×ˢ ct)","state_after":"case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ s ×ˢ t ⊆ closure cs ×ˢ closure ct","tactic":"rw [closure_prod_eq]","premises":[{"full_name":"closure_prod_eq","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[718,8],"def_end_pos":[718,23]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ s ×ˢ t ⊆ closure cs ×ˢ closure ct","state_after":"no goals","tactic":"gcongr","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","commit":"","full_name":"CategoryTheory.Limits.biprod.braiding'_eq_braiding","start":[1904,0],"end":[1905,11],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","tactics":[{"state_before":"J : Type w\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nD : Type uD\ninst✝² : Category.{uD', uD} D\ninst✝¹ : HasZeroMorphisms D\nP✝ Q✝ : C\ninst✝ : HasBinaryBiproducts C\nP Q : C\n⊢ braiding' P Q = braiding P Q","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.filter_filter","start":[1804,0],"end":[1807,37],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝¹ : DecidablePred p\nq : α → Prop\ninst✝ : DecidablePred q\ns : Multiset α\nl : List α\n⊢ filter p (filter q (Quot.mk Setoid.r l)) = filter (fun a => p a ∧ q a) (Quot.mk Setoid.r l)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/SetTheory/Ordinal/Principal.lean","commit":"","full_name":"Ordinal.principal_mul_iff_le_two_or_omega_opow_opow","start":[340,0],"end":[356,43],"file_path":"Mathlib/SetTheory/Ordinal/Principal.lean","tactics":[{"state_before":"o : Ordinal.{u_1}\n⊢ Principal (fun x x_1 => x * x_1) o ↔ o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a","state_after":"case refine_1\no : Ordinal.{u_1}\nho : Principal (fun x x_1 => x * x_1) o\n⊢ o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a\n\ncase refine_2\no : Ordinal.{u_1}\n⊢ (o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a) → Principal (fun x x_1 => x * x_1) o","tactic":"refine ⟨fun ho => ?_, ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Topology/Category/Compactum.lean","commit":"","full_name":"_private.Mathlib.Topology.Category.Compactum.0.Compactum.subset_cl","start":[199,0],"end":[200,25],"file_path":"Mathlib/Topology/Category/Compactum.lean","tactics":[{"state_before":"X : Compactum\nA : Set X.A\na : X.A\nha : a ∈ A\n⊢ X.str (X.incl a) = a","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Group/Action/Defs.lean","commit":"","full_name":"smul_mul_smul","start":[447,0],"end":[452,63],"file_path":"Mathlib/Algebra/Group/Action/Defs.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁴ : Monoid M\ninst✝³ : MulAction M α\ninst✝² : Mul α\nr s : M\nx y : α\ninst✝¹ : IsScalarTower M α α\ninst✝ : SMulCommClass M α α\n⊢ r • x * s • y = (r * s) • (x * y)","state_after":"no goals","tactic":"rw [smul_mul_assoc, mul_smul_comm, ← smul_assoc, smul_eq_mul]","premises":[{"full_name":"mul_smul_comm","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[339,6],"def_end_pos":[339,19]},{"full_name":"smul_assoc","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[219,6],"def_end_pos":[219,16]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]},{"full_name":"smul_mul_assoc","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[344,6],"def_end_pos":[344,20]}]}]}
{"url":"Mathlib/Algebra/DirectSum/Basic.lean","commit":"","full_name":"DirectSum.mk_apply_of_not_mem","start":[136,0],"end":[139,17],"file_path":"Mathlib/Algebra/DirectSum/Basic.lean","tactics":[{"state_before":"ι : Type v\nβ : ι → Type w\ninst✝¹ : (i : ι) → AddCommMonoid (β i)\ninst✝ : DecidableEq ι\ns : Finset ι\nf : (i : ↑↑s) → β ↑i\nn : ι\nhn : n ∉ s\n⊢ ((mk β s) f) n = 0","state_after":"ι : Type v\nβ : ι → Type w\ninst✝¹ : (i : ι) → AddCommMonoid (β i)\ninst✝ : DecidableEq ι\ns : Finset ι\nf : (i : ↑↑s) → β ↑i\nn : ι\nhn : n ∉ s\n⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = 0","tactic":"dsimp only [Finset.coe_sort_coe, mk, AddMonoidHom.coe_mk, ZeroHom.coe_mk, DFinsupp.mk_apply]","premises":[{"full_name":"AddMonoidHom.coe_mk","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[496,2],"def_end_pos":[496,13]},{"full_name":"DFinsupp.mk_apply","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[501,8],"def_end_pos":[501,16]},{"full_name":"DirectSum.mk","def_path":"Mathlib/Algebra/DirectSum/Basic.lean","def_pos":[110,4],"def_end_pos":[110,6]},{"full_name":"Finset.coe_sort_coe","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[258,8],"def_end_pos":[258,20]},{"full_name":"ZeroHom.coe_mk","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[484,2],"def_end_pos":[484,13]}]},{"state_before":"ι : Type v\nβ : ι → Type w\ninst✝¹ : (i : ι) → AddCommMonoid (β i)\ninst✝ : DecidableEq ι\ns : Finset ι\nf : (i : ↑↑s) → β ↑i\nn : ι\nhn : n ∉ s\n⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = 0","state_after":"no goals","tactic":"rw [dif_neg hn]","premises":[{"full_name":"dif_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[954,8],"def_end_pos":[954,15]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/RationalRoot.lean","commit":"","full_name":"scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero","start":[37,0],"end":[42,33],"file_path":"Mathlib/RingTheory/Polynomial/RationalRoot.lean","tactics":[{"state_before":"A : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁷ : CommRing A\ninst✝⁶ : Field K\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Submonoid A\ninst✝³ : Algebra A S\ninst✝² : IsLocalization M S\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : A\ns : ↥M\nhr : (aeval (mk' S r s)) p = 0\n⊢ (aeval ((algebraMap A S) r)) (p.scaleRoots ↑s) = 0","state_after":"case h.e'_2.h.e\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁷ : CommRing A\ninst✝⁶ : Field K\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Submonoid A\ninst✝³ : Algebra A S\ninst✝² : IsLocalization M S\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : A\ns : ↥M\nhr : (aeval (mk' S r s)) p = 0\n⊢ ⇑(aeval ((algebraMap A S) r)) = eval₂ (algebraMap A S) ((algebraMap A S) ↑s * mk' S r s)","tactic":"convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr","premises":[{"full_name":"Polynomial.scaleRoots_eval₂_eq_zero","def_path":"Mathlib/RingTheory/Polynomial/ScaleRoots.lean","def_pos":[145,8],"def_end_pos":[145,32]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case h.e'_2.h.e\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁷ : CommRing A\ninst✝⁶ : Field K\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Submonoid A\ninst✝³ : Algebra A S\ninst✝² : IsLocalization M S\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : A\ns : ↥M\nhr : (aeval (mk' S r s)) p = 0\n⊢ ⇑(aeval ((algebraMap A S) r)) = eval₂ (algebraMap A S) ((algebraMap A S) ↑s * mk' S r s)","state_after":"case h.e'_2.h.e.h\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁷ : CommRing A\ninst✝⁶ : Field K\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Submonoid A\ninst✝³ : Algebra A S\ninst✝² : IsLocalization M S\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : A\ns : ↥M\nhr : (aeval (mk' S r s)) p = 0\nx✝ : A[X]\n⊢ (aeval ((algebraMap A S) r)) x✝ = eval₂ (algebraMap A S) ((algebraMap A S) ↑s * mk' S r s) x✝","tactic":"funext","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h.e'_2.h.e.h\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁷ : CommRing A\ninst✝⁶ : Field K\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Submonoid A\ninst✝³ : Algebra A S\ninst✝² : IsLocalization M S\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : A\ns : ↥M\nhr : (aeval (mk' S r s)) p = 0\nx✝ : A[X]\n⊢ (aeval ((algebraMap A S) r)) x✝ = eval₂ (algebraMap A S) ((algebraMap A S) ↑s * mk' S r s) x✝","state_after":"no goals","tactic":"rw [aeval_def, mk'_spec' _ r s]","premises":[{"full_name":"IsLocalization.mk'_spec'","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[246,8],"def_end_pos":[246,17]},{"full_name":"Polynomial.aeval_def","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[242,8],"def_end_pos":[242,17]}]}]}
{"url":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","commit":"","full_name":"VectorFourier.fourierPowSMulRight_iteratedFDeriv_fourierIntegral","start":[547,0],"end":[575,79],"file_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","tactics":[{"state_before":"E : Type u_1\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : NormedSpace ℝ V\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝⁴ : SecondCountableTopology V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nμ✝ : Measure V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nK N : ℕ∞\nhf : ContDiff ℝ N f\nh'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ\nk n : ℕ\nhk : ↑k ≤ K\nhn : ↑n ≤ N\nw : W\n⊢ fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w n =\n    fourierIntegral 𝐞 μ L.toLinearMap₂ (iteratedFDeriv ℝ n fun v => fourierPowSMulRight L f v k) w","state_after":"E : Type u_1\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : NormedSpace ℝ V\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝⁴ : SecondCountableTopology V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nμ✝ : Measure V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nK N : ℕ∞\nhf : ContDiff ℝ N f\nh'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ\nk n : ℕ\nhk : ↑k ≤ K\nhn : ↑n ≤ N\nw : W\n⊢ fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w n =\n    (fun w =>\n        fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ fun v => fourierPowSMulRight L f v k) w n)\n      w\n\nE : Type u_1\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : NormedSpace ℝ V\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝⁴ : SecondCountableTopology V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nμ✝ : Measure V\ninst✝¹ : FiniteDimensional ℝ V\nμ : Measure V\ninst✝ : μ.IsAddHaarMeasure\nK N : ℕ∞\nhf : ContDiff ℝ N f\nh'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ\nk n : ℕ\nhk : ↑k ≤ K\nhn : ↑n ≤ N\nw : W\n⊢ ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n fun v => fourierPowSMulRight L f v k) μ","tactic":"rw [fourierIntegral_iteratedFDeriv (N := N) _ (hf.fourierPowSMulRight _ _) _ hn]","premises":[{"full_name":"ContDiff.fourierPowSMulRight","def_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","def_pos":[313,6],"def_end_pos":[313,41]},{"full_name":"VectorFourier.fourierIntegral_iteratedFDeriv","def_path":"Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","def_pos":[511,8],"def_end_pos":[511,38]}]}]}
{"url":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","commit":"","full_name":"MeasurableEquiv.preimage_image","start":[275,0],"end":[277,42],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\ne : α ≃ᵐ β\ns : Set α\n⊢ ⇑e ⁻¹' (⇑e '' s) = s","state_after":"no goals","tactic":"rw [← coe_toEquiv, Equiv.preimage_image]","premises":[{"full_name":"Equiv.preimage_image","def_path":"Mathlib/Logic/Equiv/Set.lean","def_pos":[90,8],"def_end_pos":[90,22]},{"full_name":"MeasurableEquiv.coe_toEquiv","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[155,8],"def_end_pos":[155,19]}]}]}
{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"disjoint_interior_frontier","start":[578,0],"end":[581,76],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\n⊢ Disjoint (interior s) (frontier s)","state_after":"no goals","tactic":"rw [disjoint_iff_inter_eq_empty, ← closure_diff_interior, diff_eq,\n    ← inter_assoc, inter_comm, ← inter_assoc, compl_inter_self, empty_inter]","premises":[{"full_name":"Set.compl_inter_self","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1310,8],"def_end_pos":[1310,24]},{"full_name":"Set.diff_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[86,8],"def_end_pos":[86,15]},{"full_name":"Set.disjoint_iff_inter_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1210,8],"def_end_pos":[1210,35]},{"full_name":"Set.empty_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[739,8],"def_end_pos":[739,19]},{"full_name":"Set.inter_assoc","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[745,8],"def_end_pos":[745,19]},{"full_name":"Set.inter_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[742,8],"def_end_pos":[742,18]},{"full_name":"closure_diff_interior","def_path":"Mathlib/Topology/Basic.lean","def_pos":[575,8],"def_end_pos":[575,29]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Splits.lean","commit":"","full_name":"Polynomial.mem_lift_of_splits_of_roots_mem_range","start":[315,0],"end":[321,66],"file_path":"Mathlib/Algebra/Polynomial/Splits.lean","tactics":[{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\n⊢ f ∈ lifts (algebraMap R K)","state_after":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\n⊢ (Multiset.map (fun a => X - C a) f.roots).prod ∈ liftsRing (algebraMap R K)","tactic":"rw [eq_prod_roots_of_monic_of_splits_id hm hs, lifts_iff_liftsRing]","premises":[{"full_name":"Polynomial.eq_prod_roots_of_monic_of_splits_id","def_path":"Mathlib/Algebra/Polynomial/Splits.lean","def_pos":[303,8],"def_end_pos":[303,43]},{"full_name":"Polynomial.lifts_iff_liftsRing","def_path":"Mathlib/Algebra/Polynomial/Lifts.lean","def_pos":[234,8],"def_end_pos":[234,27]}]},{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\n⊢ (Multiset.map (fun a => X - C a) f.roots).prod ∈ liftsRing (algebraMap R K)","state_after":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\nP : K[X]\nhP : P ∈ Multiset.map (fun a => X - C a) f.roots\n⊢ P ∈ liftsRing (algebraMap R K)","tactic":"refine Subring.multiset_prod_mem _ _ fun P hP => ?_","premises":[{"full_name":"Subring.multiset_prod_mem","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[302,18],"def_end_pos":[302,35]}]},{"state_before":"R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\nP : K[X]\nhP : P ∈ Multiset.map (fun a => X - C a) f.roots\n⊢ P ∈ liftsRing (algebraMap R K)","state_after":"case intro.intro\nR : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\nb : K\nhb : b ∈ f.roots\nhP : X - C b ∈ Multiset.map (fun a => X - C a) f.roots\n⊢ X - C b ∈ liftsRing (algebraMap R K)","tactic":"obtain ⟨b, hb, rfl⟩ := Multiset.mem_map.1 hP","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]}]},{"state_before":"case intro.intro\nR : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field F\ni : K →+* L\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : f.Monic\nhr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range\nb : K\nhb : b ∈ f.roots\nhP : X - C b ∈ Multiset.map (fun a => X - C a) f.roots\n⊢ X - C b ∈ liftsRing (algebraMap R K)","state_after":"no goals","tactic":"exact Subring.sub_mem _ (X_mem_lifts _) (C'_mem_lifts (hr _ hb))","premises":[{"full_name":"Polynomial.C'_mem_lifts","def_path":"Mathlib/Algebra/Polynomial/Lifts.lean","def_pos":[78,8],"def_end_pos":[78,20]},{"full_name":"Polynomial.X_mem_lifts","def_path":"Mathlib/Algebra/Polynomial/Lifts.lean","def_pos":[85,8],"def_end_pos":[85,19]},{"full_name":"Subring.sub_mem","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[290,18],"def_end_pos":[290,25]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Int.lean","commit":"","full_name":"norm_zsmul_le","start":[51,0],"end":[53,85],"file_path":"Mathlib/Analysis/Normed/Group/Int.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : SeminormedCommGroup α\nn : ℤ\na : α\n⊢ ‖a ^ n‖ ≤ ‖n‖ * ‖a‖","state_after":"no goals","tactic":"rcases n.eq_nat_or_neg with ⟨n, rfl | rfl⟩ <;> simpa using norm_pow_le_mul_norm n a","premises":[{"full_name":"Int.eq_nat_or_neg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[1006,8],"def_end_pos":[1006,21]},{"full_name":"norm_pow_le_mul_norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1018,8],"def_end_pos":[1018,28]}]}]}
{"url":"Mathlib/Data/Fintype/Basic.lean","commit":"","full_name":"List.toFinset_finRange","start":[707,0],"end":[708,11],"file_path":"Mathlib/Data/Fintype/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nn : ℕ\n⊢ (finRange n).toFinset = univ","state_after":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nn : ℕ\na✝ : Fin n\n⊢ a✝ ∈ (finRange n).toFinset ↔ a✝ ∈ univ","tactic":"ext","premises":[]},{"state_before":"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nn : ℕ\na✝ : Fin n\n⊢ a✝ ∈ (finRange n).toFinset ↔ a✝ ∈ univ","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieHom.mem_idealRange_iff","start":[1054,0],"end":[1058,89],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : f.IsIdealMorphism\ny : L'\n⊢ y ∈ f.idealRange ↔ ∃ x, f x = y","state_after":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : lieIdealSubalgebra R L' f.idealRange = f.range\ny : L'\n⊢ y ∈ f.idealRange ↔ ∃ x, f x = y","tactic":"rw [f.isIdealMorphism_def] at h","premises":[{"full_name":"LieHom.isIdealMorphism_def","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1015,8],"def_end_pos":[1015,27]}]},{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : lieIdealSubalgebra R L' f.idealRange = f.range\ny : L'\n⊢ y ∈ f.idealRange ↔ ∃ x, f x = y","state_after":"no goals","tactic":"rw [← LieSubmodule.mem_coe, ← LieIdeal.coe_toSubalgebra, h, f.range_coe, Set.mem_range]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LieHom.range_coe","def_path":"Mathlib/Algebra/Lie/Subalgebra.lean","def_pos":[271,8],"def_end_pos":[271,17]},{"full_name":"LieIdeal.coe_toSubalgebra","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[225,8],"def_end_pos":[225,33]},{"full_name":"LieSubmodule.mem_coe","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[107,8],"def_end_pos":[107,15]},{"full_name":"Set.mem_range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[146,16],"def_end_pos":[146,25]}]}]}
{"url":"Mathlib/NumberTheory/RamificationInertia.lean","commit":"","full_name":"Ideal.le_pow_of_le_ramificationIdx","start":[112,0],"end":[115,29],"file_path":"Mathlib/NumberTheory/RamificationInertia.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhn : n ≤ ramificationIdx f p P\n⊢ map f p ≤ P ^ n","state_after":"R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhn : ¬map f p ≤ P ^ n\n⊢ ramificationIdx f p P < n","tactic":"contrapose! hn","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhn : ¬map f p ≤ P ^ n\n⊢ ramificationIdx f p P < n","state_after":"no goals","tactic":"exact ramificationIdx_lt hn","premises":[{"full_name":"Ideal.ramificationIdx_lt","def_path":"Mathlib/NumberTheory/RamificationInertia.lean","def_pos":[90,8],"def_end_pos":[90,26]}]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","commit":"","full_name":"cfc_pow_id","start":[480,0],"end":[481,29],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : TopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Ring A\ninst✝² : StarRing A\ninst✝¹ : Algebra R A\ninst✝ : ContinuousFunctionalCalculus R p\nf g : R → R\na✝ : A\nha✝ : autoParam (p a✝) _auto✝\nhf : autoParam (ContinuousOn f (spectrum R a✝)) _auto✝\nhg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝\na : A\nn : ℕ\nha : autoParam (p a) _auto✝\n⊢ cfc (fun x => x ^ n) a = a ^ n","state_after":"no goals","tactic":"rw [cfc_pow .., cfc_id' ..]","premises":[{"full_name":"cfc_id'","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","def_pos":[329,6],"def_end_pos":[329,13]},{"full_name":"cfc_pow","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean","def_pos":[405,6],"def_end_pos":[405,13]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/SlashActions.lean","commit":"","full_name":"ModularForm.mul_slash_SL2","start":[215,0],"end":[222,49],"file_path":"Mathlib/NumberTheory/ModularForms/SlashActions.lean","tactics":[{"state_before":"Γ : Subgroup SL(2, ℤ)\nk : ℤ\nf✝ : ℍ → ℂ\nk1 k2 : ℤ\nA : SL(2, ℤ)\nf g : ℍ → ℂ\n⊢ (f * g) ∣[k1 + k2] ↑A = (↑↑↑A).det • f ∣[k1] A * g ∣[k2] A","state_after":"no goals","tactic":"apply mul_slash","premises":[{"full_name":"ModularForm.mul_slash","def_path":"Mathlib/NumberTheory/ModularForms/SlashActions.lean","def_pos":[195,8],"def_end_pos":[195,17]}]},{"state_before":"Γ : Subgroup SL(2, ℤ)\nk : ℤ\nf✝ : ℍ → ℂ\nk1 k2 : ℤ\nA : SL(2, ℤ)\nf g : ℍ → ℂ\n⊢ (↑↑↑A).det • f ∣[k1] A * g ∣[k2] A = 1 • f ∣[k1] A * g ∣[k2] A","state_after":"no goals","tactic":"rw [det_coe']","premises":[{"full_name":"UpperHalfPlane.ModularGroup.det_coe'","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[287,8],"def_end_pos":[287,16]}]},{"state_before":"Γ : Subgroup SL(2, ℤ)\nk : ℤ\nf✝ : ℍ → ℂ\nk1 k2 : ℤ\nA : SL(2, ℤ)\nf g : ℍ → ℂ\n⊢ 1 • f ∣[k1] A * g ∣[k2] A = f ∣[k1] A * g ∣[k2] A","state_after":"no goals","tactic":"rw [one_smul]","premises":[{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","commit":"","full_name":"Orientation.oangle_rotation_self_left","start":[221,0],"end":[224,52],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nhx : x ≠ 0\nθ : Real.Angle\n⊢ o.oangle ((o.rotation θ) x) x = -θ","state_after":"no goals","tactic":"simp [hx]","premises":[]}]}
{"url":"Mathlib/Algebra/GCDMonoid/Finset.lean","commit":"","full_name":"Finset.gcd_div_eq_one","start":[246,0],"end":[252,58],"file_path":"Mathlib/Algebra/GCDMonoid/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\n⊢ (s.gcd fun j => f j / s.gcd f) = 1","state_after":"case intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\ng : ι → α\nhe : ∀ b ∈ s, f b = s.gcd f * g b\nhg : s.gcd g = 1\n⊢ (s.gcd fun j => f j / s.gcd f) = 1","tactic":"obtain ⟨g, he, hg⟩ := Finset.extract_gcd f ⟨i, his⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.extract_gcd","def_path":"Mathlib/Algebra/GCDMonoid/Finset.lean","def_pos":[231,8],"def_end_pos":[231,19]}]},{"state_before":"case intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\ng : ι → α\nhe : ∀ b ∈ s, f b = s.gcd f * g b\nhg : s.gcd g = 1\n⊢ (s.gcd fun j => f j / s.gcd f) = 1","state_after":"case intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\ng : ι → α\nhe : ∀ b ∈ s, f b = s.gcd f * g b\nhg : s.gcd g = 1\na : ι\nha : a ∈ s\n⊢ f a / s.gcd f = g a","tactic":"refine (Finset.gcd_congr rfl fun a ha ↦ ?_).trans hg","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Finset.gcd_congr","def_path":"Mathlib/Algebra/GCDMonoid/Finset.lean","def_pos":[160,8],"def_end_pos":[160,17]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\ng : ι → α\nhe : ∀ b ∈ s, f b = s.gcd f * g b\nhg : s.gcd g = 1\na : ι\nha : a ∈ s\n⊢ f a / s.gcd f = g a","state_after":"case intro.intro.ha\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\ng : ι → α\nhe : ∀ b ∈ s, f b = s.gcd f * g b\nhg : s.gcd g = 1\na : ι\nha : a ∈ s\n⊢ s.gcd f ≠ 0","tactic":"rw [he a ha, mul_div_cancel_left₀]","premises":[{"full_name":"mul_div_cancel_left₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[172,14],"def_end_pos":[172,34]}]},{"state_before":"case intro.intro.ha\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ns✝ s₁ s₂ : Finset β\nf✝ : β → α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\ng : ι → α\nhe : ∀ b ∈ s, f b = s.gcd f * g b\nhg : s.gcd g = 1\na : ι\nha : a ∈ s\n⊢ s.gcd f ≠ 0","state_after":"no goals","tactic":"exact mt Finset.gcd_eq_zero_iff.1 fun h ↦ hfi <| h i his","premises":[{"full_name":"Finset.gcd_eq_zero_iff","def_path":"Mathlib/Algebra/GCDMonoid/Finset.lean","def_pos":[178,8],"def_end_pos":[178,23]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
{"url":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","commit":"","full_name":"FiniteDimensional.finrank_directSum","start":[224,0],"end":[231,33],"file_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","tactics":[{"state_before":"R S : Type u\nM✝ : Type v\nM' : Type v'\nM₁ : Type v\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝¹⁷ : Ring R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : AddCommGroup M✝\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R M✝\ninst✝¹¹ : Module R M'\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : Module.Free R M✝\ninst✝⁷ : Module.Free R M'\ninst✝⁶ : Module.Finite R M✝\ninst✝⁵ : Module.Finite R M'\nι : Type v\ninst✝⁴ : Fintype ι\nM : ι → Type w\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Module.Free R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\n⊢ finrank R (⨁ (i : ι), M i) = ∑ i : ι, finrank R (M i)","state_after":"R S : Type u\nM✝ : Type v\nM' : Type v'\nM₁ : Type v\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝¹⁷ : Ring R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : AddCommGroup M✝\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R M✝\ninst✝¹¹ : Module R M'\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : Module.Free R M✝\ninst✝⁷ : Module.Free R M'\ninst✝⁶ : Module.Finite R M✝\ninst✝⁵ : Module.Finite R M'\nι : Type v\ninst✝⁴ : Fintype ι\nM : ι → Type w\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Module.Free R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ finrank R (⨁ (i : ι), M i) = ∑ i : ι, finrank R (M i)","tactic":"letI := nontrivial_of_invariantBasisNumber R","premises":[{"full_name":"nontrivial_of_invariantBasisNumber","def_path":"Mathlib/LinearAlgebra/InvariantBasisNumber.lean","def_pos":[233,8],"def_end_pos":[233,42]}]},{"state_before":"R S : Type u\nM✝ : Type v\nM' : Type v'\nM₁ : Type v\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝¹⁷ : Ring R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : AddCommGroup M✝\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R M✝\ninst✝¹¹ : Module R M'\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : Module.Free R M✝\ninst✝⁷ : Module.Free R M'\ninst✝⁶ : Module.Finite R M✝\ninst✝⁵ : Module.Finite R M'\nι : Type v\ninst✝⁴ : Fintype ι\nM : ι → Type w\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Module.Free R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ finrank R (⨁ (i : ι), M i) = ∑ i : ι, finrank R (M i)","state_after":"no goals","tactic":"simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_directSum, ← mk_sigma,\n    mk_toNat_eq_card, card_sigma]","premises":[{"full_name":"Cardinal.mk_sigma","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[747,8],"def_end_pos":[747,16]},{"full_name":"Cardinal.mk_toNat_eq_card","def_path":"Mathlib/SetTheory/Cardinal/ToNat.lean","def_pos":[117,8],"def_end_pos":[117,24]},{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Fintype.card_sigma","def_path":"Mathlib/Data/Fintype/BigOperators.lean","def_pos":[130,21],"def_end_pos":[130,31]},{"full_name":"Module.Free.rank_eq_card_chooseBasisIndex","def_path":"Mathlib/LinearAlgebra/Dimension/Free.lean","def_pos":[77,8],"def_end_pos":[77,37]},{"full_name":"rank_directSum","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[178,8],"def_end_pos":[178,22]}]}]}
{"url":"Mathlib/Data/Real/EReal.lean","commit":"","full_name":"EReal.mul_comm","start":[200,0],"end":[203,37],"file_path":"Mathlib/Data/Real/EReal.lean","tactics":[{"state_before":"x y : EReal\n⊢ x * y = y * x","state_after":"case h_real.h_real\na✝¹ a✝ : ℝ\n⊢ ↑a✝¹ * ↑a✝ = ↑a✝ * ↑a✝¹","tactic":"induction x <;> induction y  <;>\n    try { rfl }","premises":[]},{"state_before":"case h_real.h_real\na✝¹ a✝ : ℝ\n⊢ ↑a✝¹ * ↑a✝ = ↑a✝ * ↑a✝¹","state_after":"no goals","tactic":"rw [← coe_mul, ← coe_mul, mul_comm]","premises":[{"full_name":"EReal.coe_mul","def_path":"Mathlib/Data/Real/EReal.lean","def_pos":[151,8],"def_end_pos":[151,15]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/RingTheory/Norm/Basic.lean","commit":"","full_name":"Algebra.norm_eq_prod_roots","start":[193,0],"end":[197,99],"file_path":"Mathlib/RingTheory/Norm/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Field F\ninst✝⁵ : Algebra K L\ninst✝⁴ : Algebra K F\nι : Type w\nE : Type u_7\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra.IsSeparable K L\ninst✝ : FiniteDimensional K L\nx : L\nhF : Splits (algebraMap K F) (minpoly K x)\n⊢ (algebraMap K F) ((norm K) x) = ((minpoly K x).aroots F).prod ^ finrank (↥K⟮x⟯) L","state_after":"no goals","tactic":"rw [norm_eq_norm_adjoin K x, map_pow, IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots _ hF]","premises":[{"full_name":"Algebra.norm_eq_norm_adjoin","def_path":"Mathlib/RingTheory/Norm/Basic.lean","def_pos":[134,8],"def_end_pos":[134,27]},{"full_name":"IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots","def_path":"Mathlib/RingTheory/Norm/Basic.lean","def_pos":[159,8],"def_end_pos":[159,68]},{"full_name":"map_pow","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[423,8],"def_end_pos":[423,15]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","commit":"","full_name":"Matrix.det_fromBlocks₁₁","start":[362,0],"end":[368,94],"file_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝ : Invertible A\n⊢ (fromBlocks A B C D).det = A.det * (D - C * ⅟A * B).det","state_after":"no goals","tactic":"rw [fromBlocks_eq_of_invertible₁₁ (A := A), det_mul, det_mul, det_fromBlocks_zero₂₁,\n    det_fromBlocks_zero₂₁, det_fromBlocks_zero₁₂, det_one, det_one, one_mul, one_mul, mul_one]","premises":[{"full_name":"Matrix.det_fromBlocks_zero₁₂","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[662,8],"def_end_pos":[662,29]},{"full_name":"Matrix.det_fromBlocks_zero₂₁","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[608,8],"def_end_pos":[608,29]},{"full_name":"Matrix.det_mul","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[129,8],"def_end_pos":[129,15]},{"full_name":"Matrix.det_one","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[83,8],"def_end_pos":[83,15]},{"full_name":"Matrix.fromBlocks_eq_of_invertible₁₁","def_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","def_pos":[50,8],"def_end_pos":[50,37]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]}]}]}
{"url":"Mathlib/Logic/Equiv/Basic.lean","commit":"","full_name":"Equiv.boolProdEquivSum_apply","start":[924,0],"end":[930,40],"file_path":"Mathlib/Logic/Equiv/Basic.lean","tactics":[{"state_before":"α : Type ?u.53748\n⊢ LeftInverse (Sum.elim (Prod.mk false) (Prod.mk true)) fun p => Bool.casesOn p.1 (inl p.2) (inr p.2)","state_after":"no goals","tactic":"rintro ⟨_ | _, _⟩ <;> rfl","premises":[]},{"state_before":"α : Type ?u.53748\n⊢ Function.RightInverse (Sum.elim (Prod.mk false) (Prod.mk true)) fun p => Bool.casesOn p.1 (inl p.2) (inr p.2)","state_after":"no goals","tactic":"rintro (_ | _) <;> rfl","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","commit":"","full_name":"rank_finsupp","start":[157,0],"end":[162,23],"file_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","tactics":[{"state_before":"R S : Type u\nM : Type v\nM' : Type v'\nM₁ : Type v\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝¹⁰ : Ring R\ninst✝⁹ : CommRing S\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : AddCommGroup M₁\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'\ninst✝³ : Module R M₁\ninst✝² : StrongRankCondition R\ninst✝¹ : Module.Free R M\ninst✝ : Module.Free R M'\nι : Type w\n⊢ Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} (Module.rank R M)","state_after":"case intro.mk\nR S : Type u\nM : Type v\nM' : Type v'\nM₁ : Type v\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝¹⁰ : Ring R\ninst✝⁹ : CommRing S\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : AddCommGroup M₁\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'\ninst✝³ : Module R M₁\ninst✝² : StrongRankCondition R\ninst✝¹ : Module.Free R M\ninst✝ : Module.Free R M'\nι : Type w\nfst✝ : Type v\nbs : Basis fst✝ R M\n⊢ Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} (Module.rank R M)","tactic":"obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)","premises":[{"full_name":"Module.Free.exists_basis","def_path":"Mathlib/LinearAlgebra/FreeModule/Basic.lean","def_pos":[35,2],"def_end_pos":[35,14]}]},{"state_before":"case intro.mk\nR S : Type u\nM : Type v\nM' : Type v'\nM₁ : Type v\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝¹⁰ : Ring R\ninst✝⁹ : CommRing S\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : AddCommGroup M₁\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M'\ninst✝³ : Module R M₁\ninst✝² : StrongRankCondition R\ninst✝¹ : Module.Free R M\ninst✝ : Module.Free R M'\nι : Type w\nfst✝ : Type v\nbs : Basis fst✝ R M\n⊢ Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} (Module.rank R M)","state_after":"no goals","tactic":"rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,\n    Cardinal.sum_const]","premises":[{"full_name":"Basis.mk_eq_rank''","def_path":"Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean","def_pos":[285,8],"def_end_pos":[285,26]},{"full_name":"Cardinal.mk_sigma","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[747,8],"def_end_pos":[747,16]},{"full_name":"Cardinal.sum_const","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[751,8],"def_end_pos":[751,17]},{"full_name":"Finsupp.basis","def_path":"Mathlib/LinearAlgebra/FinsuppVectorSpace.lean","def_pos":[62,14],"def_end_pos":[62,19]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.sum_of_injOn","start":[663,0],"end":[669,62],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι✝ : Type u_1\nκ✝ : Type u_2\nα✝ : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α✝\na : α✝\nf✝ g✝ : α✝ → β\ninst✝¹ : CommMonoid β\nι : Type u_6\nκ : Type u_7\nα : Type u_8\ninst✝ : CommMonoid α\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → α\ne : ι → κ\nhe : Set.InjOn e ↑s\nhest : Set.MapsTo e ↑s ↑t\nh' : ∀ i ∈ t, i ∉ e '' ↑s → g i = 1\nh : ∀ i ∈ s, f i = g (e i)\n⊢ ∏ i ∈ s, f i = ∏ j ∈ t, g j","state_after":"no goals","tactic":"classical\n  exact (prod_nbij e (fun a ↦ mem_image_of_mem e) he (by simp [Set.surjOn_image]) h).trans <|\n    prod_subset (image_subset_iff.2 hest) <| by simpa using h'","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Finset.image_subset_iff","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[403,8],"def_end_pos":[403,24]},{"full_name":"Finset.mem_image_of_mem","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[306,8],"def_end_pos":[306,24]},{"full_name":"Finset.prod_nbij","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[613,6],"def_end_pos":[613,15]},{"full_name":"Finset.prod_subset","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[521,8],"def_end_pos":[521,19]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.surjOn_image","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[753,8],"def_end_pos":[753,20]}]}]}
{"url":"Mathlib/Order/Category/NonemptyFinLinOrd.lean","commit":"","full_name":"NonemptyFinLinOrd.Iso.mk_hom","start":[91,0],"end":[102,30],"file_path":"Mathlib/Order/Category/NonemptyFinLinOrd.lean","tactics":[{"state_before":"α β : NonemptyFinLinOrd\ne : ↑α ≃o ↑β\n⊢ ↑e ≫ ↑e.symm = 𝟙 α","state_after":"case w\nα β : NonemptyFinLinOrd\ne : ↑α ≃o ↑β\nx : (forget NonemptyFinLinOrd).obj α\n⊢ (↑e ≫ ↑e.symm) x = (𝟙 α) x","tactic":"ext x","premises":[]},{"state_before":"case w\nα β : NonemptyFinLinOrd\ne : ↑α ≃o ↑β\nx : (forget NonemptyFinLinOrd).obj α\n⊢ (↑e ≫ ↑e.symm) x = (𝟙 α) x","state_after":"no goals","tactic":"exact e.symm_apply_apply x","premises":[{"full_name":"OrderIso.symm_apply_apply","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[761,8],"def_end_pos":[761,24]}]},{"state_before":"α β : NonemptyFinLinOrd\ne : ↑α ≃o ↑β\n⊢ ↑e.symm ≫ ↑e = 𝟙 β","state_after":"case w\nα β : NonemptyFinLinOrd\ne : ↑α ≃o ↑β\nx : (forget NonemptyFinLinOrd).obj β\n⊢ (↑e.symm ≫ ↑e) x = (𝟙 β) x","tactic":"ext x","premises":[]},{"state_before":"case w\nα β : NonemptyFinLinOrd\ne : ↑α ≃o ↑β\nx : (forget NonemptyFinLinOrd).obj β\n⊢ (↑e.symm ≫ ↑e) x = (𝟙 β) x","state_after":"no goals","tactic":"exact e.apply_symm_apply x","premises":[{"full_name":"OrderIso.apply_symm_apply","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[757,8],"def_end_pos":[757,24]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/List.lean","commit":"","full_name":"List.sum_eq_card_nsmul","start":[127,0],"end":[129,57],"file_path":"Mathlib/Algebra/BigOperators/Group/List.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na : M\nl : List M\nm : M\nh : ∀ x ∈ l, x = m\n⊢ l.prod = m ^ l.length","state_after":"no goals","tactic":"rw [← prod_replicate, ← List.eq_replicate.mpr ⟨rfl, h⟩]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"List.eq_replicate","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1347,8],"def_end_pos":[1347,20]},{"full_name":"List.prod_replicate","def_path":"Mathlib/Algebra/BigOperators/Group/List.lean","def_pos":[121,8],"def_end_pos":[121,22]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Order/ConditionallyCompleteLattice/Finset.lean","commit":"","full_name":"Finset.sup'_univ_eq_ciSup","start":[80,0],"end":[81,34],"file_path":"Mathlib/Order/ConditionallyCompleteLattice/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nf : ι → α\n⊢ univ.sup' ⋯ f = ⨆ i, f i","state_after":"no goals","tactic":"simp [sup'_eq_csSup_image, iSup]","premises":[{"full_name":"Finset.sup'_eq_csSup_image","def_path":"Mathlib/Order/ConditionallyCompleteLattice/Finset.lean","def_pos":[63,8],"def_end_pos":[63,27]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","commit":"","full_name":"InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero","start":[86,0],"end":[93,100],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x ≠ 0\n⊢ angle x (x + y) = Real.arctan (‖y‖ / ‖x‖)","state_after":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x ≠ 0\n⊢ Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) = Real.arcsin (‖y‖ / (‖x‖ * √(1 + (‖y‖ / ‖x‖) ^ 2)))","tactic":"rw [angle_add_eq_arcsin_of_inner_eq_zero h (Or.inl h0), Real.arctan_eq_arcsin, ←\n    div_mul_eq_div_div, norm_add_eq_sqrt_iff_real_inner_eq_zero.2 h]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[70,8],"def_end_pos":[70,44]},{"full_name":"Or.inl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[534,4],"def_end_pos":[534,7]},{"full_name":"Real.arctan_eq_arcsin","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean","def_pos":[134,8],"def_end_pos":[134,24]},{"full_name":"div_mul_eq_div_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[555,8],"def_end_pos":[555,26]},{"full_name":"norm_add_eq_sqrt_iff_real_inner_eq_zero","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1261,8],"def_end_pos":[1261,47]}]},{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x ≠ 0\n⊢ Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) = Real.arcsin (‖y‖ / (‖x‖ * √(1 + (‖y‖ / ‖x‖) ^ 2)))","state_after":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x ≠ 0\n⊢ Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) = Real.arcsin (‖y‖ / (√(‖x‖ ^ 2) * √(1 + (‖y‖ / ‖x‖) ^ 2)))","tactic":"nth_rw 3 [← Real.sqrt_sq (norm_nonneg x)]","premises":[{"full_name":"Real.sqrt_sq","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[173,8],"def_end_pos":[173,15]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]},{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x ≠ 0\n⊢ Real.arcsin (‖y‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)) = Real.arcsin (‖y‖ / (√(‖x‖ ^ 2) * √(1 + (‖y‖ / ‖x‖) ^ 2)))","state_after":"no goals","tactic":"rw_mod_cast [← Real.sqrt_mul (sq_nonneg _), div_pow, pow_two, pow_two, mul_add, mul_one, mul_div,\n    mul_comm (‖x‖ * ‖x‖), ← mul_div, div_self (mul_self_pos.2 (norm_ne_zero_iff.2 h0)).ne', mul_one]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real.sqrt_mul","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[317,8],"def_end_pos":[317,16]},{"full_name":"div_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[589,6],"def_end_pos":[589,13]},{"full_name":"div_self","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[251,14],"def_end_pos":[251,22]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"mul_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[342,8],"def_end_pos":[342,15]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"mul_self_pos","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[852,8],"def_end_pos":[852,20]},{"full_name":"norm_ne_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1212,14],"def_end_pos":[1212,30]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]},{"full_name":"sq_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[893,6],"def_end_pos":[893,15]}]}]}
{"url":"Mathlib/GroupTheory/Nilpotent.lean","commit":"","full_name":"_private.Mathlib.GroupTheory.Nilpotent.0.comap_center_subst","start":[531,0],"end":[532,90],"file_path":"Mathlib/GroupTheory/Nilpotent.lean","tactics":[{"state_before":"G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : H.Normal\nH₁ H₂ : Subgroup G\ninst✝¹ : H₁.Normal\ninst✝ : H₂.Normal\nh : H₁ = H₂\n⊢ comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₂) (center (G ⧸ H₂))","state_after":"G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : H.Normal\nH₁ : Subgroup G\ninst✝¹ inst✝ : H₁.Normal\n⊢ comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₁) (center (G ⧸ H₁))","tactic":"subst h","premises":[]},{"state_before":"G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : H.Normal\nH₁ : Subgroup G\ninst✝¹ inst✝ : H₁.Normal\n⊢ comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₁) (center (G ⧸ H₁))","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/BigOperators/Option.lean","commit":"","full_name":"Finset.prod_eraseNone","start":[31,0],"end":[38,57],"file_path":"Mathlib/Algebra/BigOperators/Option.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x ∈ eraseNone s, f x = ∏ x ∈ s, Option.elim' 1 f x","state_after":"no goals","tactic":"classical calc\n      ∏ x ∈ eraseNone s, f x = ∏ x ∈ (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x ∈ s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n      _ = ∏ x ∈ s, Option.elim' 1 f x := prod_erase _ rfl","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset.erase","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1574,4],"def_end_pos":[1574,9]},{"full_name":"Finset.eraseNone","def_path":"Mathlib/Data/Finset/Option.lean","def_pos":[81,4],"def_end_pos":[81,13]},{"full_name":"Finset.map","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[56,4],"def_end_pos":[56,7]},{"full_name":"Finset.map_some_eraseNone","def_path":"Mathlib/Data/Finset/Option.lean","def_pos":[137,8],"def_end_pos":[137,26]},{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Finset.prod_erase","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1669,8],"def_end_pos":[1669,18]},{"full_name":"Finset.prod_map","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[372,8],"def_end_pos":[372,16]},{"full_name":"Function.Embedding.some","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[192,14],"def_end_pos":[192,18]},{"full_name":"Option.elim'","def_path":"Mathlib/Data/Option/Defs.lean","def_pos":[32,14],"def_end_pos":[32,19]},{"full_name":"Option.none","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2242,4],"def_end_pos":[2242,8]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Content.lean","commit":"","full_name":"MeasureTheory.Content.innerContent_bot","start":[132,0],"end":[139,11],"file_path":"Mathlib/MeasureTheory/Measure/Content.lean","tactics":[{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ.innerContent ⊥ = 0","state_after":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ.innerContent ⊥ ≤ 0","tactic":"refine le_antisymm ?_ (zero_le _)","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"zero_le","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[105,29],"def_end_pos":[105,36]}]},{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ.innerContent ⊥ ≤ 0","state_after":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ.innerContent ⊥ ≤ (fun s => ↑(μ.toFun s)) ⊥","tactic":"rw [← μ.empty]","premises":[{"full_name":"MeasureTheory.Content.empty","def_path":"Mathlib/MeasureTheory/Measure/Content.lean","def_pos":[110,8],"def_end_pos":[110,13]}]},{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ.innerContent ⊥ ≤ (fun s => ↑(μ.toFun s)) ⊥","state_after":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK : Compacts G\nhK : ↑K ⊆ ↑⊥\n⊢ (fun s => ↑(μ.toFun s)) K ≤ (fun s => ↑(μ.toFun s)) ⊥","tactic":"refine iSup₂_le fun K hK => ?_","premises":[{"full_name":"iSup₂_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[667,8],"def_end_pos":[667,16]}]},{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK : Compacts G\nhK : ↑K ⊆ ↑⊥\n⊢ (fun s => ↑(μ.toFun s)) K ≤ (fun s => ↑(μ.toFun s)) ⊥","state_after":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK : Compacts G\nhK : ↑K ⊆ ↑⊥\nthis : K = ⊥\n⊢ (fun s => ↑(μ.toFun s)) K ≤ (fun s => ↑(μ.toFun s)) ⊥","tactic":"have : K = ⊥ := by\n    ext1\n    rw [subset_empty_iff.mp hK, Compacts.coe_bot]","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Set.subset_empty_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[460,8],"def_end_pos":[460,24]},{"full_name":"TopologicalSpace.Compacts.coe_bot","def_path":"Mathlib/Topology/Sets/Compacts.lean","def_pos":[110,8],"def_end_pos":[110,15]}]},{"state_before":"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK : Compacts G\nhK : ↑K ⊆ ↑⊥\nthis : K = ⊥\n⊢ (fun s => ↑(μ.toFun s)) K ≤ (fun s => ↑(μ.toFun s)) ⊥","state_after":"no goals","tactic":"rw [this]","premises":[]}]}
{"url":"Mathlib/Probability/Martingale/Basic.lean","commit":"","full_name":"MeasureTheory.martingale_of_condexp_sub_eq_zero_nat","start":[426,0],"end":[434,55],"file_path":"Mathlib/Probability/Martingale/Basic.lean","tactics":[{"state_before":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\n⊢ Martingale f 𝒢 μ","state_after":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\n⊢ 0 ≤ᶠ[ae μ] μ[f i - f (i + 1)|↑𝒢 i]","tactic":"refine martingale_iff.2 ⟨supermartingale_of_condexp_sub_nonneg_nat hadp hint fun i => ?_,\n    submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => (hf i).symm.le⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Filter.EventuallyEq.le","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1454,8],"def_end_pos":[1454,23]},{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.martingale_iff","def_path":"Mathlib/Probability/Martingale/Basic.lean","def_pos":[129,8],"def_end_pos":[129,22]},{"full_name":"MeasureTheory.submartingale_of_condexp_sub_nonneg_nat","def_path":"Mathlib/Probability/Martingale/Basic.lean","def_pos":[412,8],"def_end_pos":[412,47]},{"full_name":"MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat","def_path":"Mathlib/Probability/Martingale/Basic.lean","def_pos":[419,8],"def_end_pos":[419,49]}]},{"state_before":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\n⊢ 0 ≤ᶠ[ae μ] μ[f i - f (i + 1)|↑𝒢 i]","state_after":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\n⊢ 0 ≤ᶠ[ae μ] μ[-(f (i + 1) - f i)|↑𝒢 i]","tactic":"rw [← neg_sub]","premises":[{"full_name":"neg_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]}]},{"state_before":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\n⊢ 0 ≤ᶠ[ae μ] μ[-(f (i + 1) - f i)|↑𝒢 i]","state_after":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\n⊢ 0 =ᶠ[ae μ] -μ[f (i + 1) - f i|↑𝒢 i]","tactic":"refine (EventuallyEq.trans ?_ (condexp_neg _).symm).le","premises":[{"full_name":"Filter.EventuallyEq.le","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1454,8],"def_end_pos":[1454,23]},{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]},{"full_name":"MeasureTheory.condexp_neg","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[300,8],"def_end_pos":[300,19]}]},{"state_before":"Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\n⊢ 0 =ᶠ[ae μ] -μ[f (i + 1) - f i|↑𝒢 i]","state_after":"case h\nΩ : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\nx : Ω\nhx : (μ[f (i + 1) - f i|↑𝒢 i]) x = 0 x\n⊢ 0 x = (-μ[f (i + 1) - f i|↑𝒢 i]) x","tactic":"filter_upwards [hf i] with x hx","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nΩ : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf✝ g : ι → Ω → E\nℱ : Filtration ι m0\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : Adapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ), μ[f (i + 1) - f i|↑𝒢 i] =ᶠ[ae μ] 0\ni : ℕ\nx : Ω\nhx : (μ[f (i + 1) - f i|↑𝒢 i]) x = 0 x\n⊢ 0 x = (-μ[f (i + 1) - f i|↑𝒢 i]) x","state_after":"no goals","tactic":"simpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg]","premises":[{"full_name":"Pi.neg_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[125,2],"def_end_pos":[125,13]},{"full_name":"Pi.zero_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[59,2],"def_end_pos":[59,13]},{"full_name":"zero_eq_neg","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[462,2],"def_end_pos":[462,13]}]}]}
{"url":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","commit":"","full_name":"PresheafOfModules.Sheafify.zero_smul","start":[226,0],"end":[230,30],"file_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","tactics":[{"state_before":"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\n⊢ smul α φ 0 m = 0","state_after":"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\n⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),\n    (Presheaf.imageSieve φ m).arrows f → (A.val.map f.op) (smul α φ 0 m) = (A.val.map f.op) 0","tactic":"apply A.isSeparated _ _ (Presheaf.imageSieve_mem J φ m)","premises":[{"full_name":"CategoryTheory.Presheaf.imageSieve_mem","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[86,6],"def_end_pos":[86,20]},{"full_name":"CategoryTheory.Sheaf.isSeparated","def_path":"Mathlib/CategoryTheory/Sites/Whiskering.lean","def_pos":[144,6],"def_end_pos":[144,23]}]},{"state_before":"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\n⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),\n    (Presheaf.imageSieve φ m).arrows f → (A.val.map f.op) (smul α φ 0 m) = (A.val.map f.op) 0","state_after":"case a.intro\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y✝ : Cᵒᵖ\nπ : X ⟶ Y✝\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nY : C\nf : Y ⟶ Opposite.unop X\nm₀ : (forget AddCommGrp).obj (M₀.presheaf.obj (Opposite.op Y))\nhm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m\n⊢ (A.val.map f.op) (smul α φ 0 m) = (A.val.map f.op) 0","tactic":"rintro Y f ⟨m₀, hm₀⟩","premises":[]},{"state_before":"case a.intro\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y✝ : Cᵒᵖ\nπ : X ⟶ Y✝\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nY : C\nf : Y ⟶ Opposite.unop X\nm₀ : (forget AddCommGrp).obj (M₀.presheaf.obj (Opposite.op Y))\nhm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m\n⊢ (A.val.map f.op) (smul α φ 0 m) = (A.val.map f.op) 0","state_after":"no goals","tactic":"erw [map_smul_eq α φ 0 m f.op 0 (by simp) m₀ hm₀, zero_smul, map_zero,\n    (A.val.map f.op).map_zero]","premises":[{"full_name":"AddMonoidHom.map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[620,2],"def_end_pos":[620,13]},{"full_name":"CategoryTheory.Sheaf.val","def_path":"Mathlib/CategoryTheory/Sites/Sheaf.lean","def_pos":[307,2],"def_end_pos":[307,5]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]},{"full_name":"PresheafOfModules.Sheafify.map_smul_eq","def_path":"Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean","def_pos":[214,6],"def_end_pos":[214,17]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]}]}
{"url":"Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean","commit":"","full_name":"Behrend.roth_lower_bound","start":[493,0],"end":[499,41],"file_path":"Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nn d k N : ℕ\nx : Fin n → ℕ\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑(rothNumberNat N)","state_after":"case inl\nα : Type u_1\nβ : Type u_2\nn d k : ℕ\nx : Fin n → ℕ\n⊢ ↑0 * rexp (-4 * √(log ↑0)) ≤ ↑(rothNumberNat 0)\n\ncase inr\nα : Type u_1\nβ : Type u_2\nn d k N : ℕ\nx : Fin n → ℕ\nhN : N > 0\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑(rothNumberNat N)","tactic":"obtain rfl | hN := Nat.eq_zero_or_pos N","premises":[{"full_name":"Nat.eq_zero_or_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[350,8],"def_end_pos":[350,22]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\nn d k N : ℕ\nx : Fin n → ℕ\nhN : N > 0\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑(rothNumberNat N)","state_after":"case inr.inl\nα : Type u_1\nβ : Type u_2\nn d k N : ℕ\nx : Fin n → ℕ\nhN : N > 0\nh₁ : 4096 ≤ N\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑(rothNumberNat N)\n\ncase inr.inr\nα : Type u_1\nβ : Type u_2\nn d k N : ℕ\nx : Fin n → ℕ\nhN : N > 0\nh₁ : N < 4096\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑(rothNumberNat N)","tactic":"obtain h₁ | h₁ := le_or_lt 4096 N","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.erase_append_right","start":[610,0],"end":[613,54],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nh : ¬a ∈ l₁\n⊢ (l₁ ++ l₂).erase a = l₁ ++ l₂.erase a","state_after":"case a\nα : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nh : ¬a ∈ l₁\n⊢ ∀ (b : α), b ∈ l₁ → ¬(a == b) = true","tactic":"rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right]","premises":[{"full_name":"List.eraseP_append_right","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[533,8],"def_end_pos":[533,27]},{"full_name":"List.erase_eq_eraseP","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[592,8],"def_end_pos":[592,23]}]},{"state_before":"case a\nα : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nh : ¬a ∈ l₁\n⊢ ∀ (b : α), b ∈ l₁ → ¬(a == b) = true","state_after":"case a\nα : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nh : ¬a ∈ l₁\nb : α\nh' : b ∈ l₁\nh'' : (a == b) = true\n⊢ False","tactic":"intros b h' h''","premises":[]},{"state_before":"case a\nα : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nh : ¬a ∈ l₁\nb : α\nh' : b ∈ l₁\nh'' : (a == b) = true\n⊢ False","state_after":"case a\nα : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nb : α\nh : ¬b ∈ l₁\nh' : b ∈ l₁\nh'' : (a == b) = true\n⊢ False","tactic":"rw [eq_of_beq h''] at h","premises":[{"full_name":"LawfulBEq.eq_of_beq","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[625,2],"def_end_pos":[625,11]}]},{"state_before":"case a\nα : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\nl₁ l₂ : List α\nb : α\nh : ¬b ∈ l₁\nh' : b ∈ l₁\nh'' : (a == b) = true\n⊢ False","state_after":"no goals","tactic":"exact h h'","premises":[]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"AddSubgroup.comap_id","start":[1082,0],"end":[1085,5],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nK : Subgroup N\n⊢ comap (MonoidHom.id N) K = K","state_after":"case h\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nK : Subgroup N\nx✝ : N\n⊢ x✝ ∈ comap (MonoidHom.id N) K ↔ x✝ ∈ K","tactic":"ext","premises":[]},{"state_before":"case h\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type u_5\ninst✝¹ : Group N\nP : Type u_6\ninst✝ : Group P\nK : Subgroup N\nx✝ : N\n⊢ x✝ ∈ comap (MonoidHom.id N) K ↔ x✝ ∈ K","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/ModelTheory/Encoding.lean","commit":"","full_name":"FirstOrder.Language.Term.encoding_decode","start":[86,0],"end":[96,12],"file_path":"Mathlib/ModelTheory/Encoding.lean","tactics":[{"state_before":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nt : L.Term α\n⊢ (fun l =>\n        (do\n            let a ← (listDecode l).head?\n            pure (some a)).join)\n      t.listEncode =\n    some t","state_after":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nt : L.Term α\nh : listDecode ([t].bind listEncode) = [t]\n⊢ (fun l =>\n        (do\n            let a ← (listDecode l).head?\n            pure (some a)).join)\n      t.listEncode =\n    some t","tactic":"have h := listDecode_encode_list [t]","premises":[{"full_name":"FirstOrder.Language.Term.listDecode_encode_list","def_path":"Mathlib/ModelTheory/Encoding.lean","def_pos":[63,8],"def_end_pos":[63,30]},{"full_name":"List.cons","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2289,4],"def_end_pos":[2289,8]},{"full_name":"List.nil","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2286,4],"def_end_pos":[2286,7]}]},{"state_before":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nt : L.Term α\nh : listDecode ([t].bind listEncode) = [t]\n⊢ (fun l =>\n        (do\n            let a ← (listDecode l).head?\n            pure (some a)).join)\n      t.listEncode =\n    some t","state_after":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nt : L.Term α\nh : listDecode t.listEncode = [t]\n⊢ (fun l =>\n        (do\n            let a ← (listDecode l).head?\n            pure (some a)).join)\n      t.listEncode =\n    some t","tactic":"rw [bind_singleton] at h","premises":[{"full_name":"List.bind_singleton","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1263,8],"def_end_pos":[1263,22]}]},{"state_before":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nt : L.Term α\nh : listDecode t.listEncode = [t]\n⊢ (fun l =>\n        (do\n            let a ← (listDecode l).head?\n            pure (some a)).join)\n      t.listEncode =\n    some t","state_after":"no goals","tactic":"simp only [Option.join, h, head?_cons, Option.pure_def, Option.bind_eq_bind, Option.some_bind,\n      id_eq]","premises":[{"full_name":"List.head?_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[301,16],"def_end_pos":[301,26]},{"full_name":"Option.bind_eq_bind","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean","def_pos":[97,16],"def_end_pos":[97,28]},{"full_name":"Option.join","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[167,20],"def_end_pos":[167,24]},{"full_name":"Option.pure_def","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean","def_pos":[95,16],"def_end_pos":[95,24]},{"full_name":"Option.some_bind","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean","def_pos":[118,16],"def_end_pos":[118,25]},{"full_name":"id_eq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[297,16],"def_end_pos":[297,21]}]}]}
{"url":"Mathlib/LinearAlgebra/Basis.lean","commit":"","full_name":"Basis.ext_elem_iff","start":[254,0],"end":[256,63],"file_path":"Mathlib/LinearAlgebra/Basis.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_10\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_11\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nx y : M\n⊢ x = y ↔ ∀ (i : ι), (b.repr x) i = (b.repr y) i","state_after":"no goals","tactic":"simp only [← DFunLike.ext_iff, EmbeddingLike.apply_eq_iff_eq]","premises":[{"full_name":"DFunLike.ext_iff","def_path":"Mathlib/Data/FunLike/Basic.lean","def_pos":[196,8],"def_end_pos":[196,15]},{"full_name":"EmbeddingLike.apply_eq_iff_eq","def_path":"Mathlib/Data/FunLike/Embedding.lean","def_pos":[143,8],"def_end_pos":[143,23]}]}]}
{"url":"Mathlib/Algebra/GeomSum.lean","commit":"","full_name":"geom_sum_Ico","start":[320,0],"end":[322,94],"file_path":"Mathlib/Algebra/GeomSum.lean","tactics":[{"state_before":"α : Type u\ninst✝ : DivisionRing α\nx : α\nhx : x ≠ 1\nm n : ℕ\nhmn : m ≤ n\n⊢ ∑ i ∈ Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1)","state_after":"no goals","tactic":"simp only [sum_Ico_eq_sub _ hmn, geom_sum_eq hx, div_sub_div_same, sub_sub_sub_cancel_right]","premises":[{"full_name":"Finset.sum_Ico_eq_sub","def_path":"Mathlib/Algebra/BigOperators/Intervals.lean","def_pos":[146,2],"def_end_pos":[146,13]},{"full_name":"div_sub_div_same","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[128,8],"def_end_pos":[128,24]},{"full_name":"geom_sum_eq","def_path":"Mathlib/Algebra/GeomSum.lean","def_pos":[242,8],"def_end_pos":[242,19]},{"full_name":"sub_sub_sub_cancel_right","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[734,29],"def_end_pos":[734,53]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/Presheaf.lean","commit":"","full_name":"CategoryTheory.OverPresheafAux.MakesOverArrow.of_yoneda_arrow","start":[105,0],"end":[107,57],"file_path":"Mathlib/CategoryTheory/Comma/Presheaf.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nY : C\nη : yoneda.obj Y ⟶ A\nX : C\ns : yoneda.obj X ⟶ A\nf : X ⟶ Y\nhf : yoneda.map f ≫ η = s\n⊢ MakesOverArrow η s f","state_after":"no goals","tactic":"simpa only [yonedaEquiv_yoneda_map f] using of_arrow hf","premises":[{"full_name":"CategoryTheory.OverPresheafAux.MakesOverArrow.of_arrow","def_path":"Mathlib/CategoryTheory/Comma/Presheaf.lean","def_pos":[101,6],"def_end_pos":[101,14]},{"full_name":"CategoryTheory.yonedaEquiv_yoneda_map","def_path":"Mathlib/CategoryTheory/Yoneda.lean","def_pos":[304,6],"def_end_pos":[304,28]}]}]}
{"url":"Mathlib/Order/SuccPred/LinearLocallyFinite.lean","commit":"","full_name":"LinearLocallyFiniteOrder.isGLB_Ioc_of_isGLB_Ioi","start":[73,0],"end":[80,50],"file_path":"Mathlib/Order/SuccPred/LinearLocallyFinite.lean","tactics":[{"state_before":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : IsGLB (Set.Ioi i) k\n⊢ IsGLB (Set.Ioc i j) k","state_after":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\n⊢ (∀ x ∈ Set.Ioc i j, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioc i j, x ≤ x_1) → x ≤ k","tactic":"simp_rw [IsGLB, IsGreatest, mem_upperBounds, mem_lowerBounds] at h ⊢","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"IsGLB","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[72,4],"def_end_pos":[72,9]},{"full_name":"IsGreatest","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[64,4],"def_end_pos":[64,14]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"mem_lowerBounds","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[78,8],"def_end_pos":[78,23]},{"full_name":"mem_upperBounds","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[75,8],"def_end_pos":[75,23]}]},{"state_before":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\n⊢ (∀ x ∈ Set.Ioc i j, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioc i j, x ≤ x_1) → x ≤ k","state_after":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\nx : ι\nhx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1\n⊢ ∀ x_1 ∈ Set.Ioi i, x ≤ x_1","tactic":"refine ⟨fun x hx ↦ h.1 x hx.1, fun x hx ↦ h.2 x ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]},{"state_before":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\nx : ι\nhx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1\n⊢ ∀ x_1 ∈ Set.Ioi i, x ≤ x_1","state_after":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\nx : ι\nhx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1\ny : ι\nhy : y ∈ Set.Ioi i\n⊢ x ≤ y","tactic":"intro y hy","premises":[]},{"state_before":"ι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\nx : ι\nhx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1\ny : ι\nhy : y ∈ Set.Ioi i\n⊢ x ≤ y","state_after":"case inl\nι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\nx : ι\nhx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1\ny : ι\nhy : y ∈ Set.Ioi i\nh_le : y ≤ j\n⊢ x ≤ y\n\ncase inr\nι : Type u_1\ninst✝ : LinearOrder ι\ni j k : ι\nhij_lt : i < j\nh : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k\nx : ι\nhx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1\ny : ι\nhy : y ∈ Set.Ioi i\nh_lt : j < y\n⊢ x ≤ y","tactic":"rcases le_or_lt y j with h_le | h_lt","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]}]}
{"url":"Mathlib/Topology/Order/DenselyOrdered.lean","commit":"","full_name":"tendsto_comp_coe_Ioo_atTop","start":[290,0],"end":[293,51],"file_path":"Mathlib/Topology/Order/DenselyOrdered.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf : α → β\nh : a < b\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f (𝓝[<] b) l","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf : α → β\nh : a < b\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto (f ∘ Subtype.val) atTop l","tactic":"rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]","premises":[{"full_name":"Filter.tendsto_map'_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2676,8],"def_end_pos":[2676,24]},{"full_name":"map_coe_Ioo_atTop","def_path":"Mathlib/Topology/Order/DenselyOrdered.lean","def_pos":[273,8],"def_end_pos":[273,25]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf : α → β\nh : a < b\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto (f ∘ Subtype.val) atTop l","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Topology/ShrinkingLemma.lean","commit":"","full_name":"exists_subset_iUnion_closure_subset","start":[197,0],"end":[213,29],"file_path":"Mathlib/Topology/ShrinkingLemma.lean","tactics":[{"state_before":"ι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i","state_after":"ι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis : Nonempty (PartialRefinement u s)\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i","tactic":"haveI : Nonempty (PartialRefinement u s) := ⟨⟨u, ∅, uo, us, False.elim, fun _ => rfl⟩⟩","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"False.elim","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[236,20],"def_end_pos":[236,30]},{"full_name":"Nonempty","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[709,16],"def_end_pos":[709,24]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]},{"full_name":"ShrinkingLemma.PartialRefinement","def_path":"Mathlib/Topology/ShrinkingLemma.lean","def_pos":[53,17],"def_end_pos":[53,34]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"ι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i","state_after":"case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i","tactic":"rcases zorn_nonempty_partialOrder this with ⟨v, hv⟩","premises":[{"full_name":"zorn_nonempty_partialOrder","def_path":"Mathlib/Order/Zorn.lean","def_pos":[154,8],"def_end_pos":[154,34]}]},{"state_before":"case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i","state_after":"case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier","tactic":"suffices ∀ i, i ∈ v.carrier from\n    ⟨v, v.subset_iUnion, fun i => v.isOpen _, fun i => v.closure_subset (this i)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ShrinkingLemma.PartialRefinement.carrier","def_path":"Mathlib/Topology/ShrinkingLemma.lean","def_pos":[57,2],"def_end_pos":[57,9]},{"full_name":"ShrinkingLemma.PartialRefinement.closure_subset","def_path":"Mathlib/Topology/ShrinkingLemma.lean","def_pos":[63,2],"def_end_pos":[63,16]},{"full_name":"ShrinkingLemma.PartialRefinement.isOpen","def_path":"Mathlib/Topology/ShrinkingLemma.lean","def_pos":[59,12],"def_end_pos":[59,18]},{"full_name":"ShrinkingLemma.PartialRefinement.subset_iUnion","def_path":"Mathlib/Topology/ShrinkingLemma.lean","def_pos":[61,2],"def_end_pos":[61,15]}]},{"state_before":"case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier","state_after":"case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∃ i, i ∉ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v","tactic":"contrapose! hv","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∃ i, i ∉ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v","state_after":"case intro.intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : i ∉ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v","tactic":"rcases hv with ⟨i, hi⟩","premises":[]},{"state_before":"case intro.intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : i ∉ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v","state_after":"case intro.intro.intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : i ∉ v.carrier\nv' : PartialRefinement u s\nhlt : v < v'\n⊢ ∃ a, v ≤ a ∧ a ≠ v","tactic":"rcases v.exists_gt hs i hi with ⟨v', hlt⟩","premises":[{"full_name":"ShrinkingLemma.PartialRefinement.exists_gt","def_path":"Mathlib/Topology/ShrinkingLemma.lean","def_pos":[157,8],"def_end_pos":[157,17]}]},{"state_before":"case intro.intro.intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : i ∉ v.carrier\nv' : PartialRefinement u s\nhlt : v < v'\n⊢ ∃ a, v ≤ a ∧ a ≠ v","state_after":"no goals","tactic":"exact ⟨v', hlt.le, hlt.ne'⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]}]}]}
{"url":"Mathlib/Order/SymmDiff.lean","commit":"","full_name":"symmDiff_triangle","start":[181,0],"end":[183,61],"file_path":"Mathlib/Order/SymmDiff.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a ∆ c ≤ a ∆ b ⊔ b ∆ c","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a \\ b ⊔ b \\ c ⊔ (c \\ b ⊔ b \\ a) = a ∆ b ⊔ b ∆ c","tactic":"refine (sup_le_sup (sdiff_triangle a b c) <| sdiff_triangle _ b _).trans_eq ?_","premises":[{"full_name":"sdiff_triangle","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[547,8],"def_end_pos":[547,22]},{"full_name":"sup_le_sup","def_path":"Mathlib/Order/Lattice.lean","def_pos":[178,8],"def_end_pos":[178,18]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a \\ b ⊔ b \\ c ⊔ (c \\ b ⊔ b \\ a) = a ∆ b ⊔ b ∆ c","state_after":"no goals","tactic":"rw [sup_comm (c \\ b), sup_sup_sup_comm, symmDiff, symmDiff]","premises":[{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]},{"full_name":"sup_sup_sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[215,8],"def_end_pos":[215,24]},{"full_name":"symmDiff","def_path":"Mathlib/Order/SymmDiff.lean","def_pos":[58,4],"def_end_pos":[58,12]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/Basic.lean","commit":"","full_name":"CategoryTheory.Abelian.comp_coimage_π_eq_zero","start":[329,0],"end":[330,64],"file_path":"Mathlib/CategoryTheory/Abelian/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nR : C\ng : Q ⟶ R\nh : f ≫ g = 0\n⊢ (f ≫ coimage.π g) ≫ Abelian.factorThruCoimage g = 0","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","commit":"","full_name":"MeasureTheory.condexpIndL1Fin_smul'","start":[109,0],"end":[118,31],"file_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x","state_after":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(condexpIndL1Fin hm hs hμs (c • x)) =ᶠ[ae μ] ↑↑(c • condexpIndL1Fin hm hs hμs x)","tactic":"ext1","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(condexpIndL1Fin hm hs hμs (c • x)) =ᶠ[ae μ] ↑↑(c • condexpIndL1Fin hm hs hμs x)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(condexpIndSMul hm hs hμs (c • x)) =ᶠ[ae μ] ↑↑(c • condexpIndL1Fin hm hs hμs x)","tactic":"refine (Memℒp.coeFn_toLp q).trans ?_","premises":[{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]},{"full_name":"MeasureTheory.Memℒp.coeFn_toLp","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[125,8],"def_end_pos":[125,18]},{"full_name":"_private.Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.0.MeasureTheory.q","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","def_pos":[83,16],"def_end_pos":[83,17]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(condexpIndSMul hm hs hμs (c • x)) =ᶠ[ae μ] ↑↑(c • condexpIndL1Fin hm hs hμs x)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(condexpIndSMul hm hs hμs (c • x)) =ᶠ[ae μ] c • ↑↑(condexpIndL1Fin hm hs hμs x)","tactic":"refine EventuallyEq.trans ?_ (Lp.coeFn_smul _ _).symm","premises":[{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]},{"full_name":"MeasureTheory.Lp.coeFn_smul","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[451,8],"def_end_pos":[451,18]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(condexpIndSMul hm hs hμs (c • x)) =ᶠ[ae μ] c • ↑↑(condexpIndL1Fin hm hs hμs x)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(c • condexpIndSMul hm hs hμs x) =ᶠ[ae μ] c • ↑↑(condexpIndL1Fin hm hs hμs x)","tactic":"rw [condexpIndSMul_smul' hs hμs c x]","premises":[{"full_name":"MeasureTheory.condexpIndSMul_smul'","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean","def_pos":[388,8],"def_end_pos":[388,28]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ ↑↑(c • condexpIndSMul hm hs hμs x) =ᶠ[ae μ] c • ↑↑(condexpIndL1Fin hm hs hμs x)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ c • ↑↑(condexpIndSMul hm hs hμs x) =ᶠ[ae μ] c • ↑↑(condexpIndL1Fin hm hs hμs x)","tactic":"refine (Lp.coeFn_smul _ _).trans ?_","premises":[{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]},{"full_name":"MeasureTheory.Lp.coeFn_smul","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[451,8],"def_end_pos":[451,18]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\n⊢ c • ↑↑(condexpIndSMul hm hs hμs x) =ᶠ[ae μ] c • ↑↑(condexpIndL1Fin hm hs hμs x)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\ny : α\nhy : ↑↑(condexpIndL1Fin hm hs hμs x) y = ↑↑(condexpIndSMul hm hs hμs x) y\n⊢ (c • ↑↑(condexpIndSMul hm hs hμs x)) y = (c • ↑↑(condexpIndL1Fin hm hs hμs x)) y","tactic":"refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun y hy => ?_","premises":[{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean","def_pos":[74,8],"def_end_pos":[74,44]}]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n𝕜 : Type u_7\np : ℝ≥0∞\ninst✝¹⁴ : RCLike 𝕜\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\ninst✝¹¹ : NormedAddCommGroup F'\ninst✝¹⁰ : NormedSpace 𝕜 F'\ninst✝⁹ : NormedSpace ℝ F'\ninst✝⁸ : CompleteSpace F'\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace ℝ G'\ninst✝⁴ : CompleteSpace G'\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nhs : MeasurableSet s\nhμs : μ s ≠ ⊤\nc : 𝕜\nx : F\ny : α\nhy : ↑↑(condexpIndL1Fin hm hs hμs x) y = ↑↑(condexpIndSMul hm hs hμs x) y\n⊢ (c • ↑↑(condexpIndSMul hm hs hμs x)) y = (c • ↑↑(condexpIndL1Fin hm hs hμs x)) y","state_after":"no goals","tactic":"simp only [Pi.smul_apply, hy]","premises":[{"full_name":"Pi.smul_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[104,59],"def_end_pos":[104,69]}]}]}
{"url":"Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean","commit":"","full_name":"CategoryTheory.Functor.hasRightDerivedFunctor_iff","start":[160,0],"end":[164,84],"file_path":"Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean","tactics":[{"state_before":"C : Type u_1\nC' : Type ?u.80223\nD : Type u_5\nD' : Type ?u.80229\nH : Type u_2\nH' : Type ?u.80235\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{?u.80243, ?u.80223} C'\ninst✝⁴ : Category.{u_6, u_5} D\ninst✝³ : Category.{?u.80251, ?u.80229} D'\ninst✝² : Category.{u_4, u_2} H\ninst✝¹ : Category.{?u.80259, ?u.80235} H'\nRF RF' RF'' : D ⥤ H\nF F' F'' : C ⥤ H\ne : F ≅ F'\nL : C ⥤ D\nα : F ⟶ L ⋙ RF\nα' : F' ⟶ L ⋙ RF'\nα'' : F'' ⟶ L ⋙ RF''\nα'₂ : F ⟶ L ⋙ RF'\nW : MorphismProperty C\ninst✝ : L.IsLocalization W\n⊢ F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F","state_after":"C : Type u_1\nC' : Type ?u.80223\nD : Type u_5\nD' : Type ?u.80229\nH : Type u_2\nH' : Type ?u.80235\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{?u.80243, ?u.80223} C'\ninst✝⁴ : Category.{u_6, u_5} D\ninst✝³ : Category.{?u.80251, ?u.80229} D'\ninst✝² : Category.{u_4, u_2} H\ninst✝¹ : Category.{?u.80259, ?u.80235} H'\nRF RF' RF'' : D ⥤ H\nF F' F'' : C ⥤ H\ne : F ≅ F'\nL : C ⥤ D\nα : F ⟶ L ⋙ RF\nα' : F' ⟶ L ⋙ RF'\nα'' : F'' ⟶ L ⋙ RF''\nα'₂ : F ⟶ L ⋙ RF'\nW : MorphismProperty C\ninst✝ : L.IsLocalization W\nthis : F.HasRightDerivedFunctor W ↔ W.Q.HasLeftKanExtension F\n⊢ F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F","tactic":"have : HasRightDerivedFunctor F W ↔ HasLeftKanExtension W.Q F :=\n    ⟨fun h => h.hasLeftKanExtension', fun h => ⟨h⟩⟩","premises":[{"full_name":"CategoryTheory.Functor.HasLeftKanExtension","def_path":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","def_pos":[257,7],"def_end_pos":[257,26]},{"full_name":"CategoryTheory.Functor.HasRightDerivedFunctor","def_path":"Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean","def_pos":[154,6],"def_end_pos":[154,28]},{"full_name":"CategoryTheory.Functor.HasRightDerivedFunctor.hasLeftKanExtension'","def_path":"Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean","def_pos":[155,2],"def_end_pos":[155,22]},{"full_name":"CategoryTheory.MorphismProperty.Q","def_path":"Mathlib/CategoryTheory/Localization/Construction.lean","def_pos":[104,4],"def_end_pos":[104,5]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"C : Type u_1\nC' : Type ?u.80223\nD : Type u_5\nD' : Type ?u.80229\nH : Type u_2\nH' : Type ?u.80235\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{?u.80243, ?u.80223} C'\ninst✝⁴ : Category.{u_6, u_5} D\ninst✝³ : Category.{?u.80251, ?u.80229} D'\ninst✝² : Category.{u_4, u_2} H\ninst✝¹ : Category.{?u.80259, ?u.80235} H'\nRF RF' RF'' : D ⥤ H\nF F' F'' : C ⥤ H\ne : F ≅ F'\nL : C ⥤ D\nα : F ⟶ L ⋙ RF\nα' : F' ⟶ L ⋙ RF'\nα'' : F'' ⟶ L ⋙ RF''\nα'₂ : F ⟶ L ⋙ RF'\nW : MorphismProperty C\ninst✝ : L.IsLocalization W\nthis : F.HasRightDerivedFunctor W ↔ W.Q.HasLeftKanExtension F\n⊢ F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F","state_after":"no goals","tactic":"rw [this, hasLeftExtension_iff_postcomp₁ (Localization.compUniqFunctor W.Q L W) F]","premises":[{"full_name":"CategoryTheory.Functor.hasLeftExtension_iff_postcomp₁","def_path":"Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean","def_pos":[341,6],"def_end_pos":[341,36]},{"full_name":"CategoryTheory.Localization.compUniqFunctor","def_path":"Mathlib/CategoryTheory/Localization/Predicate.lean","def_pos":[439,4],"def_end_pos":[439,19]},{"full_name":"CategoryTheory.MorphismProperty.Q","def_path":"Mathlib/CategoryTheory/Localization/Construction.lean","def_pos":[104,4],"def_end_pos":[104,5]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Basic.lean","commit":"","full_name":"Finset.Icc_subset_Ici_self","start":[335,0],"end":[336,52],"file_path":"Mathlib/Order/Interval/Finset/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : LocallyFiniteOrderTop α\n⊢ Icc a b ⊆ Ici a","state_after":"no goals","tactic":"simpa [← coe_subset] using Set.Icc_subset_Ici_self","premises":[{"full_name":"Finset.coe_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[336,8],"def_end_pos":[336,18]},{"full_name":"Set.Icc_subset_Ici_self","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[391,8],"def_end_pos":[391,27]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/ZPow.lean","commit":"","full_name":"iter_deriv_inv","start":[123,0],"end":[125,86],"file_path":"Mathlib/Analysis/Calculus/Deriv/ZPow.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm : ℤ\nk : ℕ\nx : 𝕜\n⊢ deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - ↑i)) * x ^ (-1 - ↑k)","state_after":"no goals","tactic":"simpa only [zpow_neg_one, Int.cast_neg, Int.cast_one] using iter_deriv_zpow (-1) x k","premises":[{"full_name":"Int.cast_neg","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[75,8],"def_end_pos":[75,16]},{"full_name":"Int.cast_one","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[70,8],"def_end_pos":[70,16]},{"full_name":"iter_deriv_zpow","def_path":"Mathlib/Analysis/Calculus/Deriv/ZPow.lean","def_pos":[104,8],"def_end_pos":[104,23]},{"full_name":"zpow_neg_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[918,6],"def_end_pos":[918,18]}]}]}
{"url":"Mathlib/Data/W/Constructions.lean","commit":"","full_name":"WType.leftInverse_nat","start":[59,0],"end":[67,7],"file_path":"Mathlib/Data/W/Constructions.lean","tactics":[{"state_before":"f : Natβ Natα.zero → WType Natβ\n⊢ ofNat (mk Natα.zero f).toNat = mk Natα.zero f","state_after":"f : Natβ Natα.zero → WType Natβ\n⊢ mk Natα.zero Empty.elim = mk Natα.zero f","tactic":"rw [toNat, ofNat]","premises":[{"full_name":"WType.ofNat","def_path":"Mathlib/Data/W/Constructions.lean","def_pos":[49,4],"def_end_pos":[49,9]},{"full_name":"WType.toNat","def_path":"Mathlib/Data/W/Constructions.lean","def_pos":[55,4],"def_end_pos":[55,9]}]},{"state_before":"f : Natβ Natα.zero → WType Natβ\n⊢ mk Natα.zero Empty.elim = mk Natα.zero f","state_after":"case e_f\nf : Natβ Natα.zero → WType Natβ\n⊢ Empty.elim = f","tactic":"congr","premises":[]},{"state_before":"case e_f\nf : Natβ Natα.zero → WType Natβ\n⊢ Empty.elim = f","state_after":"case e_f.h\nf : Natβ Natα.zero → WType Natβ\nx : Empty\n⊢ x.elim = f x","tactic":"ext x","premises":[]},{"state_before":"case e_f.h\nf : Natβ Natα.zero → WType Natβ\nx : Empty\n⊢ x.elim = f x","state_after":"no goals","tactic":"cases x","premises":[]},{"state_before":"f : Natβ Natα.succ → WType Natβ\n⊢ ofNat (mk Natα.succ f).toNat = mk Natα.succ f","state_after":"f : Natβ Natα.succ → WType Natβ\n⊢ (fun x => f ()) = f","tactic":"simp only [toNat, ofNat, leftInverse_nat (f ()), mk.injEq, heq_eq_eq, true_and]","premises":[{"full_name":"Unit.unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[148,24],"def_end_pos":[148,33]},{"full_name":"WType.ofNat","def_path":"Mathlib/Data/W/Constructions.lean","def_pos":[49,4],"def_end_pos":[49,9]},{"full_name":"WType.toNat","def_path":"Mathlib/Data/W/Constructions.lean","def_pos":[55,4],"def_end_pos":[55,9]},{"full_name":"heq_eq_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[251,16],"def_end_pos":[251,25]},{"full_name":"true_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[105,16],"def_end_pos":[105,24]}]},{"state_before":"f : Natβ Natα.succ → WType Natβ\n⊢ (fun x => f ()) = f","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/IsLocalHomeomorph.lean","commit":"","full_name":"isLocalHomeomorph_iff_openEmbedding_restrict","start":[144,0],"end":[147,83],"file_path":"Mathlib/Topology/IsLocalHomeomorph.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ng : Y → Z\nf✝ : X → Y\ns : Set X\nt : Set Y\nf : X → Y\n⊢ IsLocalHomeomorph f ↔ ∀ (x : X), ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f)","state_after":"no goals","tactic":"simp_rw [isLocalHomeomorph_iff_isLocalHomeomorphOn_univ,\n    isLocalHomeomorphOn_iff_openEmbedding_restrict, imp_iff_right (Set.mem_univ _)]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"imp_iff_right","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1409,8],"def_end_pos":[1409,21]},{"full_name":"isLocalHomeomorphOn_iff_openEmbedding_restrict","def_path":"Mathlib/Topology/IsLocalHomeomorph.lean","def_pos":[42,8],"def_end_pos":[42,54]},{"full_name":"isLocalHomeomorph_iff_isLocalHomeomorphOn_univ","def_path":"Mathlib/Topology/IsLocalHomeomorph.lean","def_pos":[137,8],"def_end_pos":[137,54]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Basic.lean","commit":"","full_name":"InnerProductSpace.Core.inner_smul_right","start":[223,0],"end":[225,57],"file_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ ⟪x, r • y⟫_𝕜 = r * ⟪x, y⟫_𝕜","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ (starRingEnd 𝕜) ((starRingEnd 𝕜) r * ⟪y, x⟫_𝕜) = r * ⟪x, y⟫_𝕜","tactic":"rw [← inner_conj_symm, inner_smul_left]","premises":[{"full_name":"InnerProductSpace.Core.inner_conj_symm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[196,8],"def_end_pos":[196,23]},{"full_name":"InnerProductSpace.Core.inner_smul_left","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[220,8],"def_end_pos":[220,23]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ (starRingEnd 𝕜) ((starRingEnd 𝕜) r * ⟪y, x⟫_𝕜) = r * ⟪x, y⟫_𝕜","state_after":"no goals","tactic":"simp only [conj_conj, inner_conj_symm, RingHom.map_mul]","premises":[{"full_name":"InnerProductSpace.Core.inner_conj_symm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[196,8],"def_end_pos":[196,23]},{"full_name":"RingHom.map_mul","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[484,18],"def_end_pos":[484,25]}]}]}
{"url":"Mathlib/ModelTheory/Semantics.lean","commit":"","full_name":"FirstOrder.Language.BoundedFormula.realize_mapTermRel_id","start":[320,0],"end":[334,43],"file_path":"Mathlib/ModelTheory/Semantics.lean","tactics":[{"state_before":"L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn✝ l : ℕ\nφ✝ ψ : L.BoundedFormula α l\nθ : L.BoundedFormula α l.succ\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : L'.Structure M\nft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)\nfr : (n : ℕ) → L.Relations n → L'.Relations n\nn : ℕ\nφ : L.BoundedFormula α n\nv : α → M\nv' : β → M\nxs : Fin n → M\nh1 : ∀ (n : ℕ) (t : L.Term (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\n⊢ (mapTermRel ft fr (fun x => id) φ).Realize v' xs ↔ φ.Realize v xs","state_after":"case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn✝¹ l : ℕ\nφ ψ : L.BoundedFormula α l\nθ : L.BoundedFormula α l.succ\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : L'.Structure M\nft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)\nfr : (n : ℕ) → L.Relations n → L'.Relations n\nn : ℕ\nv : α → M\nv' : β → M\nh1 : ∀ (n : ℕ) (t : L.Term (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nxs : Fin n✝ → M\n⊢ (mapTermRel ft fr (fun x => id) falsum).Realize v' xs ↔ falsum.Realize v xs\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn✝¹ l : ℕ\nφ ψ : L.BoundedFormula α l\nθ : L.BoundedFormula α l.succ\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : L'.Structure M\nft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)\nfr : (n : ℕ) → L.Relations n → L'.Relations n\nn : ℕ\nv : α → M\nv' : β → M\nh1 : ∀ (n : ℕ) (t : L.Term (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nxs : Fin n✝ → M\n⊢ (mapTermRel ft fr (fun x => id) (equal t₁✝ t₂✝)).Realize v' xs ↔ (equal t₁✝ t₂✝).Realize v xs\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn✝¹ l : ℕ\nφ ψ : L.BoundedFormula α l\nθ : L.BoundedFormula α l.succ\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : L'.Structure M\nft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)\nfr : (n : ℕ) → L.Relations n → L'.Relations n\nn : ℕ\nv : α → M\nv' : β → M\nh1 : ∀ (n : ℕ) (t : L.Term (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\nxs : Fin n✝ → M\n⊢ (mapTermRel ft fr (fun x => id) (rel R✝ ts✝)).Realize v' xs ↔ (rel R✝ ts✝).Realize v xs\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn✝¹ l : ℕ\nφ ψ : L.BoundedFormula α l\nθ : L.BoundedFormula α l.succ\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : L'.Structure M\nft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)\nfr : (n : ℕ) → L.Relations n → L'.Relations n\nn : ℕ\nv : α → M\nv' : β → M\nh1 : ∀ (n : ℕ) (t : L.Term (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nf₁✝ f₂✝ : L.BoundedFormula α n✝\nih1 : ∀ {xs : Fin n✝ → M}, (mapTermRel ft fr (fun x => id) f₁✝).Realize v' xs ↔ f₁✝.Realize v xs\nih2 : ∀ {xs : Fin n✝ → M}, (mapTermRel ft fr (fun x => id) f₂✝).Realize v' xs ↔ f₂✝.Realize v xs\nxs : Fin n✝ → M\n⊢ (mapTermRel ft fr (fun x => id) (f₁✝ ⟹ f₂✝)).Realize v' xs ↔ (f₁✝ ⟹ f₂✝).Realize v xs\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn✝¹ l : ℕ\nφ ψ : L.BoundedFormula α l\nθ : L.BoundedFormula α l.succ\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : L'.Structure M\nft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)\nfr : (n : ℕ) → L.Relations n → L'.Relations n\nn : ℕ\nv : α → M\nv' : β → M\nh1 : ∀ (n : ℕ) (t : L.Term (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nf✝ : L.BoundedFormula α (n✝ + 1)\nih : ∀ {xs : Fin (n✝ + 1) → M}, (mapTermRel ft fr (fun x => id) f✝).Realize v' xs ↔ f✝.Realize v xs\nxs : Fin n✝ → M\n⊢ (mapTermRel ft fr (fun x => id) (∀'f✝)).Realize v' xs ↔ (∀'f✝).Realize v xs","tactic":"induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih","premises":[]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.Bounded.weak_left","start":[803,0],"end":[805,54],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Preorder α\no₁ : WithBot α\no₂ : WithTop α\nh : nil.Bounded o₁ o₂\n⊢ nil.Bounded ⊥ o₂","state_after":"no goals","tactic":"cases o₂ <;> trivial","premises":[{"full_name":"True.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[192,4],"def_end_pos":[192,9]}]}]}
{"url":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","commit":"","full_name":"div_le_self_iff","start":[228,0],"end":[230,23],"file_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u\ninst✝³ : Group α\ninst✝² : LE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c d a b : α\n⊢ a / b ≤ a ↔ 1 ≤ b","state_after":"no goals","tactic":"simp [div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.equation_of_equiv","start":[269,0],"end":[271,32],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → R\nh : P ≈ Q\n⊢ W'.Equation P ↔ W'.Equation Q","state_after":"case intro\nR : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nQ : Fin 3 → R\nu : Rˣ\n⊢ W'.Equation ((fun m => m • Q) u) ↔ W'.Equation Q","tactic":"rcases h with ⟨u, rfl⟩","premises":[]},{"state_before":"case intro\nR : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nQ : Fin 3 → R\nu : Rˣ\n⊢ W'.Equation ((fun m => m • Q) u) ↔ W'.Equation Q","state_after":"no goals","tactic":"exact equation_smul Q u.isUnit","premises":[{"full_name":"Units.isUnit","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[555,18],"def_end_pos":[555,30]},{"full_name":"WeierstrassCurve.Jacobian.equation_smul","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[263,6],"def_end_pos":[263,19]}]}]}
{"url":"Mathlib/CategoryTheory/Simple.lean","commit":"","full_name":"CategoryTheory.simple_of_isSimpleOrder_subobject","start":[218,0],"end":[230,51],"file_path":"Mathlib/CategoryTheory/Simple.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\n⊢ Simple X","state_after":"case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\n⊢ ∀ {Y : C} (f : Y ⟶ X) [inst : Mono f], IsIso f ↔ f ≠ 0","tactic":"constructor","premises":[]},{"state_before":"case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\n⊢ ∀ {Y : C} (f : Y ⟶ X) [inst : Mono f], IsIso f ↔ f ≠ 0","state_after":"case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f ↔ f ≠ 0","tactic":"intros Y f hf","premises":[]},{"state_before":"case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f ↔ f ≠ 0","state_after":"case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f → f ≠ 0\n\ncase mono_isIso_iff_nonzero.mpr\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ f ≠ 0 → IsIso f","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","commit":"","full_name":"SimpleGraph.incidenceSet_inter_incidenceSet_of_not_adj","start":[659,0],"end":[663,46],"file_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","tactics":[{"state_before":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : ¬G.Adj a b\nhn : a ≠ b\n⊢ G.incidenceSet a ∩ G.incidenceSet b = ∅","state_after":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : ¬G.Adj a b\nhn : a ≠ b\n⊢ ∀ x ∈ G.incidenceSet a, x ∉ G.incidenceSet b","tactic":"simp_rw [Set.eq_empty_iff_forall_not_mem, Set.mem_inter_iff, not_and]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.eq_empty_iff_forall_not_mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[463,8],"def_end_pos":[463,35]},{"full_name":"Set.mem_inter_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[718,8],"def_end_pos":[718,21]},{"full_name":"not_and","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[116,16],"def_end_pos":[116,23]}]},{"state_before":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v w : V\ne : Sym2 V\nh : ¬G.Adj a b\nhn : a ≠ b\n⊢ ∀ x ∈ G.incidenceSet a, x ∉ G.incidenceSet b","state_after":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u✝ v w : V\ne : Sym2 V\nh : ¬G.Adj a b\nhn : a ≠ b\nu : Sym2 V\nha : u ∈ G.incidenceSet a\nhb : u ∈ G.incidenceSet b\n⊢ False","tactic":"intro u ha hb","premises":[]},{"state_before":"ι : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u✝ v w : V\ne : Sym2 V\nh : ¬G.Adj a b\nhn : a ≠ b\nu : Sym2 V\nha : u ∈ G.incidenceSet a\nhb : u ∈ G.incidenceSet b\n⊢ False","state_after":"no goals","tactic":"exact h (G.adj_of_mem_incidenceSet hn ha hb)","premises":[{"full_name":"SimpleGraph.adj_of_mem_incidenceSet","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[654,8],"def_end_pos":[654,31]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Fintype.prod_unique","start":[1963,0],"end":[1965,67],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι✝ : Type u_1\nκ✝ : Type u_2\nα✝¹ : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α✝¹\na : α✝¹\nf✝ g : α✝¹ → β✝\nι : Type u_6\nκ : Type u_7\nα✝ : Type u_8\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype κ\ninst✝³ : CommMonoid α✝\nα : Type u_9\nβ : Type u_10\ninst✝² : CommMonoid β\ninst✝¹ : Unique α\ninst✝ : Fintype α\nf : α → β\n⊢ ∏ x : α, f x = f default","state_after":"no goals","tactic":"rw [univ_unique, prod_singleton]","premises":[{"full_name":"Finset.prod_singleton","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[354,8],"def_end_pos":[354,22]},{"full_name":"Finset.univ_unique","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[116,8],"def_end_pos":[116,19]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Bochner.lean","commit":"","full_name":"MeasureTheory.L1.integral_of_fun_eq_integral","start":[850,0],"end":[856,37],"file_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\nhf : Integrable f μ\n⊢ ∫ (a : α), ↑↑(Integrable.toL1 f hf) a ∂μ = ∫ (a : α), f a ∂μ","state_after":"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\nhf : Integrable f μ\nhG : CompleteSpace G\n⊢ ∫ (a : α), ↑↑(Integrable.toL1 f hf) a ∂μ = ∫ (a : α), f a ∂μ\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\nhf : Integrable f μ\nhG : ¬CompleteSpace G\n⊢ ∫ (a : α), ↑↑(Integrable.toL1 f hf) a ∂μ = ∫ (a : α), f a ∂μ","tactic":"by_cases hG : CompleteSpace G","premises":[{"full_name":"CompleteSpace","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[360,6],"def_end_pos":[360,19]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/CategoryTheory/CofilteredSystem.lean","commit":"","full_name":"CategoryTheory.Functor.toEventualRanges_nonempty","start":[294,0],"end":[299,16],"file_path":"Mathlib/CategoryTheory/CofilteredSystem.lean","tactics":[{"state_before":"J : Type u\ninst✝² : Category.{u_1, u} J\nF : J ⥤ Type v\ni j✝ k : J\ns : Set (F.obj i)\ninst✝¹ : IsCofilteredOrEmpty J\nh : F.IsMittagLeffler\ninst✝ : ∀ (j : J), Nonempty (F.obj j)\nj : J\n⊢ Nonempty (F.toEventualRanges.obj j)","state_after":"J : Type u\ninst✝² : Category.{u_1, u} J\nF : J ⥤ Type v\ni✝ j✝ k : J\ns : Set (F.obj i✝)\ninst✝¹ : IsCofilteredOrEmpty J\nh✝ : F.IsMittagLeffler\ninst✝ : ∀ (j : J), Nonempty (F.obj j)\nj i : J\nf : i ⟶ j\nh : F.eventualRange j = range (F.map f)\n⊢ Nonempty (F.toEventualRanges.obj j)","tactic":"let ⟨i, f, h⟩ := F.isMittagLeffler_iff_eventualRange.1 h j","premises":[{"full_name":"CategoryTheory.Functor.isMittagLeffler_iff_eventualRange","def_path":"Mathlib/CategoryTheory/CofilteredSystem.lean","def_pos":[143,8],"def_end_pos":[143,41]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"J : Type u\ninst✝² : Category.{u_1, u} J\nF : J ⥤ Type v\ni✝ j✝ k : J\ns : Set (F.obj i✝)\ninst✝¹ : IsCofilteredOrEmpty J\nh✝ : F.IsMittagLeffler\ninst✝ : ∀ (j : J), Nonempty (F.obj j)\nj i : J\nf : i ⟶ j\nh : F.eventualRange j = range (F.map f)\n⊢ Nonempty (F.toEventualRanges.obj j)","state_after":"J : Type u\ninst✝² : Category.{u_1, u} J\nF : J ⥤ Type v\ni✝ j✝ k : J\ns : Set (F.obj i✝)\ninst✝¹ : IsCofilteredOrEmpty J\nh✝ : F.IsMittagLeffler\ninst✝ : ∀ (j : J), Nonempty (F.obj j)\nj i : J\nf : i ⟶ j\nh : F.eventualRange j = range (F.map f)\n⊢ Nonempty ↑(range (F.map f))","tactic":"rw [toEventualRanges_obj, h]","premises":[{"full_name":"CategoryTheory.Functor.toEventualRanges_obj","def_path":"Mathlib/CategoryTheory/CofilteredSystem.lean","def_pos":[255,2],"def_end_pos":[255,7]}]},{"state_before":"J : Type u\ninst✝² : Category.{u_1, u} J\nF : J ⥤ Type v\ni✝ j✝ k : J\ns : Set (F.obj i✝)\ninst✝¹ : IsCofilteredOrEmpty J\nh✝ : F.IsMittagLeffler\ninst✝ : ∀ (j : J), Nonempty (F.obj j)\nj i : J\nf : i ⟶ j\nh : F.eventualRange j = range (F.map f)\n⊢ Nonempty ↑(range (F.map f))","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Data/Int/LeastGreatest.lean","commit":"","full_name":"Int.exists_least_of_bdd","start":[55,0],"end":[66,16],"file_path":"Mathlib/Data/Int/LeastGreatest.lean","tactics":[{"state_before":"P : ℤ → Prop\nHbdd : ∃ b, ∀ (z : ℤ), P z → b ≤ z\nHinh : ∃ z, P z\n⊢ ∃ lb, P lb ∧ ∀ (z : ℤ), P z → lb ≤ z","state_after":"no goals","tactic":"classical\n  let ⟨b , Hb⟩ := Hbdd\n  let ⟨lb , H⟩ := leastOfBdd b Hb Hinh\n  exact ⟨lb , H⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Int.leastOfBdd","def_path":"Mathlib/Data/Int/LeastGreatest.lean","def_pos":[44,4],"def_end_pos":[44,14]}]}]}
{"url":".lake/packages/batteries/Batteries/Classes/Order.lean","commit":"","full_name":"Ordering.then_eq_eq","start":[23,0],"end":[24,28],"file_path":".lake/packages/batteries/Batteries/Classes/Order.lean","tactics":[{"state_before":"o₁ o₂ : Ordering\n⊢ o₁.then o₂ = eq ↔ o₁ = eq ∧ o₂ = eq","state_after":"no goals","tactic":"cases o₁ <;> simp [«then»]","premises":[{"full_name":"Ordering.then","def_path":".lake/packages/lean4/src/lean/Init/Data/Ord.lean","def_pos":[43,20],"def_end_pos":[43,26]}]}]}
{"url":"Mathlib/Algebra/Group/Units.lean","commit":"","full_name":"isUnit_of_subsingleton","start":[541,0],"end":[543,49],"file_path":"Mathlib/Algebra/Group/Units.lean","tactics":[{"state_before":"α : Type u\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Subsingleton M\na : M\n⊢ a * a = 1","state_after":"no goals","tactic":"subsingleton","premises":[]},{"state_before":"α : Type u\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Subsingleton M\na : M\n⊢ a * a = 1","state_after":"no goals","tactic":"subsingleton","premises":[]}]}
{"url":"Mathlib/MeasureTheory/OuterMeasure/Induced.lean","commit":"","full_name":"MeasureTheory.extend_mono'","start":[117,0],"end":[121,23],"file_path":"Mathlib/MeasureTheory/OuterMeasure/Induced.lean","tactics":[{"state_before":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\n⊢ extend m s₁ ≤ extend m s₂","state_after":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\n⊢ ∀ (i : P s₂), extend m s₁ ≤ m s₂ i","tactic":"refine le_iInf ?_","premises":[{"full_name":"le_iInf","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[664,8],"def_end_pos":[664,15]}]},{"state_before":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\n⊢ ∀ (i : P s₂), extend m s₁ ≤ m s₂ i","state_after":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\nh₂ : P s₂\n⊢ extend m s₁ ≤ m s₂ h₂","tactic":"intro h₂","premises":[]},{"state_before":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\nh₂ : P s₂\n⊢ extend m s₁ ≤ m s₂ h₂","state_after":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\nh₂ : P s₂\n⊢ m s₁ h₁ ≤ m s₂ h₂","tactic":"rw [extend_eq m h₁]","premises":[{"full_name":"MeasureTheory.extend_eq","def_path":"Mathlib/MeasureTheory/OuterMeasure/Induced.lean","def_pos":[46,8],"def_end_pos":[46,17]}]},{"state_before":"α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nh₁ : P s₁\nhs : s₁ ⊆ s₂\nh₂ : P s₂\n⊢ m s₁ h₁ ≤ m s₂ h₂","state_after":"no goals","tactic":"exact m_mono h₁ h₂ hs","premises":[]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","commit":"","full_name":"HasFTaylorSeriesUpToOn.continuousOn","start":[216,0],"end":[219,75],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s\n⊢ ContinuousOn f s","state_after":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s\nthis : ContinuousOn (fun x => (continuousMultilinearCurryFin0 𝕜 E F).symm (f x)) s\n⊢ ContinuousOn f s","tactic":"have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm","premises":[{"full_name":"ContinuousOn.congr","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[744,8],"def_end_pos":[744,26]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"HasFTaylorSeriesUpToOn.cont","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[191,2],"def_end_pos":[191,6]},{"full_name":"HasFTaylorSeriesUpToOn.zero_eq'","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[193,8],"def_end_pos":[193,39]},{"full_name":"bot_le","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[192,8],"def_end_pos":[192,14]}]},{"state_before":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s\nthis : ContinuousOn (fun x => (continuousMultilinearCurryFin0 𝕜 E F).symm (f x)) s\n⊢ ContinuousOn f s","state_after":"no goals","tactic":"rwa [← (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_continuousOn_iff]","premises":[{"full_name":"LinearIsometryEquiv.comp_continuousOn_iff","def_path":"Mathlib/Analysis/Normed/Operator/LinearIsometry.lean","def_pos":[904,8],"def_end_pos":[904,29]},{"full_name":"LinearIsometryEquiv.symm","def_path":"Mathlib/Analysis/Normed/Operator/LinearIsometry.lean","def_pos":[634,4],"def_end_pos":[634,8]},{"full_name":"continuousMultilinearCurryFin0","def_path":"Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean","def_pos":[433,4],"def_end_pos":[433,34]}]}]}
{"url":"Mathlib/Algebra/Free.lean","commit":"","full_name":"FreeAddSemigroup.traverse_add","start":[586,0],"end":[596,5],"file_path":"Mathlib/Algebra/Free.lean","tactics":[{"state_before":"α β : Type u\nm : Type u → Type u\ninst✝¹ : Applicative m\nF : α → m β\ninst✝ : LawfulApplicative m\nx✝¹ y✝ : FreeSemigroup α\nx✝ : α\nL1 : List α\ny : α\nL2 : List α\nhd : α\ntl : List α\nih :\n  ∀ (x : α),\n    traverse F ({ head := x, tail := tl } * { head := y, tail := L2 }) =\n      Seq.seq ((fun x x_1 => x * x_1) <$> traverse F { head := x, tail := tl }) fun x =>\n        traverse F { head := y, tail := L2 }\nx : α\n⊢ (Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x =>\n      traverse F ({ head := hd, tail := tl } * { head := y, tail := L2 })) =\n    Seq.seq\n      ((fun x x_1 => x * x_1) <$>\n        Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x => traverse F { head := hd, tail := tl })\n      fun x => traverse F { head := y, tail := L2 }","state_after":"α β : Type u\nm : Type u → Type u\ninst✝¹ : Applicative m\nF : α → m β\ninst✝ : LawfulApplicative m\nx✝¹ y✝ : FreeSemigroup α\nx✝ : α\nL1 : List α\ny : α\nL2 : List α\nhd : α\ntl : List α\nih :\n  ∀ (x : α),\n    traverse F ({ head := x, tail := tl } * { head := y, tail := L2 }) =\n      Seq.seq ((fun x x_1 => x * x_1) <$> traverse F { head := x, tail := tl }) fun x =>\n        traverse F { head := y, tail := L2 }\nx : α\n⊢ (Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x =>\n      Seq.seq ((fun x x_1 => x * x_1) <$> traverse F { head := hd, tail := tl }) fun x =>\n        traverse F { head := y, tail := L2 }) =\n    Seq.seq\n      ((fun x x_1 => x * x_1) <$>\n        Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x => traverse F { head := hd, tail := tl })\n      fun x => traverse F { head := y, tail := L2 }","tactic":"rw [ih]","premises":[]}]}
{"url":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","commit":"","full_name":"List.formPerm_apply_mem_eq_next","start":[113,0],"end":[116,45],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\nl l' : List α\nhl : l.Nodup\nx : α\nhx : x ∈ l\n⊢ l.formPerm x = l.next x hx","state_after":"case intro\nα : Type u_1\ninst✝ : DecidableEq α\nl l' : List α\nhl : l.Nodup\nk : Fin l.length\nhx : l.get k ∈ l\n⊢ l.formPerm (l.get k) = l.next (l.get k) hx","tactic":"obtain ⟨k, rfl⟩ := get_of_mem hx","premises":[{"full_name":"List.get_of_mem","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[355,8],"def_end_pos":[355,18]}]},{"state_before":"case intro\nα : Type u_1\ninst✝ : DecidableEq α\nl l' : List α\nhl : l.Nodup\nk : Fin l.length\nhx : l.get k ∈ l\n⊢ l.formPerm (l.get k) = l.next (l.get k) hx","state_after":"no goals","tactic":"rw [next_get _ hl, formPerm_apply_get _ hl]","premises":[{"full_name":"List.formPerm_apply_get","def_path":"Mathlib/GroupTheory/Perm/List.lean","def_pos":[219,8],"def_end_pos":[219,26]},{"full_name":"List.next_get","def_path":"Mathlib/Data/List/Cycle.lean","def_pos":[236,8],"def_end_pos":[236,16]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.prod_multiset_map_count","start":[1373,0],"end":[1377,32],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\n⊢ (Multiset.map f s).prod = ∏ m ∈ s.toFinset, f m ^ Multiset.count m s","state_after":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\nl : List α\n⊢ (Multiset.map f (Quot.mk Setoid.r l)).prod =\n    ∏ m ∈ Multiset.toFinset (Quot.mk Setoid.r l), f m ^ Multiset.count m (Quot.mk Setoid.r l)","tactic":"refine Quot.induction_on s fun l => ?_","premises":[{"full_name":"Quot.induction_on","def_path":"Mathlib/Data/Quot.lean","def_pos":[39,18],"def_end_pos":[39,30]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\nl : List α\n⊢ (Multiset.map f (Quot.mk Setoid.r l)).prod =\n    ∏ m ∈ Multiset.toFinset (Quot.mk Setoid.r l), f m ^ Multiset.count m (Quot.mk Setoid.r l)","state_after":"no goals","tactic":"simp [prod_list_map_count l f]","premises":[{"full_name":"Finset.prod_list_map_count","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1338,8],"def_end_pos":[1338,27]}]}]}
{"url":"Mathlib/CategoryTheory/Shift/Basic.lean","commit":"","full_name":"CategoryTheory.shiftFunctorAdd_assoc_inv_app","start":[337,0],"end":[343,89],"file_path":"Mathlib/CategoryTheory/Shift/Basic.lean","tactics":[{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na₁ a₂ a₃ : A\nX : C\n⊢ (shiftFunctor C a₃).map ((shiftFunctorAdd C a₁ a₂).inv.app X) ≫ (shiftFunctorAdd C (a₁ + a₂) a₃).inv.app X =\n    (shiftFunctorAdd C a₂ a₃).inv.app ((shiftFunctor C a₁).obj X) ≫\n      (shiftFunctorAdd' C a₁ (a₂ + a₃) (a₁ + a₂ + a₃) ⋯).inv.app X","state_after":"no goals","tactic":"simpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd_assoc C a₁ a₂ a₃)) X","premises":[{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.NatTrans.congr_app","def_path":"Mathlib/CategoryTheory/Functor/Category.lean","def_pos":[63,8],"def_end_pos":[63,17]},{"full_name":"CategoryTheory.shiftFunctorAdd_assoc","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[260,6],"def_end_pos":[260,27]}]}]}
{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.Finite.injOn_iff_bijOn_of_mapsTo","start":[803,0],"end":[807,99],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → α\nhs : s.Finite\nhm : MapsTo f s s\n⊢ InjOn f s ↔ BijOn f s s","state_after":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → α\nhs : s.Finite\nhm : MapsTo f s s\nh : InjOn f s\n⊢ SurjOn f s s","tactic":"refine ⟨fun h ↦ ⟨hm, h, ?_⟩, BijOn.injOn⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Set.BijOn.injOn","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[877,8],"def_end_pos":[877,19]}]},{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → α\nhs : s.Finite\nhm : MapsTo f s s\nh : InjOn f s\n⊢ SurjOn f s s","state_after":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → α\nhs : s.Finite\nhm : MapsTo f s s\nh : InjOn f s\nthis : Finite ↑s\n⊢ SurjOn f s s","tactic":"have : Finite s := finite_coe_iff.mpr hs","premises":[{"full_name":"Finite","def_path":"Mathlib/Data/Finite/Defs.lean","def_pos":[79,16],"def_end_pos":[79,22]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.finite_coe_iff","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[68,8],"def_end_pos":[68,22]}]},{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → α\nhs : s.Finite\nhm : MapsTo f s s\nh : InjOn f s\nthis : Finite ↑s\n⊢ SurjOn f s s","state_after":"no goals","tactic":"exact hm.restrict_surjective_iff.mp (Finite.injective_iff_surjective.mp <| hm.restrict_inj.mpr h)","premises":[{"full_name":"Finite.injective_iff_surjective","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[567,8],"def_end_pos":[567,32]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.MapsTo.restrict_inj","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[605,8],"def_end_pos":[605,27]},{"full_name":"Set.MapsTo.restrict_surjective_iff","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[824,8],"def_end_pos":[824,38]}]}]}
{"url":"Mathlib/Logic/Equiv/PartialEquiv.lean","commit":"","full_name":"PartialEquiv.IsImage.symm_eq_on_of_inter_eq_of_eqOn","start":[400,0],"end":[406,49],"file_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ EqOn (↑e.symm) (↑e'.symm) (e.target ∩ t)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ EqOn (↑e.symm) (↑e'.symm) (↑e '' (e.source ∩ s))","tactic":"rw [← h.image_eq]","premises":[{"full_name":"PartialEquiv.IsImage.image_eq","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[354,8],"def_end_pos":[354,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ EqOn (↑e.symm) (↑e'.symm) (↑e '' (e.source ∩ s))","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx : x ∈ e.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)","tactic":"rintro y ⟨x, hx, rfl⟩","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx : x ∈ e.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx hx' : x ∈ e.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)","tactic":"have hx' := hx","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx hx' : x ∈ e.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx : x ∈ e.source ∩ s\nhx' : x ∈ e'.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)","tactic":"rw [hs] at hx'","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx : x ∈ e.source ∩ s\nhx' : x ∈ e'.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)","state_after":"no goals","tactic":"rw [e.left_inv hx.1, heq hx, e'.left_inv hx'.1]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"PartialEquiv.left_inv","def_path":"Mathlib/Logic/Equiv/PartialEquiv.lean","def_pos":[198,8],"def_end_pos":[198,16]}]}]}
{"url":"Mathlib/Combinatorics/Quiver/Covering.lean","commit":"","full_name":"Prefunctor.IsCovering.map_injective","start":[100,0],"end":[105,45],"file_path":"Mathlib/Combinatorics/Quiver/Covering.lean","tactics":[{"state_before":"U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type ?u.1952\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nu v : U\n⊢ Injective fun f => φ.map f","state_after":"U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type ?u.1952\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f => φ.map f) f = (fun f => φ.map f) g\n⊢ f = g","tactic":"rintro f g he","premises":[]},{"state_before":"U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type ?u.1952\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f => φ.map f) f = (fun f => φ.map f) g\n⊢ f = g","state_after":"U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type ?u.1952\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f => φ.map f) f = (fun f => φ.map f) g\nthis : φ.star u (Star.mk f) = φ.star u (Star.mk g)\n⊢ f = g","tactic":"have : φ.star u (Quiver.Star.mk f) = φ.star u (Quiver.Star.mk g) := by simpa using he","premises":[{"full_name":"Prefunctor.star","def_path":"Mathlib/Combinatorics/Quiver/Covering.lean","def_pos":[71,4],"def_end_pos":[71,19]},{"full_name":"Quiver.Star.mk","def_path":"Mathlib/Combinatorics/Quiver/Covering.lean","def_pos":[57,17],"def_end_pos":[57,31]}]},{"state_before":"U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type ?u.1952\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f => φ.map f) f = (fun f => φ.map f) g\nthis : φ.star u (Star.mk f) = φ.star u (Star.mk g)\n⊢ f = g","state_after":"no goals","tactic":"simpa using (hφ.star_bijective u).left this","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Prefunctor.IsCovering.star_bijective","def_path":"Mathlib/Combinatorics/Quiver/Covering.lean","def_pos":[97,2],"def_end_pos":[97,16]}]}]}
{"url":"Mathlib/GroupTheory/NoncommPiCoprod.lean","commit":"","full_name":"AddMonoidHom.independent_range_of_coprime_order","start":[208,0],"end":[242,18],"file_path":"Mathlib/GroupTheory/NoncommPiCoprod.lean","tactics":[{"state_before":"G : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\n⊢ CompleteLattice.Independent fun i => (ϕ i).range","state_after":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\n⊢ CompleteLattice.Independent fun i => (ϕ i).range","tactic":"cases nonempty_fintype ι","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\n⊢ CompleteLattice.Independent fun i => (ϕ i).range","state_after":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\n⊢ CompleteLattice.Independent fun i => (ϕ i).range","tactic":"letI := Classical.decEq ι","premises":[{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]}]},{"state_before":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\n⊢ CompleteLattice.Independent fun i => (ϕ i).range","state_after":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\n⊢ Disjoint ((fun i => (ϕ i).range) i) (⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j)","tactic":"rintro i","premises":[]},{"state_before":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\n⊢ Disjoint ((fun i => (ϕ i).range) i) (⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j)","state_after":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\n⊢ (fun i => (ϕ i).range) i ⊓ ⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j ≤ ⊥","tactic":"rw [disjoint_iff_inf_le]","premises":[{"full_name":"disjoint_iff_inf_le","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[113,8],"def_end_pos":[113,27]}]},{"state_before":"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\n⊢ (fun i => (ϕ i).range) i ⊓ ⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j ≤ ⊥","state_after":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ ↑((fun i => (ϕ i).range) i).toSubmonoid\nhxp : f ∈ ↑(⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j).toSubmonoid\n⊢ f ∈ ⊥","tactic":"rintro f ⟨hxi, hxp⟩","premises":[]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ ↑((fun i => (ϕ i).range) i).toSubmonoid\nhxp : f ∈ ↑(⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j).toSubmonoid\n⊢ f ∈ ⊥","state_after":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ j, ⨆ (_ : ¬j = i), (ϕ j).range)\n⊢ f ∈ ⊥","tactic":"dsimp at hxi hxp","premises":[]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ j, ⨆ (_ : ¬j = i), (ϕ j).range)\n⊢ f ∈ ⊥","state_after":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\nhxp : f ∈ ↑(noncommPiCoprod (fun x => ϕ ↑x) ?intro.intro.hcomm).range\n⊢ f ∈ ⊥\n\ncase intro.intro.hcomm\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\n⊢ Pairwise fun i_1 j => ∀ (x : H ↑i_1) (y : H ↑j), Commute ((ϕ ↑i_1) x) ((ϕ ↑j) y)","tactic":"rw [iSup_subtype', ← noncommPiCoprod_range] at hxp","premises":[{"full_name":"MonoidHom.noncommPiCoprod_range","def_path":"Mathlib/GroupTheory/NoncommPiCoprod.lean","def_pos":[177,8],"def_end_pos":[177,29]},{"full_name":"iSup_subtype'","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[979,8],"def_end_pos":[979,21]}]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\nhxp : f ∈ ↑(noncommPiCoprod (fun x => ϕ ↑x) ?intro.intro.hcomm).range\n⊢ f ∈ ⊥\n\ncase intro.intro.hcomm\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\n⊢ Pairwise fun i_1 j => ∀ (x : H ↑i_1) (y : H ↑j), Commute ((ϕ ↑i_1) x) ((ϕ ↑j) y)","state_after":"case intro.intro.hcomm\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\n⊢ Pairwise fun i_1 j => ∀ (x : H ↑i_1) (y : H ↑j), Commute ((ϕ ↑i_1) x) ((ϕ ↑j) y)\n\ncase intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\nhxp : f ∈ ↑(noncommPiCoprod (fun x => ϕ ↑x) ?intro.intro.hcomm).range\n⊢ f ∈ ⊥","tactic":"rotate_left","premises":[]},{"state_before":"case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\nhxp : f ∈ ↑(noncommPiCoprod (fun x => ϕ ↑x) ⋯).range\n⊢ f ∈ ⊥","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\n⊢ f ∈ ⊥","tactic":"cases' hxp with g hgf","premises":[]},{"state_before":"case intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\n⊢ f ∈ ⊥","state_after":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\n⊢ f ∈ ⊥","tactic":"cases' hxi with g' hg'f","premises":[]},{"state_before":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\n⊢ f ∈ ⊥","state_after":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ f ∈ ⊥","tactic":"have hxi : orderOf f ∣ Fintype.card (H i) := by\n    rw [← hg'f]\n    exact (orderOf_map_dvd _ _).trans orderOf_dvd_card","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[129,18],"def_end_pos":[129,25]},{"full_name":"orderOf_dvd_card","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[835,8],"def_end_pos":[835,24]},{"full_name":"orderOf_map_dvd","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[249,8],"def_end_pos":[249,23]}]},{"state_before":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ f ∈ ⊥","state_after":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f ∈ ⊥","tactic":"have hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H j) := by\n    rw [← hgf, ← Fintype.card_pi]\n    exact (orderOf_map_dvd _ _).trans orderOf_dvd_card","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Fintype.card","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[62,4],"def_end_pos":[62,8]},{"full_name":"Fintype.card_pi","def_path":"Mathlib/Data/Fintype/BigOperators.lean","def_pos":[121,14],"def_end_pos":[121,21]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Subtype","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[583,10],"def_end_pos":[583,17]},{"full_name":"orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[129,18],"def_end_pos":[129,25]},{"full_name":"orderOf_dvd_card","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[835,8],"def_end_pos":[835,24]},{"full_name":"orderOf_map_dvd","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[249,8],"def_end_pos":[249,23]}]},{"state_before":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f ∈ ⊥","state_after":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f = 1","tactic":"change f = 1","premises":[]},{"state_before":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f = 1","state_after":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ orderOf f ∣ 1","tactic":"rw [← pow_one f, ← orderOf_dvd_iff_pow_eq_one]","premises":[{"full_name":"orderOf_dvd_iff_pow_eq_one","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[217,8],"def_end_pos":[217,34]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]}]},{"state_before":"case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ orderOf f ∣ 1","state_after":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ orderOf f ∣ 1","tactic":"obtain ⟨c, hc⟩ := Nat.dvd_gcd hxp hxi","premises":[{"full_name":"Nat.dvd_gcd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[94,8],"def_end_pos":[94,15]}]},{"state_before":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ orderOf f ∣ 1","state_after":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = orderOf f * c","tactic":"use c","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = orderOf f * c","state_after":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i))","tactic":"rw [← hc]","premises":[]},{"state_before":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i))","state_after":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = 1","tactic":"symm","premises":[]},{"state_before":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = 1","state_after":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ ∀ (a : ι) (b : a ≠ i), (Fintype.card (H ↑⟨a, b⟩)).Coprime (Fintype.card (H i))","tactic":"rw [← Nat.coprime_iff_gcd_eq_one, Nat.coprime_fintype_prod_left_iff, Subtype.forall]","premises":[{"full_name":"Nat.coprime_fintype_prod_left_iff","def_path":"Mathlib/Data/Nat/GCD/BigOperators.lean","def_pos":[48,8],"def_end_pos":[48,37]},{"full_name":"Nat.coprime_iff_gcd_eq_one","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[24,8],"def_end_pos":[24,30]},{"full_name":"Subtype.forall","def_path":"Mathlib/Data/Subtype.lean","def_pos":[41,18],"def_end_pos":[41,26]}]},{"state_before":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ ∀ (a : ι) (b : a ≠ i), (Fintype.card (H ↑⟨a, b⟩)).Coprime (Fintype.card (H i))","state_after":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\nj : ι\nh : j ≠ i\n⊢ (Fintype.card (H ↑⟨j, h⟩)).Coprime (Fintype.card (H i))","tactic":"intro j h","premises":[]},{"state_before":"case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\nj : ι\nh : j ≠ i\n⊢ (Fintype.card (H ↑⟨j, h⟩)).Coprime (Fintype.card (H i))","state_after":"no goals","tactic":"exact hcoprime h","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean","commit":"","full_name":"AlgebraicGeometry.isOpenImmersion_stableUnderBaseChange","start":[64,0],"end":[67,37],"file_path":"Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean","tactics":[{"state_before":"X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ∀ (X Y S : Scheme) (f : X ⟶ S) (g : Y ⟶ S), IsOpenImmersion g → IsOpenImmersion (pullback.fst f g)","state_after":"X✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion g\n⊢ IsOpenImmersion (pullback.fst f g)","tactic":"intro X Y Z f g H","premises":[]},{"state_before":"X✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion g\n⊢ IsOpenImmersion (pullback.fst f g)","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.sub_mod_eq_zero_of_mod_eq","start":[1065,0],"end":[1068,60],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\nh : m % k = n % k\n⊢ (m - n) % k = 0","state_after":"no goals","tactic":"rw [← Nat.mod_add_div m k, ← Nat.mod_add_div n k, ← h, ← Nat.sub_sub,\n    Nat.add_sub_cancel_left, ← k.mul_sub, Nat.mul_mod_right]","premises":[{"full_name":"Nat.add_sub_cancel_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[908,18],"def_end_pos":[908,37]},{"full_name":"Nat.mod_add_div","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[201,8],"def_end_pos":[201,19]},{"full_name":"Nat.mul_mod_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[282,16],"def_end_pos":[282,29]},{"full_name":"Nat.sub_sub","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[1020,18],"def_end_pos":[1020,25]}]}]}
{"url":"Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean","commit":"","full_name":"TwoSidedIdeal.iInf_ringCon","start":[95,0],"end":[97,51],"file_path":"Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (⨅ i, I i).ringCon = ⨅ i, (I i).ringCon","state_after":"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ sInf (ringCon '' Set.range fun i => I i) = sInf (Set.range fun i => (I i).ringCon)","tactic":"simp only [iInf, sInf_ringCon]","premises":[{"full_name":"TwoSidedIdeal.sInf_ringCon","def_path":"Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean","def_pos":[92,6],"def_end_pos":[92,18]},{"full_name":"iInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[60,4],"def_end_pos":[60,8]}]},{"state_before":"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ sInf (ringCon '' Set.range fun i => I i) = sInf (Set.range fun i => (I i).ringCon)","state_after":"case h.e'_3\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (ringCon '' Set.range fun i => I i) = Set.range fun i => (I i).ringCon","tactic":"congr!","premises":[]},{"state_before":"case h.e'_3\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (ringCon '' Set.range fun i => I i) = Set.range fun i => (I i).ringCon","state_after":"case h.e'_3.h\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\nx✝ : RingCon R\n⊢ (x✝ ∈ ringCon '' Set.range fun i => I i) ↔ x✝ ∈ Set.range fun i => (I i).ringCon","tactic":"ext","premises":[]},{"state_before":"case h.e'_3.h\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\nx✝ : RingCon R\n⊢ (x✝ ∈ ringCon '' Set.range fun i => I i) ↔ x✝ ∈ Set.range fun i => (I i).ringCon","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Topology/UniformSpace/Basic.lean","commit":"","full_name":"uniformity_lift_le_swap","start":[509,0],"end":[514,74],"file_path":"Mathlib/Topology/UniformSpace/Basic.lean","tactics":[{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : ((𝓤 α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ (map Prod.swap (𝓤 α)).lift g ≤ f","state_after":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : ((𝓤 α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ (𝓤 α).lift (g ∘ preimage Prod.swap) ≤ f","tactic":"rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]","premises":[{"full_name":"Filter.map_lift_eq2","def_path":"Mathlib/Order/Filter/Lift.lean","def_pos":[113,8],"def_end_pos":[113,20]},{"full_name":"Set.image_swap_eq_preimage_swap","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[858,8],"def_end_pos":[858,35]}]},{"state_before":"α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : ((𝓤 α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ (𝓤 α).lift (g ∘ preimage Prod.swap) ≤ f","state_after":"no goals","tactic":"exact h","premises":[]}]}
{"url":"Mathlib/Topology/Category/CompHaus/Basic.lean","commit":"","full_name":"CompHaus.epi_iff_surjective","start":[177,0],"end":[218,42],"file_path":"Mathlib/Topology/Category/CompHaus/Basic.lean","tactics":[{"state_before":"X Y : CompHaus\nf : X ⟶ Y\n⊢ Epi f ↔ Function.Surjective ⇑f","state_after":"case mp\nX Y : CompHaus\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective ⇑f\n\ncase mpr\nX Y : CompHaus\nf : X ⟶ Y\n⊢ Function.Surjective ⇑f → Epi f","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finsum_mem_inter_support","start":[455,0],"end":[458,70],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\ns : Set α\n⊢ ∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f), f i = ∏ᶠ (i : α) (_ : i ∈ s), f i","state_after":"no goals","tactic":"rw [finprod_mem_def, finprod_mem_def, mulIndicator_inter_mulSupport]","premises":[{"full_name":"Set.mulIndicator_inter_mulSupport","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[179,8],"def_end_pos":[179,37]},{"full_name":"finprod_mem_def","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[323,8],"def_end_pos":[323,23]}]}]}
{"url":"Mathlib/Tactic/Ring/Basic.lean","commit":"","full_name":"Mathlib.Tactic.Ring.one_mul","start":[372,0],"end":[372,78],"file_path":"Mathlib/Tactic/Ring/Basic.lean","tactics":[{"state_before":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR : Type u_1\ninst✝ : CommSemiring R\na✝ a' a₁ a₂ a₃ b b' b₁ b₂ b₃ c c₁ c₂ a : R\n⊢ Nat.rawCast 1 * a = a","state_after":"no goals","tactic":"simp [Nat.rawCast]","premises":[{"full_name":"Nat.rawCast","def_path":"Mathlib/Tactic/NormNum/Result.lean","def_pos":[136,12],"def_end_pos":[136,30]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.dual_rotateR","start":[303,0],"end":[305,70],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ (l.rotateR x r).dual = r.dual.rotateL x l.dual","state_after":"no goals","tactic":"rw [← dual_dual (rotateL _ _ _), dual_rotateL, dual_dual, dual_dual]","premises":[{"full_name":"Ordnode.dual_dual","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[138,8],"def_end_pos":[138,17]},{"full_name":"Ordnode.dual_rotateL","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[298,8],"def_end_pos":[298,20]},{"full_name":"Ordnode.rotateL","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[225,4],"def_end_pos":[225,11]}]}]}
{"url":"Mathlib/Data/Rel.lean","commit":"","full_name":"Rel.core_univ","start":[298,0],"end":[300,30],"file_path":"Mathlib/Data/Rel.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : Rel α β\n⊢ ∀ (x : α), x ∈ r.core Set.univ ↔ x ∈ Set.univ","state_after":"no goals","tactic":"simp [mem_core]","premises":[{"full_name":"Rel.mem_core","def_path":"Mathlib/Data/Rel.lean","def_pos":[284,8],"def_end_pos":[284,16]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm","start":[330,0],"end":[335,25],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂μ","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ENNReal.ofReal ‖f x‖ ∂μ","tactic":"simp_rw [← ofReal_norm_eq_coe_nnnorm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"ofReal_norm_eq_coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[681,14],"def_end_pos":[681,39]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ENNReal.ofReal ‖f x‖ ∂μ","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\nx : α\n⊢ f x ≤ ‖f x‖","tactic":"refine lintegral_mono fun x => ENNReal.ofReal_le_ofReal ?_","premises":[{"full_name":"ENNReal.ofReal_le_ofReal","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[159,8],"def_end_pos":[159,24]},{"full_name":"MeasureTheory.lintegral_mono","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[95,8],"def_end_pos":[95,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\nx : α\n⊢ f x ≤ ‖f x‖","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\nx : α\n⊢ f x ≤ |f x|","tactic":"rw [Real.norm_eq_abs]","premises":[{"full_name":"Real.norm_eq_abs","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1132,8],"def_end_pos":[1132,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\nx : α\n⊢ f x ≤ |f x|","state_after":"no goals","tactic":"exact le_abs_self (f x)","premises":[{"full_name":"le_abs_self","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[63,2],"def_end_pos":[63,13]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Rayleigh.lean","commit":"","full_name":"LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional","start":[246,0],"end":[263,87],"file_path":"Mathlib/Analysis/InnerProductSpace/Rayleigh.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"haveI := FiniteDimensional.proper_rclike 𝕜 E","premises":[{"full_name":"FiniteDimensional.proper_rclike","def_path":"Mathlib/Analysis/RCLike/Lemmas.lean","def_pos":[49,8],"def_end_pos":[49,21]}]},{"state_before":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"let T' := hT.toSelfAdjoint","premises":[{"full_name":"LinearMap.IsSymmetric.toSelfAdjoint","def_path":"Mathlib/Analysis/InnerProductSpace/Adjoint.lean","def_pos":[291,4],"def_end_pos":[291,29]}]},{"state_before":"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"exists_ne","def_path":"Mathlib/Logic/Nontrivial/Defs.lean","def_pos":[47,8],"def_end_pos":[47,17]}]},{"state_before":"case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"have H₁ : IsCompact (sphere (0 : E) ‖x‖) := isCompact_sphere _ _","premises":[{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"Metric.sphere","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[410,4],"def_end_pos":[410,10]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"isCompact_sphere","def_path":"Mathlib/Topology/MetricSpace/ProperSpace.lean","def_pos":[41,8],"def_end_pos":[41,24]}]},{"state_before":"case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"have H₂ : (sphere (0 : E) ‖x‖).Nonempty := ⟨x, by simp⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Metric.sphere","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[410,4],"def_end_pos":[410,10]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Set.Nonempty","def_path":"Mathlib/Init/Set.lean","def_pos":[222,14],"def_end_pos":[222,22]}]},{"state_before":"case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"obtain ⟨x₀, hx₀', hTx₀⟩ :=\n    H₁.exists_isMinOn H₂ T'.val.reApplyInnerSelf_continuous.continuousOn","premises":[{"full_name":"Continuous.continuousOn","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[810,8],"def_end_pos":[810,31]},{"full_name":"ContinuousLinearMap.reApplyInnerSelf_continuous","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[2029,8],"def_end_pos":[2029,55]},{"full_name":"IsCompact.exists_isMinOn","def_path":"Mathlib/Topology/Algebra/Order/Compact.lean","def_pos":[250,8],"def_end_pos":[250,32]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"have hx₀ : ‖x₀‖ = ‖x‖ := by simpa using hx₀'","premises":[{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis✝ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\nthis : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"have : IsMinOn T'.val.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀ := by simpa only [← hx₀] using hTx₀","premises":[{"full_name":"ContinuousLinearMap.reApplyInnerSelf","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[2022,4],"def_end_pos":[2022,40]},{"full_name":"IsMinOn","def_path":"Mathlib/Order/Filter/Extr.lean","def_pos":[106,4],"def_end_pos":[106,11]},{"full_name":"Metric.sphere","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[410,4],"def_end_pos":[410,10]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis✝ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\nthis : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis✝ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\nthis : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀_ne : x₀ ≠ 0\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","tactic":"have hx₀_ne : x₀ ≠ 0 := by\n    have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne, not_false_iff]\n    simpa [← norm_eq_zero, Ne]","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"norm_eq_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1208,29],"def_end_pos":[1208,41]},{"full_name":"not_false_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1371,8],"def_end_pos":[1371,21]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis✝ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\nthis : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀_ne : x₀ ≠ 0\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)","state_after":"no goals","tactic":"exact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMinOn hx₀_ne this)","premises":[{"full_name":"IsSelfAdjoint.hasEigenvector_of_isMinOn","def_path":"Mathlib/Analysis/InnerProductSpace/Rayleigh.lean","def_pos":[199,8],"def_end_pos":[199,33]},{"full_name":"Module.End.hasEigenvalue_of_hasEigenvector","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[92,8],"def_end_pos":[92,39]},{"full_name":"Subtype.prop","def_path":"Mathlib/Data/Subtype.lean","def_pos":[37,8],"def_end_pos":[37,12]}]}]}
{"url":"Mathlib/LinearAlgebra/SymplecticGroup.lean","commit":"","full_name":"SymplecticGroup.transpose_mem_iff","start":[147,0],"end":[149,60],"file_path":"Mathlib/LinearAlgebra/SymplecticGroup.lean","tactics":[{"state_before":"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : Aᵀ ∈ symplecticGroup l R\n⊢ A ∈ symplecticGroup l R","state_after":"no goals","tactic":"simpa using transpose_mem hA","premises":[{"full_name":"SymplecticGroup.transpose_mem","def_path":"Mathlib/LinearAlgebra/SymplecticGroup.lean","def_pos":[129,8],"def_end_pos":[129,21]}]}]}
{"url":"Mathlib/Data/Finset/Pointwise.lean","commit":"","full_name":"Finset.neg_univ","start":[274,0],"end":[275,71],"file_path":"Mathlib/Data/Finset/Pointwise.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : DecidableEq α\ninst✝¹ : InvolutiveInv α\ns : Finset α\na : α\ninst✝ : Fintype α\n⊢ univ⁻¹ = univ","state_after":"case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : DecidableEq α\ninst✝¹ : InvolutiveInv α\ns : Finset α\na : α\ninst✝ : Fintype α\na✝ : α\n⊢ a✝ ∈ univ⁻¹ ↔ a✝ ∈ univ","tactic":"ext","premises":[]},{"state_before":"case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : DecidableEq α\ninst✝¹ : InvolutiveInv α\ns : Finset α\na : α\ninst✝ : Fintype α\na✝ : α\n⊢ a✝ ∈ univ⁻¹ ↔ a✝ ∈ univ","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Probability/CondCount.lean","commit":"","full_name":"ProbabilityTheory.condCount_self","start":[94,0],"end":[97,46],"file_path":"Mathlib/Probability/CondCount.lean","tactics":[{"state_before":"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ (condCount s) s = 1","state_after":"case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0\n\ncase ht\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ ⊤","tactic":"rw [condCount, cond_apply _ hs.measurableSet, Set.inter_self, ENNReal.inv_mul_cancel]","premises":[{"full_name":"ENNReal.inv_mul_cancel","def_path":"Mathlib/Data/ENNReal/Inv.lean","def_pos":[91,18],"def_end_pos":[91,32]},{"full_name":"ProbabilityTheory.condCount","def_path":"Mathlib/Probability/CondCount.lean","def_pos":[52,4],"def_end_pos":[52,13]},{"full_name":"ProbabilityTheory.cond_apply","def_path":"Mathlib/Probability/ConditionalProbability.lean","def_pos":[137,8],"def_end_pos":[137,18]},{"full_name":"Set.Finite.measurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[269,8],"def_end_pos":[269,32]},{"full_name":"Set.inter_self","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[731,8],"def_end_pos":[731,18]}]}]}
{"url":"Mathlib/MeasureTheory/Function/ContinuousMapDense.lean","commit":"","full_name":"MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le","start":[194,0],"end":[213,12],"file_path":"Mathlib/MeasureTheory/Function/ContinuousMapDense.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","state_after":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","tactic":"have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _","premises":[{"full_name":"Real.rpow_pos_of_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[105,8],"def_end_pos":[105,23]}]},{"state_before":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","state_after":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","tactic":"have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by\n    simp only [Ne, ENNReal.ofReal_eq_zero, not_le, I]","premises":[{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"ENNReal.ofReal_eq_zero","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[195,8],"def_end_pos":[195,22]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]},{"state_before":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","state_after":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","tactic":"have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hp","premises":[{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"ENNReal.ofReal_eq_zero","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[195,8],"def_end_pos":[195,22]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]},{"state_before":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\nhg : eLpNorm (f - g) (↑p.toNNReal) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","tactic":"rcases hf.exists_hasCompactSupport_eLpNorm_sub_le ENNReal.coe_ne_top A with\n    ⟨g, g_support, hg, g_cont, g_mem⟩","premises":[{"full_name":"ENNReal.coe_ne_top","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[280,16],"def_end_pos":[280,26]},{"full_name":"MeasureTheory.Memℒp.exists_hasCompactSupport_eLpNorm_sub_le","def_path":"Mathlib/MeasureTheory/Function/ContinuousMapDense.lean","def_pos":[140,8],"def_end_pos":[140,53]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\nhg : eLpNorm (f - g) (↑p.toNNReal) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","tactic":"change eLpNorm _ (ENNReal.ofReal p) _ ≤ _ at hg","premises":[{"full_name":"ENNReal.ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[187,28],"def_end_pos":[187,34]},{"full_name":"MeasureTheory.eLpNorm","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[78,4],"def_end_pos":[78,11]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ","state_after":"case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε","tactic":"refine ⟨g, g_support, ?_, g_cont, g_mem⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε","state_after":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : (∫ (a : α), ‖(f - g) a‖ ^ p ∂μ) ^ p⁻¹ ≤ ε ^ p⁻¹\n⊢ 0 ≤ ∫ (a : α), ‖(f - g) a‖ ^ p ∂μ","tactic":"rwa [(hf.sub g_mem).eLpNorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,\n    ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,\n    Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg","premises":[{"full_name":"ENNReal.coe_ne_top","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[280,16],"def_end_pos":[280,26]},{"full_name":"ENNReal.ofReal_le_ofReal_iff","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[167,8],"def_end_pos":[167,28]},{"full_name":"ENNReal.toReal_ofReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[202,8],"def_end_pos":[202,21]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.Memℒp.eLpNorm_eq_integral_rpow_norm","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1316,8],"def_end_pos":[1316,43]},{"full_name":"MeasureTheory.Memℒp.sub","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean","def_pos":[186,8],"def_end_pos":[186,17]},{"full_name":"Real.rpow_le_rpow_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[516,8],"def_end_pos":[516,24]},{"full_name":"inv_pos","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[23,14],"def_end_pos":[23,21]},{"full_name":"one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[338,8],"def_end_pos":[338,15]}]},{"state_before":"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : (∫ (a : α), ‖(f - g) a‖ ^ p ∂μ) ^ p⁻¹ ≤ ε ^ p⁻¹\n⊢ 0 ≤ ∫ (a : α), ‖(f - g) a‖ ^ p ∂μ","state_after":"no goals","tactic":"positivity","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Roots.lean","commit":"","full_name":"Polynomial.nthRoots_zero_right","start":[263,0],"end":[266,83],"file_path":"Mathlib/Algebra/Polynomial/Roots.lean","tactics":[{"state_before":"R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : IsDomain R✝\np q : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\n⊢ nthRoots n 0 = replicate n 0","state_after":"no goals","tactic":"rw [nthRoots, C.map_zero, sub_zero, roots_pow, roots_X, Multiset.nsmul_singleton]","premises":[{"full_name":"Multiset.nsmul_singleton","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[860,8],"def_end_pos":[860,23]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.nthRoots","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[251,4],"def_end_pos":[251,12]},{"full_name":"Polynomial.roots_X","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[162,8],"def_end_pos":[162,15]},{"full_name":"Polynomial.roots_pow","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[207,8],"def_end_pos":[207,17]},{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.Valid'.merge_aux","start":[1296,0],"end":[1314,72],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : Preorder α\nl r : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) l\n⊢ Valid' o₁ (l.merge r) o₂ ∧ (l.merge r).size = l.size + r.size","state_after":"case nil\nα : Type u_1\ninst✝ : Preorder α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ nil o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) nil\n⊢ Valid' o₁ (nil.merge r) o₂ ∧ (nil.merge r).size = nil.size + r.size\n\ncase node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).merge r) o₂ ∧\n    ((Ordnode.node ls ll lx lr).merge r).size = (Ordnode.node ls ll lx lr).size + r.size","tactic":"induction' l with ls ll lx lr _ IHlr generalizing o₁ o₂ r","premises":[]},{"state_before":"case node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).merge r) o₂ ∧\n    ((Ordnode.node ls ll lx lr).merge r).size = (Ordnode.node ls ll lx lr).size + r.size","state_after":"case node.nil\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ nil o₂\nsep : All (fun x => All (fun y => x < y) nil) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).merge nil) o₂ ∧\n    ((Ordnode.node ls ll lx lr).merge nil).size = (Ordnode.node ls ll lx lr).size + nil.size\n\ncase node.node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)) o₂ ∧\n    ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size","tactic":"induction' r with rs rl rx rr IHrl _ generalizing o₁ o₂","premises":[]},{"state_before":"case node.node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)) o₂ ∧\n    ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size","state_after":"case node.node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁\n      (if delta * ls < rs then ((Ordnode.node ls ll lx lr).merge rl).balanceL rx rr\n      else\n        if delta * rs < ls then ll.balanceR lx (lr.merge (Ordnode.node rs rl rx rr))\n        else (Ordnode.node ls ll lx lr).glue (Ordnode.node rs rl rx rr))\n      o₂ ∧\n    (if delta * ls < rs then ((Ordnode.node ls ll lx lr).merge rl).balanceL rx rr\n        else\n          if delta * rs < ls then ll.balanceR lx (lr.merge (Ordnode.node rs rl rx rr))\n          else (Ordnode.node ls ll lx lr).glue (Ordnode.node rs rl rx rr)).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size","tactic":"rw [merge_node]","premises":[{"full_name":"Ordnode.merge_node","def_path":"Mathlib/Data/Ordmap/Ordset.lean","def_pos":[557,8],"def_end_pos":[557,18]}]},{"state_before":"case node.node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁\n      (if delta * ls < rs then ((Ordnode.node ls ll lx lr).merge rl).balanceL rx rr\n      else\n        if delta * rs < ls then ll.balanceR lx (lr.merge (Ordnode.node rs rl rx rr))\n        else (Ordnode.node ls ll lx lr).glue (Ordnode.node rs rl rx rr))\n      o₂ ∧\n    (if delta * ls < rs then ((Ordnode.node ls ll lx lr).merge rl).balanceL rx rr\n        else\n          if delta * rs < ls then ll.balanceR lx (lr.merge (Ordnode.node rs rl rx rr))\n          else (Ordnode.node ls ll lx lr).glue (Ordnode.node rs rl rx rr)).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size","state_after":"case pos\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : delta * ls < rs\n⊢ Valid' o₁ (((Ordnode.node ls ll lx lr).merge rl).balanceL rx rr) o₂ ∧\n    (((Ordnode.node ls ll lx lr).merge rl).balanceL rx rr).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size\n\ncase pos\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\n⊢ Valid' o₁ (ll.balanceR lx (lr.merge (Ordnode.node rs rl rx rr))) o₂ ∧\n    (ll.balanceR lx (lr.merge (Ordnode.node rs rl rx rr))).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size\n\ncase neg\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : ¬delta * rs < ls\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).glue (Ordnode.node rs rl rx rr)) o₂ ∧\n    ((Ordnode.node ls ll lx lr).glue (Ordnode.node rs rl rx rr)).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size","tactic":"split_ifs with h h_1","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.count_map","start":[2226,0],"end":[2228,58],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ✝ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α✝\ns✝ : Multiset α✝\nα : Type u_3\nβ : Type u_4\nf : α → β\ns : Multiset α\ninst✝ : DecidableEq β\nb : β\n⊢ count b (map f s) = card (filter (fun a => b = f a) s)","state_after":"no goals","tactic":"simp [Bool.beq_eq_decide_eq, eq_comm, count, countP_map]","premises":[{"full_name":"Bool.beq_eq_decide_eq","def_path":".lake/packages/lean4/src/lean/Init/Data/Bool.lean","def_pos":[234,8],"def_end_pos":[234,24]},{"full_name":"Multiset.count","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[2062,4],"def_end_pos":[2062,9]},{"full_name":"Multiset.countP_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[2004,8],"def_end_pos":[2004,18]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","commit":"","full_name":"Equiv.Perm.SameCycle.exists_pow_eq'","start":[177,0],"end":[186,33],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\np : α → Prop\nx y z : α\ninst✝ : Finite α\n⊢ f.SameCycle x y → ∃ i < orderOf f, (f ^ i) x = y","state_after":"no goals","tactic":"classical\n    rintro ⟨k, rfl⟩\n    use (k % orderOf f).natAbs\n    have h₀ := Int.natCast_pos.mpr (orderOf_pos f)\n    have h₁ := Int.emod_nonneg k h₀.ne'\n    rw [← zpow_natCast, Int.natAbs_of_nonneg h₁, zpow_mod_orderOf]\n    refine ⟨?_, by rfl⟩\n    rw [← Int.ofNat_lt, Int.natAbs_of_nonneg h₁]\n    exact Int.emod_lt_of_pos _ h₀","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.emod_lt_of_pos","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[432,8],"def_end_pos":[432,22]},{"full_name":"Int.emod_nonneg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[428,8],"def_end_pos":[428,19]},{"full_name":"Int.natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[262,4],"def_end_pos":[262,10]},{"full_name":"Int.natAbs_of_nonneg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[451,8],"def_end_pos":[451,24]},{"full_name":"Int.natCast_pos","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[113,29],"def_end_pos":[113,40]},{"full_name":"Int.ofNat_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean","def_pos":[77,27],"def_end_pos":[77,35]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]},{"full_name":"orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[129,18],"def_end_pos":[129,25]},{"full_name":"orderOf_pos","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[711,6],"def_end_pos":[711,17]},{"full_name":"zpow_mod_orderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[570,6],"def_end_pos":[570,22]},{"full_name":"zpow_natCast","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[875,8],"def_end_pos":[875,20]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","commit":"","full_name":"List.Sublist.filter","start":[660,0],"end":[661,47],"file_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\np : α → Bool\nl₁ l₂ : List α\ns : l₁ <+ l₂\n⊢ List.filter p l₁ <+ List.filter p l₂","state_after":"α : Type u_1\np : α → Bool\nl₁ l₂ : List α\ns : l₁ <+ l₂\n⊢ filterMap (Option.guard fun x => p x = true) l₁ <+ filterMap (Option.guard fun x => p x = true) l₂","tactic":"rw [← filterMap_eq_filter]","premises":[{"full_name":"List.filterMap_eq_filter","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[972,8],"def_end_pos":[972,27]}]},{"state_before":"α : Type u_1\np : α → Bool\nl₁ l₂ : List α\ns : l₁ <+ l₂\n⊢ filterMap (Option.guard fun x => p x = true) l₁ <+ filterMap (Option.guard fun x => p x = true) l₂","state_after":"no goals","tactic":"apply s.filterMap","premises":[{"full_name":"List.Sublist.filterMap","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[656,18],"def_end_pos":[656,35]}]}]}
{"url":"Mathlib/Data/Fintype/Basic.lean","commit":"","full_name":"Finset.compl_ne_univ_iff_nonempty","start":[228,0],"end":[229,44],"file_path":"Mathlib/Data/Fintype/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Fintype α\ns✝ t : Finset α\ninst✝ : DecidableEq α\na : α\ns : Finset α\n⊢ sᶜ ≠ univ ↔ s.Nonempty","state_after":"no goals","tactic":"simp [eq_univ_iff_forall, Finset.Nonempty]","premises":[{"full_name":"Finset.Nonempty","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[417,14],"def_end_pos":[417,22]},{"full_name":"Finset.eq_univ_iff_forall","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[78,8],"def_end_pos":[78,26]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean","commit":"","full_name":"CategoryTheory.MonoidalCategory.pentagon_hom_inv","start":[60,0],"end":[64,11],"file_path":"Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : MonoidalCategory C\nW X Y Z : C\n⊢ (α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) = (α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom","state_after":"no goals","tactic":"coherence","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CategoryTheory.BicategoricalCoherence.hom","def_path":"Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean","def_pos":[29,2],"def_end_pos":[29,5]},{"full_name":"CategoryTheory.Bicategory.comp_whiskerLeft","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[81,2],"def_end_pos":[81,18]},{"full_name":"CategoryTheory.Bicategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[89,2],"def_end_pos":[89,19]},{"full_name":"CategoryTheory.Bicategory.id_whiskerLeft","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[77,2],"def_end_pos":[77,16]},{"full_name":"CategoryTheory.Bicategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[87,2],"def_end_pos":[87,17]},{"full_name":"CategoryTheory.Bicategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[73,2],"def_end_pos":[73,18]},{"full_name":"CategoryTheory.Bicategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[71,2],"def_end_pos":[71,16]},{"full_name":"CategoryTheory.Bicategory.whiskerRight_comp","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[97,2],"def_end_pos":[97,19]},{"full_name":"CategoryTheory.Bicategory.whiskerRight_id","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[93,2],"def_end_pos":[93,17]},{"full_name":"CategoryTheory.Bicategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Bicategory/Basic.lean","def_pos":[103,2],"def_end_pos":[103,15]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.FreeMonoidalCategory.projectMap","def_path":"Mathlib/CategoryTheory/Monoidal/Free/Basic.lean","def_pos":[312,4],"def_end_pos":[312,14]},{"full_name":"CategoryTheory.MonoidalCategory.comp_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"CategoryTheory.MonoidalCategory.id_tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[222,8],"def_end_pos":[222,20]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[237,8],"def_end_pos":[237,22]},{"full_name":"CategoryTheory.MonoidalCategory.id_whiskerRight","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[172,2],"def_end_pos":[172,17]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_def","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[157,2],"def_end_pos":[157,15]},{"full_name":"CategoryTheory.MonoidalCategory.tensorHom_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[227,8],"def_end_pos":[227,20]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_whiskerLeft","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[242,8],"def_end_pos":[242,26]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[232,8],"def_end_pos":[232,24]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerLeft_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[170,2],"def_end_pos":[170,16]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerRight_id","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[254,8],"def_end_pos":[254,23]},{"full_name":"CategoryTheory.MonoidalCategory.whiskerRight_tensor","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[259,8],"def_end_pos":[259,27]},{"full_name":"CategoryTheory.MonoidalCategory.whisker_assoc","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[266,8],"def_end_pos":[266,21]},{"full_name":"CategoryTheory.bicategoricalComp","def_path":"Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean","def_pos":[47,4],"def_end_pos":[47,21]},{"full_name":"CategoryTheory.monoidalComp","def_path":"Mathlib/Tactic/CategoryTheory/MonoidalComp.lean","def_pos":[67,4],"def_end_pos":[67,16]},{"full_name":"Mathlib.Tactic.Coherence.LiftHom.lift","def_path":"Mathlib/Tactic/CategoryTheory/Coherence.lean","def_pos":[54,12],"def_end_pos":[54,16]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
{"url":"Mathlib/CategoryTheory/Opposites.lean","commit":"","full_name":"CategoryTheory.Functor.leftOpRightOpEquiv_functor_obj_obj","start":[566,0],"end":[584,62],"file_path":"Mathlib/CategoryTheory/Opposites.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\n⊢ ∀ {X Y : (Cᵒᵖ ⥤ D)ᵒᵖ} (f : X ⟶ Y),\n    (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map f ≫ ((fun F => (unop F).rightOpLeftOpIso.op) Y).hom =\n      ((fun F => (unop F).rightOpLeftOpIso.op) X).hom ≫\n        ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯,\n                map_comp := ⋯ } ⋙\n              { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,\n                map_comp := ⋯ }).map\n          f","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map η ≫ ((fun F => (unop F).rightOpLeftOpIso.op) G).hom =\n    ((fun F => (unop F).rightOpLeftOpIso.op) F).hom ≫\n      ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯, map_comp := ⋯ } ⋙\n            { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,\n              map_comp := ⋯ }).map\n        η","tactic":"intro F G η","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map η ≫ ((fun F => (unop F).rightOpLeftOpIso.op) G).hom =\n    ((fun F => (unop F).rightOpLeftOpIso.op) F).hom ≫\n      ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯, map_comp := ⋯ } ⋙\n            { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,\n              map_comp := ⋯ }).map\n        η","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ η ≫ (unop G).rightOpLeftOpIso.hom.op =\n    (unop F).rightOpLeftOpIso.hom.op ≫ (NatTrans.leftOp (NatTrans.rightOp η.unop)).op","tactic":"dsimp","premises":[]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ η ≫ (unop G).rightOpLeftOpIso.hom.op =\n    (unop F).rightOpLeftOpIso.hom.op ≫ (NatTrans.leftOp (NatTrans.rightOp η.unop)).op","state_after":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ ((unop G).rightOpLeftOpIso.hom ≫ η.unop).op =\n    (NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom).op","tactic":"rw [show η = η.unop.op by simp, ← op_comp, ← op_comp]","premises":[{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"Quiver.Hom.op","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[136,4],"def_end_pos":[136,10]},{"full_name":"Quiver.Hom.unop","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[139,4],"def_end_pos":[139,12]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ ((unop G).rightOpLeftOpIso.hom ≫ η.unop).op =\n    (NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom).op","state_after":"case e_f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (unop G).rightOpLeftOpIso.hom ≫ η.unop =\n    NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (Cᵒᵖ ⥤ D)ᵒᵖ\nη : F ⟶ G\n⊢ (unop G).rightOpLeftOpIso.hom ≫ η.unop =\n    NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Content.lean","commit":"","full_name":"Polynomial.content_C_mul","start":[125,0],"end":[128,71],"file_path":"Mathlib/RingTheory/Polynomial/Content.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\n⊢ (C r * p).content = normalize r * p.content","state_after":"case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : r = 0\n⊢ (C r * p).content = normalize r * p.content\n\ncase neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).content = normalize r * p.content","tactic":"by_cases h0 : r = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).content = normalize r * p.content","state_after":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).support.gcd (C r * p).coeff = normalize r * p.content","tactic":"rw [content]","premises":[{"full_name":"Polynomial.content","def_path":"Mathlib/RingTheory/Polynomial/Content.lean","def_pos":[70,4],"def_end_pos":[70,11]}]},{"state_before":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).support.gcd (C r * p).coeff = normalize r * p.content","state_after":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).support.gcd (C r * p).coeff = normalize r * p.support.gcd p.coeff","tactic":"rw [content]","premises":[{"full_name":"Polynomial.content","def_path":"Mathlib/RingTheory/Polynomial/Content.lean","def_pos":[70,4],"def_end_pos":[70,11]}]},{"state_before":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).support.gcd (C r * p).coeff = normalize r * p.support.gcd p.coeff","state_after":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).support.gcd (C r * p).coeff = p.support.gcd fun x => r * p.coeff x","tactic":"rw [← Finset.gcd_mul_left]","premises":[{"full_name":"Finset.gcd_mul_left","def_path":"Mathlib/Algebra/GCDMonoid/Finset.lean","def_pos":[208,15],"def_end_pos":[208,27]}]},{"state_before":"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\np : R[X]\nh0 : ¬r = 0\n⊢ (C r * p).support.gcd (C r * p).coeff = p.support.gcd fun x => r * p.coeff x","state_after":"no goals","tactic":"refine congr (congr rfl ?_) ?_ <;> ext <;> simp [h0, mem_support_iff]","premises":[{"full_name":"Polynomial.mem_support_iff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[607,8],"def_end_pos":[607,23]},{"full_name":"congr","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[372,8],"def_end_pos":[372,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/NumberTheory/Modular.lean","commit":"","full_name":"ModularGroup.normSq_S_smul_lt_one","start":[352,0],"end":[354,77],"file_path":"Mathlib/NumberTheory/Modular.lean","tactics":[{"state_before":"g : SL(2, ℤ)\nz : ℍ\nh : 1 < normSq ↑z\n⊢ normSq ↑(S • z) < 1","state_after":"no goals","tactic":"simpa [coe_S, num, denom] using (inv_lt_inv z.normSq_pos zero_lt_one).mpr h","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"ModularGroup.coe_S","def_path":"Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean","def_pos":[463,8],"def_end_pos":[463,13]},{"full_name":"UpperHalfPlane.denom","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[180,4],"def_end_pos":[180,9]},{"full_name":"UpperHalfPlane.normSq_pos","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[164,8],"def_end_pos":[164,18]},{"full_name":"UpperHalfPlane.num","def_path":"Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean","def_pos":[175,4],"def_end_pos":[175,7]},{"full_name":"inv_lt_inv","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[167,8],"def_end_pos":[167,18]},{"full_name":"zero_lt_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[34,14],"def_end_pos":[34,25]}]}]}
{"url":"Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean","commit":"","full_name":"minpolyDiv_eval_eq_zero_of_ne_of_aeval_eq_zero","start":[62,0],"end":[66,47],"file_path":"Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean","tactics":[{"state_before":"R : Type u_2\nK : Type ?u.30406\nL : Type ?u.30409\nS : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra K L\nx : S\nhx : IsIntegral R x\ninst✝ : IsDomain S\ny : S\nhxy : y ≠ x\nhy : (aeval y) (minpoly R x) = 0\n⊢ eval y (minpolyDiv R x) = 0","state_after":"R : Type u_2\nK : Type ?u.30406\nL : Type ?u.30409\nS : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra K L\nx : S\nhx : IsIntegral R x\ninst✝ : IsDomain S\ny : S\nhxy : y ≠ x\nhy : eval y (minpolyDiv R x * (X - C x)) = 0\n⊢ eval y (minpolyDiv R x) = 0","tactic":"rw [aeval_def, ← eval_map, ← minpolyDiv_spec R x] at hy","premises":[{"full_name":"Polynomial.aeval_def","def_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","def_pos":[242,8],"def_end_pos":[242,17]},{"full_name":"Polynomial.eval_map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[807,8],"def_end_pos":[807,16]},{"full_name":"minpolyDiv_spec","def_path":"Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean","def_pos":[31,6],"def_end_pos":[31,21]}]},{"state_before":"R : Type u_2\nK : Type ?u.30406\nL : Type ?u.30409\nS : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra K L\nx : S\nhx : IsIntegral R x\ninst✝ : IsDomain S\ny : S\nhxy : y ≠ x\nhy : eval y (minpolyDiv R x * (X - C x)) = 0\n⊢ eval y (minpolyDiv R x) = 0","state_after":"R : Type u_2\nK : Type ?u.30406\nL : Type ?u.30409\nS : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra K L\nx : S\nhx : IsIntegral R x\ninst✝ : IsDomain S\ny : S\nhxy : y ≠ x\nhy : eval y (minpolyDiv R x) = 0 ∨ y - x = 0\n⊢ eval y (minpolyDiv R x) = 0","tactic":"simp only [eval_mul, eval_sub, eval_X, eval_C, mul_eq_zero] at hy","premises":[{"full_name":"Polynomial.eval_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[308,8],"def_end_pos":[308,14]},{"full_name":"Polynomial.eval_X","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[324,8],"def_end_pos":[324,14]},{"full_name":"Polynomial.eval_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[938,8],"def_end_pos":[938,16]},{"full_name":"Polynomial.eval_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1119,8],"def_end_pos":[1119,16]},{"full_name":"mul_eq_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[248,8],"def_end_pos":[248,19]}]},{"state_before":"R : Type u_2\nK : Type ?u.30406\nL : Type ?u.30409\nS : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra K L\nx : S\nhx : IsIntegral R x\ninst✝ : IsDomain S\ny : S\nhxy : y ≠ x\nhy : eval y (minpolyDiv R x) = 0 ∨ y - x = 0\n⊢ eval y (minpolyDiv R x) = 0","state_after":"no goals","tactic":"exact hy.resolve_right (by rwa [sub_eq_zero])","premises":[{"full_name":"Or.resolve_right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[556,8],"def_end_pos":[556,24]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]}]}
{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Int.map_floor","start":[1443,0],"end":[1444,60],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : LinearOrderedRing α\ninst✝⁴ : LinearOrderedRing β\ninst✝³ : FloorRing α\ninst✝² : FloorRing β\ninst✝¹ : FunLike F α β\ninst✝ : RingHomClass F α β\na✝ : α\nb : β\nf : F\nhf : StrictMono ⇑f\na : α\nn : ℤ\n⊢ ↑n ≤ f a ↔ ↑n ≤ a","state_after":"no goals","tactic":"rw [← map_intCast f, hf.le_iff_le]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"StrictMono.le_iff_le","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[725,8],"def_end_pos":[725,28]},{"full_name":"map_intCast","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[359,8],"def_end_pos":[359,19]}]}]}
{"url":"Mathlib/LinearAlgebra/Semisimple.lean","commit":"","full_name":"Module.End.isSemisimple_neg","start":[77,0],"end":[77,97],"file_path":"Mathlib/LinearAlgebra/Semisimple.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\n⊢ (-f).IsSemisimple ↔ f.IsSemisimple","state_after":"no goals","tactic":"simp [isSemisimple_iff]","premises":[{"full_name":"Module.End.isSemisimple_iff","def_path":"Mathlib/LinearAlgebra/Semisimple.lean","def_pos":[61,6],"def_end_pos":[61,22]}]}]}
{"url":"Mathlib/Data/Sym/Sym2.lean","commit":"","full_name":"Sym2.eq_iff","start":[169,0],"end":[170,6],"file_path":"Mathlib/Data/Sym/Sym2.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nx y z w : α\n⊢ s(x, y) = s(z, w) ↔ x = z ∧ y = w ∨ x = w ∧ y = z","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/FieldTheory/PurelyInseparable.lean","commit":"","full_name":"isPurelyInseparable_iff_minpoly_eq_X_sub_C_pow","start":[462,0],"end":[469,53],"file_path":"Mathlib/FieldTheory/PurelyInseparable.lean","tactics":[{"state_before":"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\nq : ℕ\nhF : ExpChar F q\n⊢ IsPurelyInseparable F E ↔ ∀ (x : E), ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n","state_after":"no goals","tactic":"simp_rw [isPurelyInseparable_iff_natSepDegree_eq_one,\n    minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow q]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"isPurelyInseparable_iff_natSepDegree_eq_one","def_path":"Mathlib/FieldTheory/PurelyInseparable.lean","def_pos":[441,8],"def_end_pos":[441,51]},{"full_name":"minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[619,8],"def_end_pos":[619,46]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","commit":"","full_name":"MeasureTheory.measure_eq_iInf'","start":[155,0],"end":[160,53],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ μ s = ⨅ t, μ ↑t","state_after":"no goals","tactic":"simp_rw [iInf_subtype, iInf_and, ← measure_eq_iInf]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.measure_eq_iInf","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[152,8],"def_end_pos":[152,23]},{"full_name":"iInf_and","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1088,8],"def_end_pos":[1088,16]},{"full_name":"iInf_subtype","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[976,8],"def_end_pos":[976,20]}]}]}
{"url":"Mathlib/Algebra/Homology/HomologicalBicomplex.lean","commit":"","full_name":"HomologicalComplex₂.flip_d_f","start":[144,0],"end":[156,30],"file_path":"Mathlib/Algebra/Homology/HomologicalBicomplex.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{?u.12556, u_1} C\ninst✝ : HasZeroMorphisms C\nI₁ : Type u_2\nI₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni i' : I₂\nw : ¬c₂.Rel i i'\n⊢ (fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i' = 0","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{?u.12556, u_1} C\ninst✝ : HasZeroMorphisms C\nI₁ : Type u_2\nI₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni i' : I₂\nw : ¬c₂.Rel i i'\nj : I₁\n⊢ ((fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i').f j = Hom.f 0 j","tactic":"ext j","premises":[]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{?u.12556, u_1} C\ninst✝ : HasZeroMorphisms C\nI₁ : Type u_2\nI₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni i' : I₂\nw : ¬c₂.Rel i i'\nj : I₁\n⊢ ((fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i').f j = Hom.f 0 j","state_after":"no goals","tactic":"exact (K.X j).shape i i' w","premises":[{"full_name":"HomologicalComplex.X","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[56,2],"def_end_pos":[56,3]},{"full_name":"HomologicalComplex.shape","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[58,2],"def_end_pos":[58,7]}]}]}
{"url":"Mathlib/NumberTheory/PythagoreanTriples.lean","commit":"","full_name":"_private.Mathlib.NumberTheory.PythagoreanTriples.0.coprime_sq_sub_mul_of_even_odd","start":[338,0],"end":[369,45],"file_path":"Mathlib/NumberTheory/PythagoreanTriples.lean","tactics":[{"state_before":"m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\n⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1","state_after":"m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\n⊢ False","tactic":"by_contra H","premises":[{"full_name":"Decidable.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[880,8],"def_end_pos":[880,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\n⊢ False","state_after":"case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : p ∣ (m ^ 2 - n ^ 2).natAbs\nhp2 : p ∣ (2 * m * n).natAbs\n⊢ False","tactic":"obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp H","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.Prime.not_coprime_iff_dvd","def_path":"Mathlib/Data/Nat/Prime/Basic.lean","def_pos":[121,8],"def_end_pos":[121,33]}]},{"state_before":"case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : p ∣ (m ^ 2 - n ^ 2).natAbs\nhp2 : p ∣ (2 * m * n).natAbs\n⊢ False","state_after":"case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ False","tactic":"rw [← Int.natCast_dvd] at hp1 hp2","premises":[{"full_name":"Int.natCast_dvd","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[458,6],"def_end_pos":[458,17]}]},{"state_before":"case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ False","state_after":"case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\n⊢ False","tactic":"have hnp : ¬(p : ℤ) ∣ Int.gcd m n := by\n    rw [h]\n    norm_cast\n    exact mt Nat.dvd_one.mp (Nat.Prime.ne_one hp)","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.gcd","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Gcd.lean","def_pos":[20,4],"def_end_pos":[20,7]},{"full_name":"Nat.Prime.ne_one","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[67,8],"def_end_pos":[67,20]},{"full_name":"Nat.dvd_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[122,16],"def_end_pos":[122,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]},{"state_before":"case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\n⊢ False","state_after":"case intro.intro.intro.inl\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\nhp2m : p ∣ (2 * m).natAbs\n⊢ False\n\ncase intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\nhpn : p ∣ n.natAbs\n⊢ False","tactic":"cases' Int.Prime.dvd_mul hp hp2 with hp2m hpn","premises":[{"full_name":"Int.Prime.dvd_mul","def_path":"Mathlib/RingTheory/Int/Basic.lean","def_pos":[82,8],"def_end_pos":[82,25]}]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\nhpn : p ∣ n.natAbs\n⊢ False","state_after":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ False","tactic":"rw [Int.gcd_comm] at hnp","premises":[{"full_name":"Int.gcd_comm","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[191,8],"def_end_pos":[191,16]}]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ False","state_after":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m","tactic":"apply mt (Int.dvd_gcd (Int.natCast_dvd.mpr hpn)) hnp","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.dvd_gcd","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[184,8],"def_end_pos":[184,15]},{"full_name":"Int.natCast_dvd","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[458,6],"def_end_pos":[458,17]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m","state_after":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m ∨ ↑p ∣ m","tactic":"apply or_self_iff.mp","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"or_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[74,8],"def_end_pos":[74,19]}]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m ∨ ↑p ∣ m","state_after":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m * m","tactic":"apply Int.Prime.dvd_mul' hp","premises":[{"full_name":"Int.Prime.dvd_mul'","def_path":"Mathlib/RingTheory/Int/Basic.lean","def_pos":[86,8],"def_end_pos":[86,26]}]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m * m","state_after":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m ^ 2 - n ^ 2 + n * n","tactic":"rw [(by ring : m * m = m ^ 2 - n ^ 2 + n * n)]","premises":[]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m ^ 2 - n ^ 2 + n * n","state_after":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ n * n","tactic":"apply dvd_add hp1","premises":[{"full_name":"dvd_add","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[44,8],"def_end_pos":[44,15]}]},{"state_before":"case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ n * n","state_after":"no goals","tactic":"exact (Int.natCast_dvd.mpr hpn).mul_right n","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.natCast_dvd","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[458,6],"def_end_pos":[458,17]}]}]}
{"url":"Mathlib/RingTheory/Smooth/Basic.lean","commit":"","full_name":"Algebra.FormallySmooth.of_equiv","start":[138,0],"end":[143,19],"file_path":"Mathlib/RingTheory/Smooth/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝⁵ : CommSemiring R\nA B : Type u\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\ne : A ≃ₐ[R] B\n⊢ FormallySmooth R B","state_after":"case comp_surjective\nR : Type u\ninst✝⁵ : CommSemiring R\nA B : Type u\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\ne : A ≃ₐ[R] B\n⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),\n    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp","tactic":"constructor","premises":[]},{"state_before":"case comp_surjective\nR : Type u\ninst✝⁵ : CommSemiring R\nA B : Type u\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\ne : A ≃ₐ[R] B\n⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),\n    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp","state_after":"case comp_surjective\nR : Type u\ninst✝⁷ : CommSemiring R\nA B : Type u\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : FormallySmooth R A\ne : A ≃ₐ[R] B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","tactic":"intro C _ _ I hI f","premises":[]},{"state_before":"case comp_surjective\nR : Type u\ninst✝⁷ : CommSemiring R\nA B : Type u\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : FormallySmooth R A\ne : A ≃ₐ[R] B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f","state_after":"case h\nR : Type u\ninst✝⁷ : CommSemiring R\nA B : Type u\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : FormallySmooth R A\ne : A ≃ₐ[R] B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\n⊢ (Ideal.Quotient.mkₐ R I).comp ((lift I ⋯ (f.comp ↑e)).comp ↑e.symm) = f","tactic":"use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm","premises":[{"full_name":"AlgEquiv.symm","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[275,4],"def_end_pos":[275,8]},{"full_name":"AlgHom","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[28,10],"def_end_pos":[28,16]},{"full_name":"AlgHom.comp","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[274,4],"def_end_pos":[274,8]},{"full_name":"Algebra.FormallySmooth.lift","def_path":"Mathlib/RingTheory/Smooth/Basic.lean","def_pos":[89,18],"def_end_pos":[89,22]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"HasQuotient.Quotient","def_path":"Mathlib/Algebra/Quotient.lean","def_pos":[56,7],"def_end_pos":[56,27]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nR : Type u\ninst✝⁷ : CommSemiring R\nA B : Type u\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : FormallySmooth R A\ne : A ≃ₐ[R] B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\n⊢ (Ideal.Quotient.mkₐ R I).comp ((lift I ⋯ (f.comp ↑e)).comp ↑e.symm) = f","state_after":"no goals","tactic":"rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,\n    AlgHom.comp_id]","premises":[{"full_name":"AlgEquiv.comp_symm","def_path":"Mathlib/Algebra/Algebra/Equiv.lean","def_pos":[374,8],"def_end_pos":[374,17]},{"full_name":"AlgHom.comp_assoc","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[297,8],"def_end_pos":[297,18]},{"full_name":"AlgHom.comp_id","def_path":"Mathlib/Algebra/Algebra/Hom.lean","def_pos":[290,8],"def_end_pos":[290,15]},{"full_name":"Algebra.FormallySmooth.comp_lift","def_path":"Mathlib/RingTheory/Smooth/Basic.lean","def_pos":[94,8],"def_end_pos":[94,17]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","commit":"","full_name":"CategoryTheory.CostructuredArrow.mkPrecomp_id","start":[497,0],"end":[497,97],"file_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nf : S.obj Y ⟶ T\n⊢ mk (S.map (𝟙 Y) ≫ f) = mk f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nf : S.obj Y ⟶ T\n⊢ mkPrecomp f (𝟙 Y) = eqToHom ⋯","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/FieldTheory/KummerExtension.lean","commit":"","full_name":"root_X_pow_sub_C_eq_zero_iff","start":[120,0],"end":[126,57],"file_path":"Mathlib/FieldTheory/KummerExtension.lean","tactics":[{"state_before":"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\n⊢ root (X ^ n - C a) = 0 ↔ a = 0","state_after":"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nhn : 0 < n\n⊢ root (X ^ n - C a) = 0 ↔ a = 0","tactic":"have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.pos_iff_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[516,18],"def_end_pos":[516,33]},{"full_name":"ne_zero_of_irreducible_X_pow_sub_C","def_path":"Mathlib/FieldTheory/KummerExtension.lean","def_pos":[108,6],"def_end_pos":[108,40]}]},{"state_before":"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nhn : 0 < n\n⊢ root (X ^ n - C a) = 0 ↔ a = 0","state_after":"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nhn : 0 < n\n⊢ a = 0 → root (X ^ n - C a) = 0","tactic":"refine ⟨not_imp_not.mp (root_X_pow_sub_C_ne_zero' hn), ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"not_imp_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[290,8],"def_end_pos":[290,19]},{"full_name":"root_X_pow_sub_C_ne_zero'","def_path":"Mathlib/FieldTheory/KummerExtension.lean","def_pos":[63,6],"def_end_pos":[63,31]}]},{"state_before":"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nhn : 0 < n\n⊢ a = 0 → root (X ^ n - C a) = 0","state_after":"K : Type u\ninst✝ : Field K\nn : ℕ\nhn : 0 < n\nH : Irreducible (X ^ n - C 0)\n⊢ root (X ^ n - C 0) = 0","tactic":"rintro rfl","premises":[]},{"state_before":"K : Type u\ninst✝ : Field K\nn : ℕ\nhn : 0 < n\nH : Irreducible (X ^ n - C 0)\n⊢ root (X ^ n - C 0) = 0","state_after":"K : Type u\ninst✝ : Field K\nn : ℕ\nhn : 0 < n\nH : Irreducible (X ^ n - C 0)\nthis : n = 1\n⊢ root (X ^ n - C 0) = 0","tactic":"have := not_imp_not.mp (fun hn ↦ ne_zero_of_irreducible_X_pow_sub_C' hn H) rfl","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"ne_zero_of_irreducible_X_pow_sub_C'","def_path":"Mathlib/FieldTheory/KummerExtension.lean","def_pos":[114,6],"def_end_pos":[114,41]},{"full_name":"not_imp_not","def_path":"Mathlib/Logic/Basic.lean","def_pos":[290,8],"def_end_pos":[290,19]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"K : Type u\ninst✝ : Field K\nn : ℕ\nhn : 0 < n\nH : Irreducible (X ^ n - C 0)\nthis : n = 1\n⊢ root (X ^ n - C 0) = 0","state_after":"no goals","tactic":"rw [this, pow_one, map_zero, sub_zero, ← mk_X, mk_self]","premises":[{"full_name":"AdjoinRoot.mk_X","def_path":"Mathlib/RingTheory/AdjoinRoot.lean","def_pos":[187,8],"def_end_pos":[187,12]},{"full_name":"AdjoinRoot.mk_self","def_path":"Mathlib/RingTheory/AdjoinRoot.lean","def_pos":[179,8],"def_end_pos":[179,15]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]},{"full_name":"pow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[571,6],"def_end_pos":[571,13]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]}]}
{"url":"Mathlib/Analysis/NormedSpace/Exponential.lean","commit":"","full_name":"NormedSpace.expSeries_apply_eq","start":[115,0],"end":[116,77],"file_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","tactics":[{"state_before":"𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : Field 𝕂\ninst✝³ : Ring 𝔸\ninst✝² : Algebra 𝕂 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\nn : ℕ\n⊢ ((expSeries 𝕂 𝔸 n) fun x_1 => x) = (↑n !)⁻¹ • x ^ n","state_after":"no goals","tactic":"simp [expSeries]","premises":[{"full_name":"NormedSpace.expSeries","def_path":"Mathlib/Analysis/NormedSpace/Exponential.lean","def_pos":[99,4],"def_end_pos":[99,13]}]}]}
{"url":"Mathlib/Algebra/Star/Unitary.lean","commit":"","full_name":"unitary.map_mem","start":[152,0],"end":[154,77],"file_path":"Mathlib/Algebra/Star/Unitary.lean","tactics":[{"state_before":"R✝ : Type u_1\nF : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁶ : Monoid R\ninst✝⁵ : StarMul R\ninst✝⁴ : Monoid S\ninst✝³ : StarMul S\ninst✝² : FunLike F R S\ninst✝¹ : StarHomClass F R S\ninst✝ : MonoidHomClass F R S\nf : F\nr : R\nhr : r ∈ unitary R\n⊢ f r ∈ unitary S","state_after":"R✝ : Type u_1\nF : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁶ : Monoid R\ninst✝⁵ : StarMul R\ninst✝⁴ : Monoid S\ninst✝³ : StarMul S\ninst✝² : FunLike F R S\ninst✝¹ : StarHomClass F R S\ninst✝ : MonoidHomClass F R S\nf : F\nr : R\nhr : star r * r = 1 ∧ r * star r = 1\n⊢ f r ∈ unitary S","tactic":"rw [unitary.mem_iff] at hr","premises":[{"full_name":"unitary.mem_iff","def_path":"Mathlib/Algebra/Star/Unitary.lean","def_pos":[50,8],"def_end_pos":[50,15]}]},{"state_before":"R✝ : Type u_1\nF : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁶ : Monoid R\ninst✝⁵ : StarMul R\ninst✝⁴ : Monoid S\ninst✝³ : StarMul S\ninst✝² : FunLike F R S\ninst✝¹ : StarHomClass F R S\ninst✝ : MonoidHomClass F R S\nf : F\nr : R\nhr : star r * r = 1 ∧ r * star r = 1\n⊢ f r ∈ unitary S","state_after":"no goals","tactic":"simpa [map_star, map_mul] using And.intro congr(f $(hr.1)) congr(f $(hr.2))","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"StarHomClass.map_star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[435,2],"def_end_pos":[435,10]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","commit":"","full_name":"InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero","start":[194,0],"end":[201,100],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖y‖ / Real.tan (angle x (x + y)) = ‖x‖","state_after":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖","tactic":"rw [tan_angle_add_of_inner_eq_zero h]","premises":[{"full_name":"InnerProductGeometry.tan_angle_add_of_inner_eq_zero","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[140,8],"def_end_pos":[140,38]}]},{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖","state_after":"case inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0\n⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖\n\ncase inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : y ≠ 0\n⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖","tactic":"rcases h0 with (h0 | h0)","premises":[]}]}
{"url":"Mathlib/Data/Fin/Tuple/Basic.lean","commit":"","full_name":"Fin.removeNth_zero","start":[766,0],"end":[767,29],"file_path":"Mathlib/Data/Fin/Tuple/Basic.lean","tactics":[{"state_before":"m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nf : (i : Fin (n + 1)) → α i\n⊢ removeNth 0 f = tail f","state_after":"case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nf : (i : Fin (n + 1)) → α i\nx✝ : Fin n\n⊢ removeNth 0 f x✝ = tail f x✝","tactic":"ext","premises":[]},{"state_before":"case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nf : (i : Fin (n + 1)) → α i\nx✝ : Fin n\n⊢ removeNth 0 f x✝ = tail f x✝","state_after":"no goals","tactic":"simp [tail, removeNth]","premises":[{"full_name":"Fin.removeNth","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[732,4],"def_end_pos":[732,13]},{"full_name":"Fin.tail","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[101,4],"def_end_pos":[101,8]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","commit":"","full_name":"MeasureTheory.prob_compl_le_one_sub_of_le_prob","start":[270,0],"end":[272,68],"file_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν ν₁ ν₂ : Measure α\ns t : Set α\ninst✝ : IsProbabilityMeasure μ\np✝ : α → Prop\nf : β → α\np : ℝ≥0∞\nhμs : p ≤ μ s\ns_mble : MeasurableSet s\n⊢ μ sᶜ ≤ 1 - p","state_after":"no goals","tactic":"simpa [prob_compl_eq_one_sub s_mble] using tsub_le_tsub_left hμs 1","premises":[{"full_name":"MeasureTheory.prob_compl_eq_one_sub","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[260,8],"def_end_pos":[260,29]},{"full_name":"tsub_le_tsub_left","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[110,18],"def_end_pos":[110,35]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","commit":"","full_name":"InnerProductGeometry.angle_add_pos_of_inner_eq_zero","start":[95,0],"end":[103,21],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","tactics":[{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ 0 < angle x (x + y)","state_after":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1","tactic":"rw [angle_add_eq_arccos_of_inner_eq_zero h, Real.arccos_pos,\n    norm_add_eq_sqrt_iff_real_inner_eq_zero.2 h]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean","def_pos":[63,8],"def_end_pos":[63,44]},{"full_name":"Real.arccos_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean","def_pos":[299,8],"def_end_pos":[299,18]},{"full_name":"norm_add_eq_sqrt_iff_real_inner_eq_zero","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1261,8],"def_end_pos":[1261,47]}]},{"state_before":"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1","state_after":"case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : x = 0\n⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1\n\ncase neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : ¬x = 0\n⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1","tactic":"by_cases hx : x = 0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : ¬x = 0\n⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1","state_after":"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : ¬x = 0\n⊢ ‖x‖ * ‖x‖ < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖","tactic":"rw [div_lt_one (Real.sqrt_pos.2 (Left.add_pos_of_pos_of_nonneg (mul_self_pos.2\n    (norm_ne_zero_iff.2 hx)) (mul_self_nonneg _))), Real.lt_sqrt (norm_nonneg _), pow_two]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Left.add_pos_of_pos_of_nonneg","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[647,14],"def_end_pos":[647,43]},{"full_name":"Real.lt_sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[348,8],"def_end_pos":[348,15]},{"full_name":"Real.sqrt_pos","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[261,8],"def_end_pos":[261,16]},{"full_name":"div_lt_one","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[303,8],"def_end_pos":[303,18]},{"full_name":"mul_self_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[909,6],"def_end_pos":[909,21]},{"full_name":"mul_self_pos","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[852,8],"def_end_pos":[852,20]},{"full_name":"norm_ne_zero_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1212,14],"def_end_pos":[1212,30]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]}]},{"state_before":"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : ¬x = 0\n⊢ ‖x‖ * ‖x‖ < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖","state_after":"no goals","tactic":"simpa [hx] using h0","premises":[]}]}
{"url":"Mathlib/RingTheory/Jacobson.lean","commit":"","full_name":"Ideal.Polynomial.isJacobson_polynomial_iff_isJacobson","start":[431,0],"end":[435,70],"file_path":"Mathlib/RingTheory/Jacobson.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : IsDomain S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹ : CommRing Rₘ\ninst✝ : CommRing Sₘ\n⊢ IsJacobson R[X] ↔ IsJacobson R","state_after":"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : IsDomain S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹ : CommRing Rₘ\ninst✝ : CommRing Sₘ\n⊢ IsJacobson R[X] → IsJacobson R","tactic":"refine ⟨?_, isJacobson_polynomial_of_isJacobson⟩","premises":[{"full_name":"Ideal.Polynomial.isJacobson_polynomial_of_isJacobson","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[403,8],"def_end_pos":[403,43]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : IsDomain S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹ : CommRing Rₘ\ninst✝ : CommRing Sₘ\n⊢ IsJacobson R[X] → IsJacobson R","state_after":"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : IsDomain S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹ : CommRing Rₘ\ninst✝ : CommRing Sₘ\nH : IsJacobson R[X]\n⊢ IsJacobson R","tactic":"intro H","premises":[]},{"state_before":"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : IsDomain S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹ : CommRing Rₘ\ninst✝ : CommRing Sₘ\nH : IsJacobson R[X]\n⊢ IsJacobson R","state_after":"no goals","tactic":"exact isJacobson_of_surjective ⟨eval₂RingHom (RingHom.id _) 1, fun x =>\n    ⟨C x, by simp only [coe_eval₂RingHom, RingHom.id_apply, eval₂_C]⟩⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.isJacobson_of_surjective","def_path":"Mathlib/RingTheory/Jacobson.lean","def_pos":[96,8],"def_end_pos":[96,32]},{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.coe_eval₂RingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[241,8],"def_end_pos":[241,24]},{"full_name":"Polynomial.eval₂RingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[235,4],"def_end_pos":[235,16]},{"full_name":"Polynomial.eval₂_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[61,8],"def_end_pos":[61,15]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"RingHom.id_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[549,8],"def_end_pos":[549,16]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","commit":"","full_name":"CategoryTheory.Limits.coprod.map_swap","start":[766,0],"end":[769,96],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ Y✝ A B X Y : C\nf : A ⟶ B\ng : X ⟶ Y\ninst✝ : HasColimitsOfShape (Discrete WalkingPair) C\n⊢ map (𝟙 X) f ≫ map g (𝟙 B) = map g (𝟙 A) ≫ map (𝟙 Y) f","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean","commit":"","full_name":"LinearMap.BilinForm.orthogonal_orthogonal","start":[317,0],"end":[322,7],"file_path":"Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\n⊢ B.orthogonal (B.orthogonal W) = W","state_after":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\n⊢ finrank K ↥(B.orthogonal (B.orthogonal W)) ≤ finrank K ↥W","tactic":"apply (eq_of_le_of_finrank_le (LinearMap.BilinForm.le_orthogonal_orthogonal hB₀) _).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FiniteDimensional.eq_of_le_of_finrank_le","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[461,8],"def_end_pos":[461,30]},{"full_name":"LinearMap.BilinForm.le_orthogonal_orthogonal","def_path":"Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean","def_pos":[148,8],"def_end_pos":[148,32]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\n⊢ finrank K ↥(B.orthogonal (B.orthogonal W)) ≤ finrank K ↥W","state_after":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\n⊢ finrank K V - (finrank K V - finrank K ↥W) ≤ finrank K ↥W","tactic":"simp only [finrank_orthogonal hB hB₀]","premises":[{"full_name":"LinearMap.BilinForm.finrank_orthogonal","def_path":"Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean","def_pos":[310,6],"def_end_pos":[310,24]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\n⊢ finrank K V - (finrank K V - finrank K ↥W) ≤ finrank K ↥W","state_after":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\nthis : finrank K ↥W ≤ finrank K V\n⊢ finrank K V - (finrank K V - finrank K ↥W) ≤ finrank K ↥W","tactic":"have : finrank K W ≤ finrank K V := finrank_le W","premises":[{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"Submodule.finrank_le","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[374,8],"def_end_pos":[374,28]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM₂' : Type u_7\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R M₂'\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nhB₀ : B.IsRefl\nW : Submodule K V\nthis : finrank K ↥W ≤ finrank K V\n⊢ finrank K V - (finrank K V - finrank K ↥W) ≤ finrank K ↥W","state_after":"no goals","tactic":"omega","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/QuadraticForm/Prod.lean","commit":"","full_name":"QuadraticMap.anisotropic_of_prod","start":[134,0],"end":[145,31],"file_path":"Mathlib/LinearAlgebra/QuadraticForm/Prod.lean","tactics":[{"state_before":"ι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : (Q₁.prod Q₂).Anisotropic\n⊢ Q₁.Anisotropic ∧ Q₂.Anisotropic","state_after":"ι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Q₁.Anisotropic ∧ Q₂.Anisotropic","tactic":"simp_rw [Anisotropic, prod_apply, Prod.forall, Prod.mk_eq_zero] at h","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Prod.forall","def_path":"Mathlib/Data/Prod/Basic.lean","def_pos":[28,8],"def_end_pos":[28,16]},{"full_name":"Prod.mk_eq_zero","def_path":"Mathlib/Algebra/Group/Prod.lean","def_pos":[92,2],"def_end_pos":[92,13]},{"full_name":"QuadraticMap.Anisotropic","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[1006,4],"def_end_pos":[1006,15]},{"full_name":"QuadraticMap.prod_apply","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Prod.lean","def_pos":[52,2],"def_end_pos":[52,8]}]},{"state_before":"ι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Q₁.Anisotropic ∧ Q₂.Anisotropic","state_after":"case left\nι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Q₁.Anisotropic\n\ncase right\nι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0\n⊢ Q₂.Anisotropic","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Data/Sym/Sym2.lean","commit":"","full_name":"Sym2.other_spec","start":[345,0],"end":[347,33],"file_path":"Mathlib/Data/Sym/Sym2.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\na : α\nz : Sym2 α\nh : a ∈ z\n⊢ s(a, Mem.other h) = z","state_after":"no goals","tactic":"erw [← Classical.choose_spec h]","premises":[{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
{"url":"Mathlib/Data/Finsupp/Order.lean","commit":"","full_name":"Finsupp.support_sup","start":[254,0],"end":[258,40],"file_path":"Mathlib/Data/Finsupp/Order.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CanonicallyLinearOrderedAddCommMonoid α\ninst✝ : DecidableEq ι\nf g : ι →₀ α\n⊢ (f ⊔ g).support = f.support ∪ g.support","state_after":"case a\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CanonicallyLinearOrderedAddCommMonoid α\ninst✝ : DecidableEq ι\nf g : ι →₀ α\na✝ : ι\n⊢ a✝ ∈ (f ⊔ g).support ↔ a✝ ∈ f.support ∪ g.support","tactic":"ext","premises":[]},{"state_before":"case a\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CanonicallyLinearOrderedAddCommMonoid α\ninst✝ : DecidableEq ι\nf g : ι →₀ α\na✝ : ι\n⊢ a✝ ∈ (f ⊔ g).support ↔ a✝ ∈ f.support ∪ g.support","state_after":"case a\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CanonicallyLinearOrderedAddCommMonoid α\ninst✝ : DecidableEq ι\nf g : ι →₀ α\na✝ : ι\n⊢ ¬f a✝ ⊔ g a✝ = ⊥ ↔ ¬f a✝ = ⊥ ∨ ¬g a✝ = ⊥","tactic":"simp only [Finset.mem_union, mem_support_iff, sup_apply, Ne, ← bot_eq_zero]","premises":[{"full_name":"Finset.mem_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1191,8],"def_end_pos":[1191,17]},{"full_name":"Finsupp.mem_support_iff","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[150,8],"def_end_pos":[150,23]},{"full_name":"Finsupp.sup_apply","def_path":"Mathlib/Data/Finsupp/Order.lean","def_pos":[100,8],"def_end_pos":[100,17]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"bot_eq_zero","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[109,2],"def_end_pos":[109,13]}]},{"state_before":"case a\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : CanonicallyLinearOrderedAddCommMonoid α\ninst✝ : DecidableEq ι\nf g : ι →₀ α\na✝ : ι\n⊢ ¬f a✝ ⊔ g a✝ = ⊥ ↔ ¬f a✝ = ⊥ ∨ ¬g a✝ = ⊥","state_after":"no goals","tactic":"rw [_root_.sup_eq_bot_iff, not_and_or]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]},{"full_name":"sup_eq_bot_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[365,8],"def_end_pos":[365,22]}]}]}
{"url":"Mathlib/NumberTheory/MulChar/Basic.lean","commit":"","full_name":"MulChar.IsQuadratic.comp","start":[479,0],"end":[483,47],"file_path":"Mathlib/NumberTheory/MulChar/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nf : R' →+* R''\n⊢ (χ.ringHomComp f).IsQuadratic","state_after":"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nf : R' →+* R''\na : R\n⊢ (χ.ringHomComp f) a = 0 ∨ (χ.ringHomComp f) a = 1 ∨ (χ.ringHomComp f) a = -1","tactic":"intro a","premises":[]},{"state_before":"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nf : R' →+* R''\na : R\n⊢ (χ.ringHomComp f) a = 0 ∨ (χ.ringHomComp f) a = 1 ∨ (χ.ringHomComp f) a = -1","state_after":"no goals","tactic":"rcases hχ a with (ha | ha | ha) <;> simp [ha]","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/EraseLead.lean","commit":"","full_name":"Polynomial.card_support_eraseLead'","start":[113,0],"end":[115,56],"file_path":"Mathlib/Algebra/Polynomial/EraseLead.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nc : ℕ\nfc : f.support.card = c + 1\n⊢ f.eraseLead.support.card = c","state_after":"no goals","tactic":"rw [card_support_eraseLead, fc, add_tsub_cancel_right]","premises":[{"full_name":"Polynomial.card_support_eraseLead","def_path":"Mathlib/Algebra/Polynomial/EraseLead.lean","def_pos":[108,8],"def_end_pos":[108,30]},{"full_name":"add_tsub_cancel_right","def_path":"Mathlib/Algebra/Order/Sub/Defs.lean","def_pos":[305,8],"def_end_pos":[305,29]}]}]}
{"url":"Mathlib/LinearAlgebra/PiTensorProduct.lean","commit":"","full_name":"PiTensorProduct.nonempty_lifts","start":[323,0],"end":[329,19],"file_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","tactics":[{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ x.lifts.Nonempty","state_after":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ Quotient.out x ∈ x.lifts","tactic":"existsi @Quotient.out _ (addConGen (PiTensorProduct.Eqv R s)).toSetoid x","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"PiTensorProduct.Eqv","def_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","def_pos":[83,10],"def_end_pos":[83,13]},{"full_name":"Quotient.out","def_path":"Mathlib/Data/Quot.lean","def_pos":[339,18],"def_end_pos":[339,30]},{"full_name":"addConGen","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[98,14],"def_end_pos":[98,23]}]},{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ Quotient.out x ∈ x.lifts","state_after":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ ↑(Quotient.out x) = x","tactic":"simp only [lifts, Set.mem_setOf_eq]","premises":[{"full_name":"PiTensorProduct.lifts","def_path":"Mathlib/LinearAlgebra/PiTensorProduct.lean","def_pos":[312,4],"def_end_pos":[312,9]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ ↑(Quotient.out x) = x","state_after":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ Quot.mk (⇑(addConGen (Eqv R s))) (Quotient.out x) = x","tactic":"rw [← AddCon.quot_mk_eq_coe]","premises":[{"full_name":"AddCon.quot_mk_eq_coe","def_path":"Mathlib/GroupTheory/Congruence/Basic.lean","def_pos":[242,2],"def_end_pos":[242,13]}]},{"state_before":"ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx : ⨂[R] (i : ι), s i\n⊢ Quot.mk (⇑(addConGen (Eqv R s))) (Quotient.out x) = x","state_after":"no goals","tactic":"erw [Quot.out_eq]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Quot.out_eq","def_path":"Mathlib/Data/Quot.lean","def_pos":[334,8],"def_end_pos":[334,19]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"IntervalIntegrable.comp_add_right","start":[291,0],"end":[300,30],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\n⊢ IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)","state_after":"case inr\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nthis :\n  ∀ {a b : ℝ}, IntervalIntegrable f volume a b → a ≤ b → IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)\nh : ¬a ≤ b\n⊢ IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)\n\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntervalIntegrable f volume a b\nh : a ≤ b\n⊢ IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)","tactic":"wlog h : a ≤ b generalizing a b","premises":[]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntervalIntegrable f volume a b\nh : a ≤ b\n⊢ IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c)","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] volume\nh : a ≤ b\n⊢ IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] volume","tactic":"rw [intervalIntegrable_iff'] at hf ⊢","premises":[{"full_name":"intervalIntegrable_iff'","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[92,8],"def_end_pos":[92,31]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] volume\nh : a ≤ b\n⊢ IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] volume","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] volume\nh : a ≤ b\nA : MeasurableEmbedding fun x => x + c\n⊢ IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] volume","tactic":"have A : MeasurableEmbedding fun x => x + c :=\n    (Homeomorph.addRight c).closedEmbedding.measurableEmbedding","premises":[{"full_name":"ClosedEmbedding.measurableEmbedding","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[633,18],"def_end_pos":[633,53]},{"full_name":"Homeomorph.addRight","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[84,2],"def_end_pos":[84,13]},{"full_name":"Homeomorph.closedEmbedding","def_path":"Mathlib/Topology/Homeomorph.lean","def_pos":[337,18],"def_end_pos":[337,33]},{"full_name":"MeasurableEmbedding","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean","def_pos":[55,10],"def_end_pos":[55,29]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] volume\nh : a ≤ b\nA : MeasurableEmbedding fun x => x + c\n⊢ IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] volume","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] (Measure.map (fun x => x + c) volume)\nh : a ≤ b\nA : MeasurableEmbedding fun x => x + c\n⊢ IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] volume","tactic":"rw [← map_add_right_eq_self volume c] at hf","premises":[{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"MeasureTheory.map_add_right_eq_self","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[78,2],"def_end_pos":[78,13]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] (Measure.map (fun x => x + c) volume)\nh : a ≤ b\nA : MeasurableEmbedding fun x => x + c\n⊢ IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] volume","state_after":"case h.e'_6\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] (Measure.map (fun x => x + c) volume)\nh : a ≤ b\nA : MeasurableEmbedding fun x => x + c\n⊢ [[a - c, b - c]] = (fun x => x + c) ⁻¹' [[a, b]]","tactic":"convert (MeasurableEmbedding.integrableOn_map_iff A).mp hf using 1","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasurableEmbedding.integrableOn_map_iff","def_path":"Mathlib/MeasureTheory/Integral/IntegrableOn.lean","def_pos":[202,8],"def_end_pos":[202,55]}]},{"state_before":"case h.e'_6\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na✝ b✝ : ℝ\nμ : Measure ℝ\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] (Measure.map (fun x => x + c) volume)\nh : a ≤ b\nA : MeasurableEmbedding fun x => x + c\n⊢ [[a - c, b - c]] = (fun x => x + c) ⁻¹' [[a, b]]","state_after":"no goals","tactic":"rw [preimage_add_const_uIcc]","premises":[{"full_name":"Set.preimage_add_const_uIcc","def_path":"Mathlib/Data/Set/Pointwise/Interval.lean","def_pos":[445,8],"def_end_pos":[445,31]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Cover/Open.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.OpenCover.add_map","start":[153,0],"end":[162,58],"file_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","tactics":[{"state_before":"X✝ Y✝ Z : Scheme\n𝒰✝ : X✝.OpenCover\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝¹ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nX Y : Scheme\n𝒰 : X.OpenCover\nf : Y ⟶ X\ninst✝ : IsOpenImmersion f\n⊢ ∀ (x : Option 𝒰.J), IsOpenImmersion ((fun i => Option.rec f 𝒰.map i) x)","state_after":"no goals","tactic":"rintro (_ | _) <;> dsimp <;> infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean","commit":"","full_name":"Complex.cos_eq_one_iff","start":[99,0],"end":[101,42],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean","tactics":[{"state_before":"x : ℂ\n⊢ cos x = 1 ↔ ∃ k, ↑k * (2 * ↑π) = x","state_after":"x : ℂ\n⊢ (∃ k, x = 2 * ↑k * ↑π + 0 ∨ x = 2 * ↑k * ↑π - 0) ↔ ∃ k, ↑k * (2 * ↑π) = x","tactic":"rw [← cos_zero, eq_comm, cos_eq_cos_iff]","premises":[{"full_name":"Complex.cos_eq_cos_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean","def_pos":[81,8],"def_end_pos":[81,22]},{"full_name":"Complex.cos_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[452,8],"def_end_pos":[452,16]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]},{"state_before":"x : ℂ\n⊢ (∃ k, x = 2 * ↑k * ↑π + 0 ∨ x = 2 * ↑k * ↑π - 0) ↔ ∃ k, ↑k * (2 * ↑π) = x","state_after":"no goals","tactic":"simp [mul_assoc, mul_left_comm, eq_comm]","premises":[{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_left_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[152,8],"def_end_pos":[152,21]}]}]}
{"url":"Mathlib/Analysis/SumOverResidueClass.lean","commit":"","full_name":"not_summable_indicator_mod_of_antitone_of_neg","start":[59,0],"end":[67,93],"file_path":"Mathlib/Analysis/SumOverResidueClass.lean","tactics":[{"state_before":"m : ℕ\nhm : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nk : ZMod m\n⊢ ¬Summable ({n | ↑n = k}.indicator f)","state_after":"m : ℕ\nhm : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nk : ZMod m\n⊢ ¬Summable fun n => f (m * n + k.val)","tactic":"rw [← ZMod.natCast_zmod_val k, summable_indicator_mod_iff_summable]","premises":[{"full_name":"ZMod.natCast_zmod_val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[188,8],"def_end_pos":[188,24]},{"full_name":"summable_indicator_mod_iff_summable","def_path":"Mathlib/Analysis/SumOverResidueClass.lean","def_pos":[30,6],"def_end_pos":[30,41]}]},{"state_before":"m : ℕ\nhm : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nk : ZMod m\n⊢ ¬Summable fun n => f (m * n + k.val)","state_after":"no goals","tactic":"exact not_summable_of_antitone_of_neg\n    (hf.comp_monotone <| (Covariant.monotone_of_const m).add_const k.val) <|\n    (hf <| (Nat.le_mul_of_pos_left n Fin.size_pos').trans <| Nat.le_add_right ..).trans_lt hn","premises":[{"full_name":"Antitone.comp_monotone","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[596,8],"def_end_pos":[596,30]},{"full_name":"Covariant.monotone_of_const","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean","def_pos":[229,8],"def_end_pos":[229,35]},{"full_name":"Fin.size_pos'","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[29,8],"def_end_pos":[29,17]},{"full_name":"Monotone.add_const","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[1073,14],"def_end_pos":[1073,23]},{"full_name":"Nat.le_add_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[395,8],"def_end_pos":[395,20]},{"full_name":"Nat.le_mul_of_pos_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[465,18],"def_end_pos":[465,36]},{"full_name":"ZMod.val","def_path":"Mathlib/Data/ZMod/Basic.lean","def_pos":[45,4],"def_end_pos":[45,7]},{"full_name":"not_summable_of_antitone_of_neg","def_path":"Mathlib/Analysis/SumOverResidueClass.lean","def_pos":[48,6],"def_end_pos":[48,37]}]}]}
{"url":"Mathlib/Topology/ContinuousOn.lean","commit":"","full_name":"continuousWithinAt_iff_continuousAt","start":[756,0],"end":[758,74],"file_path":"Mathlib/Topology/ContinuousOn.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nι : Type u_5\nπ : ι → Type u_6\ninst✝³ : (i : ι) → TopologicalSpace (π i)\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nx : α\nh : s ∈ 𝓝 x\n⊢ ContinuousWithinAt f s x ↔ ContinuousAt f x","state_after":"no goals","tactic":"rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.univ_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[803,8],"def_end_pos":[803,18]},{"full_name":"continuousWithinAt_inter","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[643,8],"def_end_pos":[643,32]},{"full_name":"continuousWithinAt_univ","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[445,8],"def_end_pos":[445,31]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"div_mul_eq_mul_div","start":[557,0],"end":[558,61],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a / b * c = a * c / b","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Dual.lean","commit":"","full_name":"Subspace.dualAnnihilator_dualAnnihilator_eq","start":[1072,0],"end":[1076,32],"file_path":"Mathlib/LinearAlgebra/Dual.lean","tactics":[{"state_before":"K : Type u\nV : Type v\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW✝ : Subspace K V\ninst✝ : FiniteDimensional K V\nW : Subspace K V\n⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W","state_after":"K : Type u\nV : Type v\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW✝ : Subspace K V\ninst✝ : FiniteDimensional K V\nW : Subspace K V\nthis : (dualAnnihilator W).dualCoannihilator = W\n⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W","tactic":"have : _ = W := Subspace.dualAnnihilator_dualCoannihilator_eq","premises":[{"full_name":"Subspace.dualAnnihilator_dualCoannihilator_eq","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[949,8],"def_end_pos":[949,44]}]},{"state_before":"K : Type u\nV : Type v\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW✝ : Subspace K V\ninst✝ : FiniteDimensional K V\nW : Subspace K V\nthis : (dualAnnihilator W).dualCoannihilator = W\n⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W","state_after":"K : Type u\nV : Type v\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW✝ : Subspace K V\ninst✝ : FiniteDimensional K V\nW : Subspace K V\nthis : (Module.mapEvalEquiv K V).symm (dualAnnihilator W).dualAnnihilator = W\n⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W","tactic":"rw [dualCoannihilator, ← Module.mapEvalEquiv_symm_apply] at this","premises":[{"full_name":"Module.mapEvalEquiv_symm_apply","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[609,8],"def_end_pos":[609,31]},{"full_name":"Submodule.dualCoannihilator","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[827,4],"def_end_pos":[827,21]}]},{"state_before":"K : Type u\nV : Type v\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW✝ : Subspace K V\ninst✝ : FiniteDimensional K V\nW : Subspace K V\nthis : (Module.mapEvalEquiv K V).symm (dualAnnihilator W).dualAnnihilator = W\n⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W","state_after":"no goals","tactic":"rwa [← OrderIso.symm_apply_eq]","premises":[{"full_name":"OrderIso.symm_apply_eq","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[771,8],"def_end_pos":[771,21]}]}]}
{"url":"Mathlib/Algebra/Polynomial/RingDivision.lean","commit":"","full_name":"Polynomial.eq_zero_of_dvd_of_degree_lt","start":[151,0],"end":[154,58],"file_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nh₁ : p ∣ q\nh₂ : q.degree < p.degree\n⊢ q = 0","state_after":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nh₁ : p ∣ q\nh₂ : q.degree < p.degree\nhc : ¬q = 0\n⊢ False","tactic":"by_contra hc","premises":[{"full_name":"Classical.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[129,8],"def_end_pos":[129,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nh₁ : p ∣ q\nh₂ : q.degree < p.degree\nhc : ¬q = 0\n⊢ False","state_after":"no goals","tactic":"exact (lt_iff_not_ge _ _).mp h₂ (degree_le_of_dvd h₁ hc)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Polynomial.degree_le_of_dvd","def_path":"Mathlib/Algebra/Polynomial/RingDivision.lean","def_pos":[147,8],"def_end_pos":[147,24]},{"full_name":"lt_iff_not_ge","def_path":"Mathlib/Order/Defs.lean","def_pos":[308,8],"def_end_pos":[308,21]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","commit":"","full_name":"ContinuousLinearMap.hasStrictDerivAt_of_bilinear","start":[60,0],"end":[63,96],"file_path":"Mathlib/Analysis/Calculus/Deriv/Mul.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nB : E →L[𝕜] F →L[𝕜] G\nu : 𝕜 → E\nv : 𝕜 → F\nu' : E\nv' : F\nhu : HasStrictDerivAt u u' x\nhv : HasStrictDerivAt v v' x\n⊢ HasStrictDerivAt (fun x => (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x","state_after":"no goals","tactic":"simpa using\n    (B.hasStrictFDerivAt_of_bilinear hu.hasStrictFDerivAt hv.hasStrictFDerivAt).hasStrictDerivAt","premises":[{"full_name":"ContinuousLinearMap.hasStrictFDerivAt_of_bilinear","def_path":"Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean","def_pos":[129,8],"def_end_pos":[129,57]},{"full_name":"HasStrictFDerivAt.hasStrictDerivAt","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[189,18],"def_end_pos":[189,52]}]}]}
{"url":"Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean","commit":"","full_name":"GenContFract.coe_of_h_rat_eq","start":[192,0],"end":[195,6],"file_path":"Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean","tactics":[{"state_before":"K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ ↑(of q).h = (of v).h","state_after":"K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ ↑(match (IntFractPair.of q, Stream'.Seq.tail ⟨IntFractPair.stream q, ⋯⟩) with\n        | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h =\n    (match (IntFractPair.of v, Stream'.Seq.tail ⟨IntFractPair.stream v, ⋯⟩) with\n      | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h","tactic":"unfold of IntFractPair.seq1","premises":[{"full_name":"GenContFract.IntFractPair.seq1","def_path":"Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean","def_pos":[161,14],"def_end_pos":[161,18]},{"full_name":"GenContFract.of","def_path":"Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean","def_pos":[182,14],"def_end_pos":[182,16]}]},{"state_before":"K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ ↑(match (IntFractPair.of q, Stream'.Seq.tail ⟨IntFractPair.stream q, ⋯⟩) with\n        | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h =\n    (match (IntFractPair.of v, Stream'.Seq.tail ⟨IntFractPair.stream v, ⋯⟩) with\n      | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h","state_after":"K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ ↑(match (IntFractPair.of q, Stream'.Seq.tail ⟨IntFractPair.stream q, ⋯⟩) with\n        | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h =\n    (match (IntFractPair.mapFr Rat.cast (IntFractPair.of q), Stream'.Seq.tail ⟨IntFractPair.stream v, ⋯⟩) with\n      | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h","tactic":"rw [← IntFractPair.coe_of_rat_eq v_eq_q]","premises":[{"full_name":"GenContFract.IntFractPair.coe_of_rat_eq","def_path":"Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean","def_pos":[157,8],"def_end_pos":[157,21]}]},{"state_before":"K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ ↑(match (IntFractPair.of q, Stream'.Seq.tail ⟨IntFractPair.stream q, ⋯⟩) with\n        | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h =\n    (match (IntFractPair.mapFr Rat.cast (IntFractPair.of q), Stream'.Seq.tail ⟨IntFractPair.stream v, ⋯⟩) with\n      | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).h","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","commit":"","full_name":"CategoryTheory.Presheaf.isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective","start":[177,0],"end":[196,64],"file_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","tactic":"let S := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂","premises":[{"full_name":"CategoryTheory.Presheaf.imageSieve","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[45,4],"def_end_pos":[45,14]},{"full_name":"Inf.inf","def_path":"Mathlib/Order/Notation.lean","def_pos":[53,2],"def_end_pos":[53,5]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\nhS : S ∈ J.sieves (unop X)\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","tactic":"have hS : S ∈ J X.unop := by\n      apply J.intersection_covering\n      all_goals apply imageSieve_mem","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.intersection_covering","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[155,8],"def_end_pos":[155,29]},{"full_name":"CategoryTheory.Presheaf.imageSieve_mem","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[86,6],"def_end_pos":[86,20]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\nhS : S ∈ J.sieves (unop X)\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\nhS : S ∈ J.sieves (unop X)\nT : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y :=\n  fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","tactic":"let T : ∀ ⦃Y : C⦄ (f : Y ⟶ X.unop) (_ : S f), Sieve Y := fun Y f hf =>\n      equalizerSieve (localPreimage f₁ x₁ f hf.1) (localPreimage f₁ x₂ f hf.2)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"CategoryTheory.Presheaf.equalizerSieve","def_path":"Mathlib/CategoryTheory/Sites/LocallyInjective.lean","def_pos":[43,4],"def_end_pos":[43,18]},{"full_name":"CategoryTheory.Presheaf.localPreimage","def_path":"Mathlib/CategoryTheory/Sites/LocallySurjective.lean","def_pos":[70,18],"def_end_pos":[70,31]},{"full_name":"CategoryTheory.Sieve","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[245,10],"def_end_pos":[245,15]},{"full_name":"Opposite.unop","def_path":"Mathlib/Data/Opposite.lean","def_pos":[37,2],"def_end_pos":[37,6]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\nhS : S ∈ J.sieves (unop X)\nT : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y :=\n  fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)\n⊢ equalizerSieve x₁ x₂ ∈ J.sieves (unop X)","state_after":"case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\nhS : S ∈ J.sieves (unop X)\nT : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y :=\n  fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)\n⊢ Sieve.bind S.arrows T ≤ equalizerSieve x₁ x₂\n\ncase refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : ConcreteCategory A\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)\ninst✝ : IsLocallySurjective J f₁\nX : Cᵒᵖ\nx₁ x₂ : (forget A).obj (F₂.obj X)\nh : (f₂.app X) x₁ = (f₂.app X) x₂\nS : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂\nhS : S ∈ J.sieves (unop X)\nT : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y :=\n  fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop X⦄, S.arrows f → Sieve.pullback f (Sieve.bind S.arrows T) ∈ J.sieves Y","tactic":"refine J.superset_covering ?_ (J.transitive hS (Sieve.bind S.1 T) ?_)","premises":[{"full_name":"CategoryTheory.GrothendieckTopology.superset_covering","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[143,8],"def_end_pos":[143,25]},{"full_name":"CategoryTheory.GrothendieckTopology.transitive","def_path":"Mathlib/CategoryTheory/Sites/Grothendieck.lean","def_pos":[132,8],"def_end_pos":[132,18]},{"full_name":"CategoryTheory.Sieve.arrows","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[247,2],"def_end_pos":[247,8]},{"full_name":"CategoryTheory.Sieve.bind","def_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","def_pos":[371,4],"def_end_pos":[371,8]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","commit":"","full_name":"Real.convexOn_Gamma","start":[119,0],"end":[127,99],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","tactics":[{"state_before":"⊢ ConvexOn ℝ (Ioi 0) Gamma","state_after":"⊢ Convex ℝ (log ∘ Gamma '' Ioi 0)","tactic":"refine\n    ((convexOn_exp.subset (subset_univ _) ?_).comp convexOn_log_Gamma\n          (exp_monotone.monotoneOn _)).congr\n      fun x hx => exp_log (Gamma_pos_of_pos hx)","premises":[{"full_name":"ConvexOn.comp","def_path":"Mathlib/Analysis/Convex/Function.lean","def_pos":[135,8],"def_end_pos":[135,21]},{"full_name":"ConvexOn.congr","def_path":"Mathlib/Analysis/Convex/Function.lean","def_pos":[97,8],"def_end_pos":[97,22]},{"full_name":"ConvexOn.subset","def_path":"Mathlib/Analysis/Convex/Function.lean","def_pos":[119,8],"def_end_pos":[119,23]},{"full_name":"Monotone.monotoneOn","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[367,18],"def_end_pos":[367,37]},{"full_name":"Real.Gamma_pos_of_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[524,8],"def_end_pos":[524,24]},{"full_name":"Real.convexOn_log_Gamma","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean","def_pos":[108,8],"def_end_pos":[108,26]},{"full_name":"Real.exp_log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[53,8],"def_end_pos":[53,15]},{"full_name":"Real.exp_monotone","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[1006,8],"def_end_pos":[1006,20]},{"full_name":"Set.subset_univ","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[550,8],"def_end_pos":[550,19]},{"full_name":"convexOn_exp","def_path":"Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean","def_pos":[59,8],"def_end_pos":[59,20]}]},{"state_before":"⊢ Convex ℝ (log ∘ Gamma '' Ioi 0)","state_after":"⊢ IsPreconnected (log ∘ Gamma '' Ioi 0)","tactic":"rw [convex_iff_isPreconnected]","premises":[{"full_name":"Real.convex_iff_isPreconnected","def_path":"Mathlib/Analysis/Convex/Topology.lean","def_pos":[29,8],"def_end_pos":[29,38]}]},{"state_before":"⊢ IsPreconnected (log ∘ Gamma '' Ioi 0)","state_after":"x : ℝ\nhx : x ∈ Ioi 0\n⊢ ContinuousAt (log ∘ Gamma) x","tactic":"refine isPreconnected_Ioi.image _ fun x hx => ContinuousAt.continuousWithinAt ?_","premises":[{"full_name":"ContinuousAt.continuousWithinAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[752,8],"def_end_pos":[752,39]},{"full_name":"IsPreconnected.image","def_path":"Mathlib/Topology/Connected/Basic.lean","def_pos":[278,18],"def_end_pos":[278,38]},{"full_name":"isPreconnected_Ioi","def_path":"Mathlib/Topology/Order/IntermediateValue.lean","def_pos":[397,8],"def_end_pos":[397,26]}]},{"state_before":"x : ℝ\nhx : x ∈ Ioi 0\n⊢ ContinuousAt (log ∘ Gamma) x","state_after":"x : ℝ\nhx : x ∈ Ioi 0\nm : ℕ\n⊢ x ≠ -↑m","tactic":"refine (differentiableAt_Gamma fun m => ?_).continuousAt.log (Gamma_pos_of_pos hx).ne'","premises":[{"full_name":"ContinuousAt.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[384,15],"def_end_pos":[384,31]},{"full_name":"DifferentiableAt.continuousAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[710,8],"def_end_pos":[710,37]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Real.Gamma_pos_of_pos","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[524,8],"def_end_pos":[524,24]},{"full_name":"Real.differentiableAt_Gamma","def_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","def_pos":[604,8],"def_end_pos":[604,30]}]},{"state_before":"x : ℝ\nhx : x ∈ Ioi 0\nm : ℕ\n⊢ x ≠ -↑m","state_after":"no goals","tactic":"exact (neg_lt_iff_pos_add.mpr (add_pos_of_pos_of_nonneg (mem_Ioi.mp hx) (Nat.cast_nonneg m))).ne'","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Nat.cast_nonneg","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[29,8],"def_end_pos":[29,19]},{"full_name":"Set.mem_Ioi","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[126,8],"def_end_pos":[126,15]},{"full_name":"neg_lt_iff_pos_add","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[180,2],"def_end_pos":[180,13]}]}]}
{"url":"Mathlib/Topology/VectorBundle/Basic.lean","commit":"","full_name":"Pretrivialization.linearEquivAt_apply","start":[95,0],"end":[105,47],"file_path":"Mathlib/Topology/VectorBundle/Basic.lean","tactics":[{"state_before":"R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ne✝ : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb✝ : B\ny : E b✝\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst✝ : Pretrivialization.IsLinear R e\nb : B\nhb : b ∈ e.baseSet\nv : F\n⊢ { toFun := fun y => (↑e { proj := b, snd := y }).2, map_add' := ⋯, map_smul' := ⋯ }.toFun (e.symm b v) = v","state_after":"no goals","tactic":"simp_rw [e.apply_mk_symm hb v]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Pretrivialization.apply_mk_symm","def_path":"Mathlib/Topology/FiberBundle/Trivialization.lean","def_pos":[250,8],"def_end_pos":[250,21]}]}]}
{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.nonempty_of_encard_ne_zero","start":[97,0],"end":[98,51],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\nh : s.encard ≠ 0\n⊢ s.Nonempty","state_after":"no goals","tactic":"rwa [nonempty_iff_ne_empty, Ne, ← encard_eq_zero]","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Set.encard_eq_zero","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[89,16],"def_end_pos":[89,30]},{"full_name":"Set.nonempty_iff_ne_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[485,8],"def_end_pos":[485,29]}]}]}
{"url":"Mathlib/Data/Seq/Seq.lean","commit":"","full_name":"Stream'.Seq.join_cons_cons","start":[670,0],"end":[672,36],"file_path":"Mathlib/Data/Seq/Seq.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na b : α\ns : Seq α\nS : Seq (Seq1 α)\n⊢ (cons (a, cons b s) S).join.destruct = some (a, (cons (b, s) S).join)","state_after":"no goals","tactic":"simp [join]","premises":[{"full_name":"Stream'.Seq.join","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[474,4],"def_end_pos":[474,8]}]}]}
{"url":"Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean","commit":"","full_name":"SimplicialObject.Splitting.decomposition_id","start":[56,0],"end":[64,8],"file_path":"Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\n⊢ 𝟙 (X.obj Δ) = ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\n⊢ ∀ (A : IndexSet Δ),\n    (s.cofan Δ).inj A ≫ 𝟙 (X.obj Δ) = (s.cofan Δ).inj A ≫ ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A","tactic":"apply s.hom_ext'","premises":[{"full_name":"SimplicialObject.Splitting.hom_ext'","def_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","def_pos":[246,8],"def_end_pos":[246,16]}]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\n⊢ ∀ (A : IndexSet Δ),\n    (s.cofan Δ).inj A ≫ 𝟙 (X.obj Δ) = (s.cofan Δ).inj A ≫ ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A","state_after":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ (s.cofan Δ).inj A ≫ 𝟙 (X.obj Δ) = (s.cofan Δ).inj A ≫ ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A","tactic":"intro A","premises":[]},{"state_before":"case h\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ (s.cofan Δ).inj A ≫ 𝟙 (X.obj Δ) = (s.cofan Δ).inj A ≫ ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A","state_after":"case h.h₀\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ ∀ b ∈ Finset.univ, b ≠ A → (s.cofan Δ).inj A ≫ s.πSummand b ≫ (s.cofan Δ).inj b = 0\n\ncase h.h₁\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ A ∉ Finset.univ → (s.cofan Δ).inj A ≫ s.πSummand A ≫ (s.cofan Δ).inj A = 0","tactic":"erw [comp_id, comp_sum, Finset.sum_eq_single A, cofan_inj_πSummand_eq_id_assoc]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Preadditive.comp_sum","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[162,8],"def_end_pos":[162,16]},{"full_name":"Finset.sum_eq_single","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[866,2],"def_end_pos":[866,13]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
{"url":"Mathlib/Order/BoundedOrder.lean","commit":"","full_name":"Subtype.coe_eq_top_iff","start":[645,0],"end":[648,38],"file_path":"Mathlib/Order/BoundedOrder.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type u_1\nδ : Type u_2\np : α → Prop\ninst✝² : PartialOrder α\ninst✝¹ : OrderTop α\ninst✝ : OrderTop (Subtype p)\nhtop : p ⊤\nx : { x // p x }\n⊢ ↑x = ⊤ ↔ x = ⊤","state_after":"no goals","tactic":"rw [← coe_top htop, Subtype.ext_iff]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Subtype.coe_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[637,8],"def_end_pos":[637,15]},{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]}]}
{"url":"Mathlib/CategoryTheory/Category/Basic.lean","commit":"","full_name":"CategoryTheory.eq_whisker","start":[180,0],"end":[181,85],"file_path":"Mathlib/CategoryTheory/Category/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf g : X ⟶ Y\nw : f = g\nh : Y ⟶ Z\n⊢ f ≫ h = g ≫ h","state_after":"no goals","tactic":"rw [w]","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","commit":"","full_name":"lp.norm_sum_single","start":[936,0],"end":[945,70],"file_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\n⊢ ‖∑ i ∈ s, lp.single p i (f i)‖ ^ p.toReal = ∑ i ∈ s, ‖f i‖ ^ p.toReal","state_after":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\n⊢ HasSum (fun i => ‖↑(∑ i ∈ s, lp.single p i (f i)) i‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"refine (hasSum_norm hp (∑ i ∈ s, lp.single p i (f i))).unique ?_","premises":[{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"HasSum.unique","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[159,2],"def_end_pos":[159,13]},{"full_name":"lp.hasSum_norm","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[383,8],"def_end_pos":[383,19]},{"full_name":"lp.single","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[896,14],"def_end_pos":[896,20]}]},{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\n⊢ HasSum (fun i => ‖↑(∑ i ∈ s, lp.single p i (f i)) i‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\n⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"simp only [lp.single_apply, coeFn_sum, Finset.sum_apply, Finset.sum_dite_eq]","premises":[{"full_name":"Finset.sum_apply","def_path":"Mathlib/Algebra/BigOperators/Pi.lean","def_pos":[32,2],"def_end_pos":[32,13]},{"full_name":"Finset.sum_dite_eq","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1101,2],"def_end_pos":[1101,13]},{"full_name":"lp.coeFn_sum","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[343,8],"def_end_pos":[343,17]},{"full_name":"lp.single_apply","def_path":"Mathlib/Analysis/Normed/Lp/lpSpace.lean","def_pos":[906,18],"def_end_pos":[906,30]}]},{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\n⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\nh : ∀ i ∉ s, ‖if i ∈ s then f i else 0‖ ^ p.toReal = 0\n⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"have h : ∀ i ∉ s, ‖ite (i ∈ s) (f i) 0‖ ^ p.toReal = 0 := fun i hi ↦ by\n    simp [if_neg hi, Real.zero_rpow hp.ne']","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]},{"full_name":"Real.zero_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[114,8],"def_end_pos":[114,17]},{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]},{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\nh : ∀ i ∉ s, ‖if i ∈ s then f i else 0‖ ^ p.toReal = 0\n⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\nh : ∀ i ∉ s, ‖if i ∈ s then f i else 0‖ ^ p.toReal = 0\nh' : ∀ i ∈ s, ‖f i‖ ^ p.toReal = ‖if i ∈ s then f i else 0‖ ^ p.toReal\n⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","tactic":"have h' : ∀ i ∈ s, ‖f i‖ ^ p.toReal = ‖ite (i ∈ s) (f i) 0‖ ^ p.toReal := by\n    intro i hi\n    rw [if_pos hi]","premises":[{"full_name":"ENNReal.toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[184,14],"def_end_pos":[184,20]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]},{"full_name":"ite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[970,20],"def_end_pos":[970,23]}]},{"state_before":"α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\nhp : 0 < p.toReal\nf : (i : α) → E i\ns : Finset α\nh : ∀ i ∉ s, ‖if i ∈ s then f i else 0‖ ^ p.toReal = 0\nh' : ∀ i ∈ s, ‖f i‖ ^ p.toReal = ‖if i ∈ s then f i else 0‖ ^ p.toReal\n⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal)","state_after":"no goals","tactic":"simpa [Finset.sum_congr rfl h'] using hasSum_sum_of_ne_finset_zero h","premises":[{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"hasSum_sum_of_ne_finset_zero","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[142,2],"def_end_pos":[142,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Data/Sign.lean","commit":"","full_name":"sign_pos","start":[315,0],"end":[316,73],"file_path":"Mathlib/Data/Sign.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nha : 0 < a\n⊢ sign a = 1","state_after":"no goals","tactic":"rwa [sign_apply, if_pos]","premises":[{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]},{"full_name":"sign_apply","def_path":"Mathlib/Data/Sign.lean","def_pos":[309,8],"def_end_pos":[309,18]}]}]}
{"url":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","commit":"","full_name":"Finset.pluennecke_ruzsa_inequality_pow_div_pow_div","start":[200,0],"end":[205,60],"file_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : A.Nonempty\nB : Finset α\nm n : ℕ\n⊢ ↑(B ^ m / B ^ n).card ≤ (↑(A / B).card / ↑A.card) ^ (m + n) * ↑A.card","state_after":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : A.Nonempty\nB : Finset α\nm n : ℕ\n⊢ ↑(B⁻¹ ^ m / B⁻¹ ^ n).card ≤ (↑(A * B⁻¹).card / ↑A.card) ^ (m + n) * ↑A.card","tactic":"rw [← card_inv, inv_div', ← inv_pow, ← inv_pow, div_eq_mul_inv A]","premises":[{"full_name":"Finset.card_inv","def_path":"Mathlib/Data/Finset/Pointwise.lean","def_pos":[268,8],"def_end_pos":[268,16]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"inv_div'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[519,8],"def_end_pos":[519,16]},{"full_name":"inv_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[418,6],"def_end_pos":[418,13]}]},{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : A.Nonempty\nB : Finset α\nm n : ℕ\n⊢ ↑(B⁻¹ ^ m / B⁻¹ ^ n).card ≤ (↑(A * B⁻¹).card / ↑A.card) ^ (m + n) * ↑A.card","state_after":"no goals","tactic":"exact pluennecke_ruzsa_inequality_pow_div_pow_mul hA _ _ _","premises":[{"full_name":"Finset.pluennecke_ruzsa_inequality_pow_div_pow_mul","def_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","def_pos":[182,8],"def_end_pos":[182,51]}]}]}
{"url":"Mathlib/Data/List/InsertNth.lean","commit":"","full_name":"List.mem_insertNth","start":[86,0],"end":[92,56],"file_path":"Mathlib/Data/List/InsertNth.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\na✝ a b : α\nas : List α\nx✝ : 0 ≤ as.length\n⊢ a ∈ insertNth 0 b as ↔ a = b ∨ a ∈ as","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\na✝ a b : α\nn : ℕ\na' : α\nas : List α\nh : n + 1 ≤ (a' :: as).length\n⊢ a ∈ insertNth (n + 1) b (a' :: as) ↔ a = b ∨ a ∈ a' :: as","state_after":"no goals","tactic":"rw [List.insertNth_succ_cons, mem_cons, mem_insertNth (Nat.le_of_succ_le_succ h),\n      ← or_assoc, @or_comm (a = a'), or_assoc, mem_cons]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"List.insertNth_succ_cons","def_path":"Mathlib/Data/List/InsertNth.lean","def_pos":[39,8],"def_end_pos":[39,27]},{"full_name":"List.mem_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[285,16],"def_end_pos":[285,24]},{"full_name":"Nat.le_of_succ_le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1713,8],"def_end_pos":[1713,30]},{"full_name":"or_assoc","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[176,8],"def_end_pos":[176,16]},{"full_name":"or_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[823,8],"def_end_pos":[823,15]}]}]}
{"url":"Mathlib/NumberTheory/ArithmeticFunction.lean","commit":"","full_name":"ArithmeticFunction.intCoe_mul","start":[278,0],"end":[282,6],"file_path":"Mathlib/NumberTheory/ArithmeticFunction.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\n⊢ ↑(f * g) = ↑f * ↑g","state_after":"case h\nR : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\nn : ℕ\n⊢ ↑(f * g) n = (↑f * ↑g) n","tactic":"ext n","premises":[]},{"state_before":"case h\nR : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\nn : ℕ\n⊢ ↑(f * g) n = (↑f * ↑g) n","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean","commit":"","full_name":"MeasureTheory.condexp_indicator","start":[69,0],"end":[108,52],"file_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean","tactics":[{"state_before":"α : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])\n\ncase neg\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : ¬m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","tactic":"by_cases hm : m ≤ m0","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])\n\ncase neg\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : ¬m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case neg\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : ¬m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])\n\ncase pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","tactic":"swap","premises":[]},{"state_before":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])\n\ncase neg\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","tactic":"by_cases hμm : SigmaFinite (μ.trim hm)","premises":[{"full_name":"MeasureTheory.Measure.trim","def_path":"Mathlib/MeasureTheory/Measure/Trim.lean","def_pos":[32,4],"def_end_pos":[32,16]},{"full_name":"MeasureTheory.SigmaFinite","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[555,6],"def_end_pos":[555,17]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])\n\ncase neg\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case neg\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])\n\ncase pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","tactic":"swap","premises":[]},{"state_before":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm : SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this : SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","tactic":"haveI : SigmaFinite (μ.trim hm) := hμm","premises":[{"full_name":"MeasureTheory.Measure.trim","def_path":"Mathlib/MeasureTheory/Measure/Trim.lean","def_pos":[32,4],"def_end_pos":[32,16]},{"full_name":"MeasureTheory.SigmaFinite","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[555,6],"def_end_pos":[555,17]}]},{"state_before":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this : SigmaFinite (μ.trim hm)\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator (μ[f|m]) =ᶠ[ae μ] s.indicator (μ[s.indicator f + sᶜ.indicator f|m])\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","tactic":"have : s.indicator (μ[f|m]) =ᵐ[μ] s.indicator (μ[s.indicator f + sᶜ.indicator f|m]) := by\n    rw [Set.indicator_self_add_compl s f]","premises":[{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.condexp","def_path":"Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean","def_pos":[90,30],"def_end_pos":[90,37]},{"full_name":"Set.indicator","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[45,2],"def_end_pos":[45,13]},{"full_name":"Set.indicator_self_add_compl","def_path":"Mathlib/Algebra/Group/Indicator.lean","def_pos":[323,2],"def_end_pos":[323,13]}]},{"state_before":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator (μ[f|m]) =ᶠ[ae μ] s.indicator (μ[s.indicator f + sᶜ.indicator f|m])\n⊢ μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])","state_after":"case pos\nα : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator (μ[f|m]) =ᶠ[ae μ] s.indicator (μ[s.indicator f + sᶜ.indicator f|m])\n⊢ s.indicator (μ[s.indicator f + sᶜ.indicator f|m]) =ᶠ[ae μ] μ[s.indicator f|m]","tactic":"refine (this.trans ?_).symm","premises":[{"full_name":"Filter.EventuallyEq.symm","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1307,8],"def_end_pos":[1307,25]},{"full_name":"Filter.EventuallyEq.trans","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1311,8],"def_end_pos":[1311,26]}]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean","commit":"","full_name":"EuclideanGeometry.collinear_iff_eq_or_eq_or_sin_eq_zero","start":[456,0],"end":[461,59],"file_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean","tactics":[{"state_before":"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np p₀ p₁✝ p₂✝ p₁ p₂ p₃ : P\n⊢ Collinear ℝ {p₁, p₂, p₃} ↔ p₁ = p₂ ∨ p₃ = p₂ ∨ sin (∠ p₁ p₂ p₃) = 0","state_after":"no goals","tactic":"rw [sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi,\n    collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi]","premises":[{"full_name":"EuclideanGeometry.collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean","def_pos":[376,8],"def_end_pos":[376,62]},{"full_name":"EuclideanGeometry.sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi","def_path":"Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean","def_pos":[447,15],"def_end_pos":[447,59]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.map₂Right_eq_zipWith","start":[2564,0],"end":[2567,61],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : Option α → β → γ\na : α\nas : List α\nb : β\nbs : List β\nh : bs.length ≤ as.length\n⊢ map₂Right f as bs = zipWith (fun a b => f (some a) b) as bs","state_after":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : Option α → β → γ\na : α\nas : List α\nb : β\nbs : List β\nh : bs.length ≤ as.length\nthis : (fun a b => flip f a (some b)) = flip fun a b => f (some a) b\n⊢ map₂Right f as bs = zipWith (fun a b => f (some a) b) as bs","tactic":"have : (fun a b => flip f a (some b)) = flip fun a b => f (some a) b := rfl","premises":[{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]},{"full_name":"flip","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[30,14],"def_end_pos":[30,18]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : Option α → β → γ\na : α\nas : List α\nb : β\nbs : List β\nh : bs.length ≤ as.length\nthis : (fun a b => flip f a (some b)) = flip fun a b => f (some a) b\n⊢ map₂Right f as bs = zipWith (fun a b => f (some a) b) as bs","state_after":"no goals","tactic":"simp only [map₂Right, map₂Left_eq_zipWith, zipWith_flip, *]","premises":[{"full_name":"List.map₂Left_eq_zipWith","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2526,8],"def_end_pos":[2526,27]},{"full_name":"List.map₂Right","def_path":"Mathlib/Data/List/Defs.lean","def_pos":[363,4],"def_end_pos":[363,13]},{"full_name":"List.zipWith_flip","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[1220,8],"def_end_pos":[1220,20]}]}]}
{"url":"Mathlib/RingTheory/LocalProperties.lean","commit":"","full_name":"localization_isReduced","start":[294,0],"end":[311,37],"file_path":"Mathlib/RingTheory/LocalProperties.lean","tactics":[{"state_before":"R S : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst✝³ : CommRing R'\ninst✝² : CommRing S'\nf : R →+* S\ninst✝¹ : Algebra R R'\ninst✝ : Algebra S S'\n⊢ LocalizationPreserves fun R hR => IsReduced R","state_after":"R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\n⊢ IsReduced S","tactic":"introv R _ _","premises":[]},{"state_before":"R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\n⊢ IsReduced S","state_after":"case eq_zero\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\n⊢ ∀ (x : S), IsNilpotent x → x = 0","tactic":"constructor","premises":[]},{"state_before":"case eq_zero\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\n⊢ ∀ (x : S), IsNilpotent x → x = 0","state_after":"case eq_zero.intro.zero\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\ne : x ^ 0 = 0\n⊢ x = 0\n\ncase eq_zero.intro.succ\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\n⊢ x = 0","tactic":"rintro x ⟨_ | n, e⟩","premises":[]},{"state_before":"case eq_zero.intro.succ\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑(y, m).2 = (algebraMap R S) (y, m).1\n⊢ x = 0","tactic":"obtain ⟨⟨y, m⟩, hx⟩ := IsLocalization.surj M x","premises":[{"full_name":"IsLocalization.surj","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[120,8],"def_end_pos":[120,12]}]},{"state_before":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑(y, m).2 = (algebraMap R S) (y, m).1\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\n⊢ x = 0","tactic":"dsimp only at hx","premises":[]},{"state_before":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (fun x => x ^ n.succ) (x * (algebraMap R S) ↑m) = (fun x => x ^ n.succ) ((algebraMap R S) y) :=\n  congr_arg (fun x => x ^ n.succ) hx\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : 0 = (algebraMap R S) (y ^ n.succ)\n⊢ x = 0","tactic":"simp only [mul_pow, e, zero_mul, ← RingHom.map_pow] at hx'","premises":[{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"RingHom.map_pow","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[622,16],"def_end_pos":[622,31]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]}]},{"state_before":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : 0 = (algebraMap R S) (y ^ n.succ)\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\n⊢ x = 0","tactic":"rw [← (algebraMap R S).map_zero] at hx'","premises":[{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case eq_zero.intro.succ.intro.mk\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : ↑m' * 0 = ↑m' * y ^ n.succ\n⊢ x = 0","tactic":"obtain ⟨m', hm'⟩ := (IsLocalization.eq_iff_exists M S).mp hx'","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsLocalization.eq_iff_exists","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[125,8],"def_end_pos":[125,21]}]},{"state_before":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : ↑m' * 0 * ↑m' ^ n = ↑m' * y ^ n.succ * ↑m' ^ n\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : 0 = ↑m' * (y ^ n.succ * ↑m' ^ n)\n⊢ x = 0","tactic":"simp only [mul_assoc, zero_mul, mul_zero] at hm'","premises":[{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : 0 = ↑m' * (y ^ n.succ * ↑m' ^ n)\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : 0 = (y * ↑m') ^ n.succ\n⊢ x = 0","tactic":"rw [← mul_left_comm, ← pow_succ', ← mul_pow] at hm'","premises":[{"full_name":"mul_left_comm","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[152,8],"def_end_pos":[152,21]},{"full_name":"mul_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[222,31],"def_end_pos":[222,38]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : 0 = (y * ↑m') ^ n.succ\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : y * ↑m' = 0\n⊢ x = 0","tactic":"replace hm' := IsNilpotent.eq_zero ⟨_, hm'.symm⟩","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsNilpotent.eq_zero","def_path":"Mathlib/RingTheory/Nilpotent/Defs.lean","def_pos":[179,8],"def_end_pos":[179,27]}]},{"state_before":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : y * ↑m' = 0\n⊢ x = 0","state_after":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : y * ↑m' = 0\n⊢ ∃ m, ↑m * y = 0","tactic":"rw [← (IsLocalization.map_units S m).mul_left_inj, hx, zero_mul,\n    IsLocalization.map_eq_zero_iff M]","premises":[{"full_name":"IsLocalization.map_eq_zero_iff","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[215,8],"def_end_pos":[215,23]},{"full_name":"IsLocalization.map_units","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[115,8],"def_end_pos":[115,17]},{"full_name":"IsUnit.mul_left_inj","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[686,8],"def_end_pos":[686,20]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]}]},{"state_before":"case eq_zero.intro.succ.intro.mk.intro\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\nM✝ : Submonoid R✝\nN : Submonoid S✝\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\nf : R✝ →+* S✝\ninst✝³ : Algebra R✝ R'\ninst✝² : Algebra S✝ S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraMap R S) (y ^ n.succ)\nm' : ↥M\nhm' : y * ↑m' = 0\n⊢ ∃ m, ↑m * y = 0","state_after":"no goals","tactic":"exact ⟨m', by rw [← hm', mul_comm]⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/GroupTheory/Exponent.lean","commit":"","full_name":"mul_not_mem_of_orderOf_eq_two","start":[645,0],"end":[650,7],"file_path":"Mathlib/GroupTheory/Exponent.lean","tactics":[{"state_before":"G : Type u\ninst✝ : Group G\nx y : G\nhx : orderOf x = 2\nhy : orderOf y = 2\nhxy : x ≠ y\n⊢ x * y ∉ {x, y, 1}","state_after":"G : Type u\ninst✝ : Group G\nx y : G\nhx : orderOf x = 2\nhy : orderOf y = 2\nhxy : x ≠ y\n⊢ ¬y = 1 ∧ ¬x = 1 ∧ ¬x = y","tactic":"simp only [Set.mem_singleton_iff, Set.mem_insert_iff, mul_right_eq_self, mul_left_eq_self,\n    mul_eq_one_iff_eq_inv, inv_eq_self_of_orderOf_eq_two hy, not_or]","premises":[{"full_name":"Set.mem_insert_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[895,8],"def_end_pos":[895,22]},{"full_name":"Set.mem_singleton_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[999,8],"def_end_pos":[999,25]},{"full_name":"inv_eq_self_of_orderOf_eq_two","def_path":"Mathlib/GroupTheory/Exponent.lean","def_pos":[634,6],"def_end_pos":[634,35]},{"full_name":"mul_eq_one_iff_eq_inv","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[638,8],"def_end_pos":[638,29]},{"full_name":"mul_left_eq_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[254,8],"def_end_pos":[254,24]},{"full_name":"mul_right_eq_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[233,8],"def_end_pos":[233,25]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]},{"state_before":"G : Type u\ninst✝ : Group G\nx y : G\nhx : orderOf x = 2\nhy : orderOf y = 2\nhxy : x ≠ y\n⊢ ¬y = 1 ∧ ¬x = 1 ∧ ¬x = y","state_after":"no goals","tactic":"aesop","premises":[]}]}
{"url":"Mathlib/AlgebraicTopology/SimplicialObject.lean","commit":"","full_name":"CategoryTheory.CosimplicialObject.δ_comp_σ_succ'","start":[493,0],"end":[497,20],"file_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = i.succ\n⊢ X.δ j ≫ X.σ i = 𝟙 (X.obj [n])","state_after":"C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 1)\n⊢ X.δ i.succ ≫ X.σ i = 𝟙 (X.obj [n])","tactic":"subst H","premises":[]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 1)\n⊢ X.δ i.succ ≫ X.σ i = 𝟙 (X.obj [n])","state_after":"no goals","tactic":"rw [δ_comp_σ_succ]","premises":[{"full_name":"CategoryTheory.CosimplicialObject.δ_comp_σ_succ","def_path":"Mathlib/AlgebraicTopology/SimplicialObject.lean","def_pos":[489,8],"def_end_pos":[489,21]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"IsOpen.closure_div","start":[1222,0],"end":[1224,46],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\ns✝ t : Set G\nht : IsOpen t\ns : Set G\n⊢ closure s / t = s / t","state_after":"no goals","tactic":"simp_rw [div_eq_mul_inv, ht.inv.closure_mul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"IsOpen.closure_mul","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[1214,8],"def_end_pos":[1214,26]},{"full_name":"IsOpen.inv","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[319,8],"def_end_pos":[319,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","commit":"","full_name":"Real.sin_eq_sqrt_one_sub_cos_sq","start":[457,0],"end":[459,92],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","tactics":[{"state_before":"x : ℝ\nhl : 0 ≤ x\nhu : x ≤ π\n⊢ sin x = √(1 - cos x ^ 2)","state_after":"no goals","tactic":"rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)]","premises":[{"full_name":"Real.abs_sin_eq_sqrt_one_sub_cos_sq","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[826,8],"def_end_pos":[826,38]},{"full_name":"Real.sin_nonneg_of_nonneg_of_le_pi","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[409,8],"def_end_pos":[409,37]},{"full_name":"abs_of_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[76,2],"def_end_pos":[76,13]}]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","commit":"","full_name":"ContinuousLinearEquiv.comp_contDiffAt_iff","start":[307,0],"end":[311,66],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ne : F ≃L[𝕜] G\n⊢ ContDiffAt 𝕜 n (⇑e ∘ f) x ↔ ContDiffAt 𝕜 n f x","state_after":"no goals","tactic":"simp only [← contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff]","premises":[{"full_name":"ContinuousLinearEquiv.comp_contDiffWithinAt_iff","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[300,8],"def_end_pos":[300,55]},{"full_name":"contDiffWithinAt_univ","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[1247,8],"def_end_pos":[1247,29]}]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","commit":"","full_name":"ContDiffWithinAt.div_const","start":[1398,0],"end":[1400,65],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁵ : NormedAddCommGroup D\ninst✝¹⁴ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedSpace 𝕜 G\nX : Type u_2\ninst✝⁷ : NormedAddCommGroup X\ninst✝⁶ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc✝ : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n𝔸 : Type u_3\n𝔸' : Type u_4\nι : Type u_5\n𝕜' : Type u_6\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedAlgebra 𝕜 𝔸\ninst✝³ : NormedCommRing 𝔸'\ninst✝² : NormedAlgebra 𝕜 𝔸'\ninst✝¹ : NormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nf : E → 𝕜'\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n f s x\nc : 𝕜'\n⊢ ContDiffWithinAt 𝕜 n (fun x => f x / c) s x","state_after":"no goals","tactic":"simpa only [div_eq_mul_inv] using hf.mul contDiffWithinAt_const","premises":[{"full_name":"ContDiffWithinAt.mul","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1332,8],"def_end_pos":[1332,28]},{"full_name":"contDiffWithinAt_const","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[95,8],"def_end_pos":[95,30]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","commit":"","full_name":"PartialHomeomorph.isOpen_extend_preimage","start":[861,0],"end":[863,64],"file_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns✝ t : Set M\ns : Set E\nhs : IsOpen s\n⊢ IsOpen (f.source ∩ ↑(f.extend I) ⁻¹' s)","state_after":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns✝ t : Set M\ns : Set E\nhs : IsOpen s\n⊢ IsOpen ((f.extend I).source ∩ ↑(f.extend I) ⁻¹' s)","tactic":"rw [← extend_source f I]","premises":[{"full_name":"PartialHomeomorph.extend_source","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[758,8],"def_end_pos":[758,21]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns✝ t : Set M\ns : Set E\nhs : IsOpen s\n⊢ IsOpen ((f.extend I).source ∩ ↑(f.extend I) ⁻¹' s)","state_after":"no goals","tactic":"exact isOpen_extend_preimage' f I hs","premises":[{"full_name":"PartialHomeomorph.isOpen_extend_preimage'","def_path":"Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean","def_pos":[857,8],"def_end_pos":[857,31]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","commit":"","full_name":"hasDerivAtFilter_iff_tendsto_slope","start":[46,0],"end":[61,62],"file_path":"Mathlib/Analysis/Calculus/Deriv/Slope.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\n⊢ HasDerivAtFilter f f' x L ↔ Tendsto (fun y => slope f x y - (y - x)⁻¹ • (y - x) • f') L (𝓝 0)","state_after":"no goals","tactic":"simp only [hasDerivAtFilter_iff_tendsto, ← norm_inv, ← norm_smul,\n          ← tendsto_zero_iff_norm_tendsto_zero, slope_def_module, smul_sub]","premises":[{"full_name":"hasDerivAtFilter_iff_tendsto","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[238,8],"def_end_pos":[238,36]},{"full_name":"norm_inv","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[716,8],"def_end_pos":[716,16]},{"full_name":"norm_smul","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[79,8],"def_end_pos":[79,17]},{"full_name":"slope_def_module","def_path":"Mathlib/LinearAlgebra/AffineSpace/Slope.lean","def_pos":[44,8],"def_end_pos":[44,24]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]},{"full_name":"tendsto_zero_iff_norm_tendsto_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[715,2],"def_end_pos":[715,13]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\n⊢ ∀ a ∉ {x}ᶜ, slope f x a - (a - x)⁻¹ • (a - x) • f' = 0","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\n⊢ (fun y => slope f x y - (y - x)⁻¹ • (y - x) • f') =ᶠ[L ⊓ 𝓟 {x}ᶜ] fun y => slope f x y - f'","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\ny : 𝕜\nhy : y ∈ {x}ᶜ\n⊢ slope f x y - (y - x)⁻¹ • (y - x) • f' = slope f x y - f'","tactic":"refine (EqOn.eventuallyEq fun y hy ↦ ?_).filter_mono inf_le_right","premises":[{"full_name":"Filter.EventuallyEq.filter_mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1295,8],"def_end_pos":[1295,32]},{"full_name":"Set.EqOn.eventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2823,8],"def_end_pos":[2823,29]},{"full_name":"inf_le_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[312,8],"def_end_pos":[312,20]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\ny : 𝕜\nhy : y ∈ {x}ᶜ\n⊢ slope f x y - (y - x)⁻¹ • (y - x) • f' = slope f x y - f'","state_after":"no goals","tactic":"rw [inv_smul_smul₀ (sub_ne_zero.2 hy) f']","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"inv_smul_smul₀","def_path":"Mathlib/GroupTheory/GroupAction/Group.lean","def_pos":[35,8],"def_end_pos":[35,22]},{"full_name":"sub_ne_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[746,2],"def_end_pos":[746,13]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\n⊢ Tendsto (fun y => slope f x y - f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) ↔ Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f')","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\n⊢ Tendsto (fun y => slope f x y - f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) ↔ Tendsto ((fun x => x - f') ∘ slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 0)","tactic":"rw [← nhds_translation_sub f', tendsto_comap_iff]","premises":[{"full_name":"Filter.tendsto_comap_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2686,8],"def_end_pos":[2686,25]},{"full_name":"nhds_translation_sub","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[1031,2],"def_end_pos":[1031,13]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL✝ L₁ L₂ : Filter 𝕜\nx : 𝕜\nL : Filter 𝕜\n⊢ Tendsto (fun y => slope f x y - f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) ↔ Tendsto ((fun x => x - f') ∘ slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 0)","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Algebra/Module/Injective.lean","commit":"","full_name":"Module.Baer.ExtensionOf.ext","start":[101,0],"end":[109,42],"file_path":"Mathlib/Algebra/Module/Injective.lean","tactics":[{"state_before":"R : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\na b : ExtensionOf i f\ndomain_eq : a.domain = b.domain\nto_fun_eq : ∀ ⦃x : ↥a.domain⦄ ⦃y : ↥b.domain⦄, ↑x = ↑y → ↑a.toLinearPMap x = ↑b.toLinearPMap y\n⊢ a = b","state_after":"case mk\nR : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\nb : ExtensionOf i f\na : N →ₗ.[R] Q\na_le : LinearMap.range i ≤ a.domain\ne1 : ∀ (m : M), f m = ↑a ⟨i m, ⋯⟩\ndomain_eq : { toLinearPMap := a, le := a_le, is_extension := e1 }.domain = b.domain\nto_fun_eq :\n  ∀ ⦃x : ↥{ toLinearPMap := a, le := a_le, is_extension := e1 }.domain⦄ ⦃y : ↥b.domain⦄,\n    ↑x = ↑y → ↑{ toLinearPMap := a, le := a_le, is_extension := e1 }.toLinearPMap x = ↑b.toLinearPMap y\n⊢ { toLinearPMap := a, le := a_le, is_extension := e1 } = b","tactic":"rcases a with ⟨a, a_le, e1⟩","premises":[]},{"state_before":"case mk\nR : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\nb : ExtensionOf i f\na : N →ₗ.[R] Q\na_le : LinearMap.range i ≤ a.domain\ne1 : ∀ (m : M), f m = ↑a ⟨i m, ⋯⟩\ndomain_eq : { toLinearPMap := a, le := a_le, is_extension := e1 }.domain = b.domain\nto_fun_eq :\n  ∀ ⦃x : ↥{ toLinearPMap := a, le := a_le, is_extension := e1 }.domain⦄ ⦃y : ↥b.domain⦄,\n    ↑x = ↑y → ↑{ toLinearPMap := a, le := a_le, is_extension := e1 }.toLinearPMap x = ↑b.toLinearPMap y\n⊢ { toLinearPMap := a, le := a_le, is_extension := e1 } = b","state_after":"case mk.mk\nR : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\na : N →ₗ.[R] Q\na_le : LinearMap.range i ≤ a.domain\ne1 : ∀ (m : M), f m = ↑a ⟨i m, ⋯⟩\nb : N →ₗ.[R] Q\nb_le : LinearMap.range i ≤ b.domain\ne2 : ∀ (m : M), f m = ↑b ⟨i m, ⋯⟩\ndomain_eq :\n  { toLinearPMap := a, le := a_le, is_extension := e1 }.domain =\n    { toLinearPMap := b, le := b_le, is_extension := e2 }.domain\nto_fun_eq :\n  ∀ ⦃x : ↥{ toLinearPMap := a, le := a_le, is_extension := e1 }.domain⦄\n    ⦃y : ↥{ toLinearPMap := b, le := b_le, is_extension := e2 }.domain⦄,\n    ↑x = ↑y →\n      ↑{ toLinearPMap := a, le := a_le, is_extension := e1 }.toLinearPMap x =\n        ↑{ toLinearPMap := b, le := b_le, is_extension := e2 }.toLinearPMap y\n⊢ { toLinearPMap := a, le := a_le, is_extension := e1 } = { toLinearPMap := b, le := b_le, is_extension := e2 }","tactic":"rcases b with ⟨b, b_le, e2⟩","premises":[]},{"state_before":"case mk.mk\nR : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\na : N →ₗ.[R] Q\na_le : LinearMap.range i ≤ a.domain\ne1 : ∀ (m : M), f m = ↑a ⟨i m, ⋯⟩\nb : N →ₗ.[R] Q\nb_le : LinearMap.range i ≤ b.domain\ne2 : ∀ (m : M), f m = ↑b ⟨i m, ⋯⟩\ndomain_eq :\n  { toLinearPMap := a, le := a_le, is_extension := e1 }.domain =\n    { toLinearPMap := b, le := b_le, is_extension := e2 }.domain\nto_fun_eq :\n  ∀ ⦃x : ↥{ toLinearPMap := a, le := a_le, is_extension := e1 }.domain⦄\n    ⦃y : ↥{ toLinearPMap := b, le := b_le, is_extension := e2 }.domain⦄,\n    ↑x = ↑y →\n      ↑{ toLinearPMap := a, le := a_le, is_extension := e1 }.toLinearPMap x =\n        ↑{ toLinearPMap := b, le := b_le, is_extension := e2 }.toLinearPMap y\n⊢ { toLinearPMap := a, le := a_le, is_extension := e1 } = { toLinearPMap := b, le := b_le, is_extension := e2 }","state_after":"case mk.mk.e_toLinearPMap\nR : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\na : N →ₗ.[R] Q\na_le : LinearMap.range i ≤ a.domain\ne1 : ∀ (m : M), f m = ↑a ⟨i m, ⋯⟩\nb : N →ₗ.[R] Q\nb_le : LinearMap.range i ≤ b.domain\ne2 : ∀ (m : M), f m = ↑b ⟨i m, ⋯⟩\ndomain_eq :\n  { toLinearPMap := a, le := a_le, is_extension := e1 }.domain =\n    { toLinearPMap := b, le := b_le, is_extension := e2 }.domain\nto_fun_eq :\n  ∀ ⦃x : ↥{ toLinearPMap := a, le := a_le, is_extension := e1 }.domain⦄\n    ⦃y : ↥{ toLinearPMap := b, le := b_le, is_extension := e2 }.domain⦄,\n    ↑x = ↑y →\n      ↑{ toLinearPMap := a, le := a_le, is_extension := e1 }.toLinearPMap x =\n        ↑{ toLinearPMap := b, le := b_le, is_extension := e2 }.toLinearPMap y\n⊢ a = b","tactic":"congr","premises":[]},{"state_before":"case mk.mk.e_toLinearPMap\nR : Type u\ninst✝⁶ : Ring R\nQ : Type v\ninst✝⁵ : AddCommGroup Q\ninst✝⁴ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\na : N →ₗ.[R] Q\na_le : LinearMap.range i ≤ a.domain\ne1 : ∀ (m : M), f m = ↑a ⟨i m, ⋯⟩\nb : N →ₗ.[R] Q\nb_le : LinearMap.range i ≤ b.domain\ne2 : ∀ (m : M), f m = ↑b ⟨i m, ⋯⟩\ndomain_eq :\n  { toLinearPMap := a, le := a_le, is_extension := e1 }.domain =\n    { toLinearPMap := b, le := b_le, is_extension := e2 }.domain\nto_fun_eq :\n  ∀ ⦃x : ↥{ toLinearPMap := a, le := a_le, is_extension := e1 }.domain⦄\n    ⦃y : ↥{ toLinearPMap := b, le := b_le, is_extension := e2 }.domain⦄,\n    ↑x = ↑y →\n      ↑{ toLinearPMap := a, le := a_le, is_extension := e1 }.toLinearPMap x =\n        ↑{ toLinearPMap := b, le := b_le, is_extension := e2 }.toLinearPMap y\n⊢ a = b","state_after":"no goals","tactic":"exact LinearPMap.ext domain_eq to_fun_eq","premises":[{"full_name":"LinearPMap.ext","def_path":"Mathlib/LinearAlgebra/LinearPMap.lean","def_pos":[61,8],"def_end_pos":[61,11]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","commit":"","full_name":"Polynomial.Chebyshev.T_derivative_eq_U","start":[246,0],"end":[264,62],"file_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℤ\n⊢ derivative (T R n) = ↑n * U R (n - 1)","state_after":"no goals","tactic":"induction n using Polynomial.Chebyshev.induct with\n  | zero => simp\n  | one =>\n    simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]\n  | add_two n ih1 ih2 =>\n    have h₁ := congr_arg derivative (T_add_two R n)\n    have h₂ := U_sub_one R n\n    have h₃ := T_eq_U_sub_X_mul_U R (n + 1)\n    simp only [derivative_sub, derivative_mul, derivative_ofNat, derivative_X] at h₁\n    linear_combination (norm := (push_cast; ring_nf))\n      h₁ - ih2 + 2 * (X : R[X]) * ih1 + 2 * h₃ - n * h₂\n  | neg_add_one n ih1 ih2 =>\n    have h₁ := congr_arg derivative (T_sub_one R (-n))\n    have h₂ := U_sub_two R (-n)\n    have h₃ := T_eq_U_sub_X_mul_U R (-n)\n    simp only [derivative_sub, derivative_mul, derivative_ofNat, derivative_X] at h₁\n    linear_combination (norm := (push_cast; ring_nf))\n      -ih2 + 2 * (X : R[X]) * ih1 + h₁ + 2 * h₃ + (n + 1) * h₂","premises":[{"full_name":"Mathlib.Tactic.LinearCombination.add_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[41,8],"def_end_pos":[41,14]},{"full_name":"Mathlib.Tactic.LinearCombination.c_mul_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[47,8],"def_end_pos":[47,16]},{"full_name":"Mathlib.Tactic.LinearCombination.eq_of_add","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[111,8],"def_end_pos":[111,17]},{"full_name":"Mathlib.Tactic.LinearCombination.neg_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[45,8],"def_end_pos":[45,14]},{"full_name":"Mathlib.Tactic.LinearCombination.sub_pf","def_path":"Mathlib/Tactic/LinearCombination.lean","def_pos":[44,8],"def_end_pos":[44,14]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.Chebyshev.T_add_two","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[81,8],"def_end_pos":[81,17]},{"full_name":"Polynomial.Chebyshev.T_eq_U_sub_X_mul_U","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[207,8],"def_end_pos":[207,26]},{"full_name":"Polynomial.Chebyshev.T_sub_one","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[91,8],"def_end_pos":[91,17]},{"full_name":"Polynomial.Chebyshev.T_two","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[105,8],"def_end_pos":[105,13]},{"full_name":"Polynomial.Chebyshev.U_one","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[159,8],"def_end_pos":[159,13]},{"full_name":"Polynomial.Chebyshev.U_sub_one","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[149,8],"def_end_pos":[149,17]},{"full_name":"Polynomial.Chebyshev.U_sub_two","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[146,8],"def_end_pos":[146,17]},{"full_name":"Polynomial.Chebyshev.induct","def_path":"Mathlib/RingTheory/Polynomial/Chebyshev.lean","def_pos":[71,18],"def_end_pos":[71,24]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.derivative","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[38,4],"def_end_pos":[38,14]},{"full_name":"Polynomial.derivative_X","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[109,8],"def_end_pos":[109,20]},{"full_name":"Polynomial.derivative_X_pow","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[95,8],"def_end_pos":[95,24]},{"full_name":"Polynomial.derivative_mul","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[238,8],"def_end_pos":[238,22]},{"full_name":"Polynomial.derivative_ofNat","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[190,8],"def_end_pos":[190,24]},{"full_name":"Polynomial.derivative_one","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[113,8],"def_end_pos":[113,22]},{"full_name":"Polynomial.derivative_sub","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[503,8],"def_end_pos":[503,22]}]}]}
{"url":"Mathlib/Order/Partition/Finpartition.lean","commit":"","full_name":"Finpartition.sum_card_parts","start":[504,0],"end":[507,5],"file_path":"Mathlib/Order/Partition/Finpartition.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\n⊢ ∑ i ∈ P.parts, i.card = s.card","state_after":"case h.e'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\n⊢ ∑ i ∈ P.parts, i.card = (P.parts.biUnion id).card","tactic":"convert congr_arg Finset.card P.biUnion_parts","premises":[{"full_name":"Finpartition.biUnion_parts","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[454,8],"def_end_pos":[454,21]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]}]},{"state_before":"case h.e'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\n⊢ ∑ i ∈ P.parts, i.card = (P.parts.biUnion id).card","state_after":"case h.e'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\n⊢ ∑ i ∈ P.parts, i.card = ∑ u ∈ P.parts, (id u).card","tactic":"rw [card_biUnion P.supIndep.pairwiseDisjoint]","premises":[{"full_name":"Finpartition.supIndep","def_path":"Mathlib/Order/Partition/Finpartition.lean","def_pos":[67,2],"def_end_pos":[67,10]},{"full_name":"Finset.SupIndep.pairwiseDisjoint","def_path":"Mathlib/Order/SupIndep.lean","def_pos":[81,8],"def_end_pos":[81,33]},{"full_name":"Finset.card_biUnion","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1848,8],"def_end_pos":[1848,20]}]},{"state_before":"case h.e'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\n⊢ ∑ i ∈ P.parts, i.card = ∑ u ∈ P.parts, (id u).card","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Orientation.lean","commit":"","full_name":"OrthonormalBasis.det_eq_neg_det_of_opposite_orientation","start":[84,0],"end":[91,41],"file_path":"Mathlib/Analysis/InnerProductSpace/Orientation.lean","tactics":[{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : e.toBasis.orientation ≠ f.toBasis.orientation\n⊢ e.toBasis.det = -f.toBasis.det","state_after":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : e.toBasis.orientation ≠ f.toBasis.orientation\n⊢ e.toBasis.det ⇑f.toBasis • f.toBasis.det = -f.toBasis.det","tactic":"rw [e.toBasis.det.eq_smul_basis_det f.toBasis]","premises":[{"full_name":"AlternatingMap.eq_smul_basis_det","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[515,8],"def_end_pos":[515,40]},{"full_name":"Basis.det","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[451,11],"def_end_pos":[451,20]},{"full_name":"OrthonormalBasis.toBasis","def_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","def_pos":[362,14],"def_end_pos":[362,21]}]},{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : e.toBasis.orientation ≠ f.toBasis.orientation\n⊢ e.toBasis.det ⇑f.toBasis • f.toBasis.det = -f.toBasis.det","state_after":"no goals","tactic":"simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,\n    neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]","premises":[{"full_name":"AlternatingMap","def_path":"Mathlib/LinearAlgebra/Alternating/Basic.lean","def_pos":[62,10],"def_end_pos":[62,24]},{"full_name":"OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation","def_path":"Mathlib/Analysis/InnerProductSpace/Orientation.lean","def_pos":[62,8],"def_end_pos":[62,62]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"neg_one_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[241,8],"def_end_pos":[241,20]}]}]}
{"url":"Mathlib/Algebra/Associated/Basic.lean","commit":"","full_name":"Associated.of_pow_associated_of_prime","start":[666,0],"end":[672,31],"file_path":"Mathlib/Algebra/Associated/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CancelCommMonoidWithZero α\np₁ p₂ : α\nk₁ k₂ : ℕ\nhp₁ : Prime p₁\nhp₂ : Prime p₂\nhk₁ : 0 < k₁\nh : p₁ ^ k₁ ~ᵤ p₂ ^ k₂\n⊢ p₁ ~ᵤ p₂","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CancelCommMonoidWithZero α\np₁ p₂ : α\nk₁ k₂ : ℕ\nhp₁ : Prime p₁\nhp₂ : Prime p₂\nhk₁ : 0 < k₁\nh : p₁ ^ k₁ ~ᵤ p₂ ^ k₂\nthis : p₁ ∣ p₂ ^ k₂\n⊢ p₁ ~ᵤ p₂","tactic":"have : p₁ ∣ p₂ ^ k₂ := by\n    rw [← h.dvd_iff_dvd_right]\n    apply dvd_pow_self _ hk₁.ne'","premises":[{"full_name":"Associated.dvd_iff_dvd_right","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[535,8],"def_end_pos":[535,36]},{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"dvd_pow_self","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[146,6],"def_end_pos":[146,18]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CancelCommMonoidWithZero α\np₁ p₂ : α\nk₁ k₂ : ℕ\nhp₁ : Prime p₁\nhp₂ : Prime p₂\nhk₁ : 0 < k₁\nh : p₁ ^ k₁ ~ᵤ p₂ ^ k₂\nthis : p₁ ∣ p₂ ^ k₂\n⊢ p₁ ~ᵤ p₂","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CancelCommMonoidWithZero α\np₁ p₂ : α\nk₁ k₂ : ℕ\nhp₁ : Prime p₁\nhp₂ : Prime p₂\nhk₁ : 0 < k₁\nh : p₁ ^ k₁ ~ᵤ p₂ ^ k₂\nthis : p₁ ∣ p₂ ^ k₂\n⊢ p₁ ∣ p₂","tactic":"rw [← hp₁.dvd_prime_iff_associated hp₂]","premises":[{"full_name":"Prime.dvd_prime_iff_associated","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[618,8],"def_end_pos":[618,38]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CancelCommMonoidWithZero α\np₁ p₂ : α\nk₁ k₂ : ℕ\nhp₁ : Prime p₁\nhp₂ : Prime p₂\nhk₁ : 0 < k₁\nh : p₁ ^ k₁ ~ᵤ p₂ ^ k₂\nthis : p₁ ∣ p₂ ^ k₂\n⊢ p₁ ∣ p₂","state_after":"no goals","tactic":"exact hp₁.dvd_of_dvd_pow this","premises":[{"full_name":"Prime.dvd_of_dvd_pow","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[70,8],"def_end_pos":[70,22]}]}]}
{"url":"Mathlib/Data/ZMod/Basic.lean","commit":"","full_name":"ZMod.neg_eq_self_mod_two","start":[965,0],"end":[966,71],"file_path":"Mathlib/Data/ZMod/Basic.lean","tactics":[{"state_before":"m n : ℕ\na : ZMod 2\n⊢ -a = a","state_after":"case tail.head.h\nm n : ℕ\n⊢ ↑(-1) = 1","tactic":"fin_cases a <;> apply Fin.ext <;> simp [Fin.coe_neg, Int.natMod]","premises":[{"full_name":"Fin.coe_neg","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[1427,18],"def_end_pos":[1427,25]},{"full_name":"Fin.ext","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[40,15],"def_end_pos":[40,18]},{"full_name":"Int.natMod","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[582,4],"def_end_pos":[582,10]}]},{"state_before":"case tail.head.h\nm n : ℕ\n⊢ ↑(-1) = 1","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","commit":"","full_name":"SimpleGraph.cliqueFree_map_iff","start":[326,0],"end":[332,42],"file_path":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\nm n : ℕ\ns : Finset α\nf : α ↪ β\ninst✝ : Nonempty α\n⊢ (SimpleGraph.map f G).CliqueFree n ↔ G.CliqueFree n","state_after":"case inl\nα : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\nm n : ℕ\ns : Finset α\nf : α ↪ β\ninst✝ : Nonempty α\nhle : n ≤ 1\n⊢ (SimpleGraph.map f G).CliqueFree n ↔ G.CliqueFree n\n\ncase inr\nα : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\nm n : ℕ\ns : Finset α\nf : α ↪ β\ninst✝ : Nonempty α\nhlt : 1 < n\n⊢ (SimpleGraph.map f G).CliqueFree n ↔ G.CliqueFree n","tactic":"obtain (hle | hlt) := le_or_lt n 1","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]},{"state_before":"case inr\nα : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\nm n : ℕ\ns : Finset α\nf : α ↪ β\ninst✝ : Nonempty α\nhlt : 1 < n\n⊢ (SimpleGraph.map f G).CliqueFree n ↔ G.CliqueFree n","state_after":"no goals","tactic":"simp [CliqueFree, isNClique_map_iff hlt]","premises":[{"full_name":"SimpleGraph.CliqueFree","def_path":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","def_pos":[261,4],"def_end_pos":[261,14]},{"full_name":"SimpleGraph.isNClique_map_iff","def_path":"Mathlib/Combinatorics/SimpleGraph/Clique.lean","def_pos":[191,8],"def_end_pos":[191,25]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/FixedPoint.lean","commit":"","full_name":"Ordinal.mul_eq_right_iff_opow_omega_dvd","start":[583,0],"end":[596,53],"file_path":"Mathlib/SetTheory/Ordinal/FixedPoint.lean","tactics":[{"state_before":"a b : Ordinal.{u_1}\n⊢ a * b = b ↔ a ^ ω ∣ b","state_after":"case inl\na b : Ordinal.{u_1}\nha : a = 0\n⊢ a * b = b ↔ a ^ ω ∣ b\n\ncase inr\na b : Ordinal.{u_1}\nha : 0 < a\n⊢ a * b = b ↔ a ^ ω ∣ b","tactic":"rcases eq_zero_or_pos a with ha | ha","premises":[{"full_name":"Ordinal.eq_zero_or_pos","def_path":"Mathlib/SetTheory/Ordinal/Basic.lean","def_pos":[359,8],"def_end_pos":[359,22]}]},{"state_before":"case inr\na b : Ordinal.{u_1}\nha : 0 < a\n⊢ a * b = b ↔ a ^ ω ∣ b","state_after":"case inr.refine_1\na b : Ordinal.{u_1}\nha : 0 < a\nhab : a * b = b\n⊢ a ^ ω ∣ b\n\ncase inr.refine_2\na b : Ordinal.{u_1}\nha : 0 < a\nh : a ^ ω ∣ b\n⊢ a * b = b","tactic":"refine ⟨fun hab => ?_, fun h => ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"case inr.refine_2\na b : Ordinal.{u_1}\nha : 0 < a\nh : a ^ ω ∣ b\n⊢ a * b = b","state_after":"case inr.refine_2.intro\na b : Ordinal.{u_1}\nha : 0 < a\nc : Ordinal.{u_1}\nhc : b = a ^ ω * c\n⊢ a * b = b","tactic":"cases' h with c hc","premises":[]},{"state_before":"case inr.refine_2.intro\na b : Ordinal.{u_1}\nha : 0 < a\nc : Ordinal.{u_1}\nhc : b = a ^ ω * c\n⊢ a * b = b","state_after":"no goals","tactic":"rw [hc, ← mul_assoc, ← opow_one_add, one_add_omega]","premises":[{"full_name":"Ordinal.one_add_omega","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[525,8],"def_end_pos":[525,21]},{"full_name":"Ordinal.opow_one_add","def_path":"Mathlib/SetTheory/Ordinal/Exponential.lean","def_pos":[185,8],"def_end_pos":[185,20]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
{"url":"Mathlib/Order/SuccPred/Relation.lean","commit":"","full_name":"reflTransGen_of_succ_of_ge","start":[34,0],"end":[39,49],"file_path":"Mathlib/Order/SuccPred/Relation.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nr : α → α → Prop\nn m : α\nh : ∀ i ∈ Ico m n, r (succ i) i\nhmn : m ≤ n\n⊢ ReflTransGen r n m","state_after":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nr : α → α → Prop\nn m : α\nh : ∀ i ∈ Ico m n, r (succ i) i\nhmn : m ≤ n\n⊢ ReflTransGen (swap r) m n","tactic":"rw [← reflTransGen_swap]","premises":[{"full_name":"Relation.reflTransGen_swap","def_path":"Mathlib/Logic/Relation.lean","def_pos":[546,8],"def_end_pos":[546,25]}]},{"state_before":"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nr : α → α → Prop\nn m : α\nh : ∀ i ∈ Ico m n, r (succ i) i\nhmn : m ≤ n\n⊢ ReflTransGen (swap r) m n","state_after":"no goals","tactic":"exact reflTransGen_of_succ_of_le (swap r) h hmn","premises":[{"full_name":"Function.swap","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[67,7],"def_end_pos":[67,11]},{"full_name":"reflTransGen_of_succ_of_le","def_path":"Mathlib/Order/SuccPred/Relation.lean","def_pos":[23,8],"def_end_pos":[23,34]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","commit":"","full_name":"EisensteinSeries.summand_bound_of_mem_verticalStrip","start":[137,0],"end":[142,23],"file_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","tactics":[{"state_before":"z : ℍ\nk : ℝ\nhk : 0 ≤ k\nx : Fin 2 → ℤ\nA B : ℝ\nhB : 0 < B\nhz : z ∈ verticalStrip A B\n⊢ Complex.abs (↑(x 0) * ↑z + ↑(x 1)) ^ (-k) ≤ r ⟨{ re := A, im := B }, hB⟩ ^ (-k) * ‖x‖ ^ (-k)","state_after":"z : ℍ\nk : ℝ\nhk : 0 ≤ k\nx : Fin 2 → ℤ\nA B : ℝ\nhB : 0 < B\nhz : z ∈ verticalStrip A B\n⊢ r z ^ (-k) ≤ r ⟨{ re := A, im := B }, hB⟩ ^ (-k)","tactic":"refine (summand_bound z hk x).trans (mul_le_mul_of_nonneg_right ?_ (by positivity))","premises":[{"full_name":"EisensteinSeries.summand_bound","def_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","def_pos":[125,6],"def_end_pos":[125,19]},{"full_name":"mul_le_mul_of_nonneg_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[194,8],"def_end_pos":[194,34]}]},{"state_before":"z : ℍ\nk : ℝ\nhk : 0 ≤ k\nx : Fin 2 → ℤ\nA B : ℝ\nhB : 0 < B\nhz : z ∈ verticalStrip A B\n⊢ r z ^ (-k) ≤ r ⟨{ re := A, im := B }, hB⟩ ^ (-k)","state_after":"no goals","tactic":"exact Real.rpow_le_rpow_of_nonpos (r_pos _) (r_lower_bound_on_verticalStrip z hB hz)\n    (neg_nonpos.mpr hk)","premises":[{"full_name":"EisensteinSeries.r_lower_bound_on_verticalStrip","def_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","def_pos":[76,6],"def_end_pos":[76,36]},{"full_name":"EisensteinSeries.r_pos","def_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","def_pos":[73,6],"def_end_pos":[73,11]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Real.rpow_le_rpow_of_nonpos","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[507,6],"def_end_pos":[507,28]}]}]}
{"url":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","commit":"","full_name":"lt_inv'","start":[253,0],"end":[254,72],"file_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","tactics":[{"state_before":"α : Type u\ninst✝³ : Group α\ninst✝² : LT α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : α\n⊢ a < b⁻¹ ↔ b < a⁻¹","state_after":"no goals","tactic":"rw [← inv_lt_inv_iff, inv_inv]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"inv_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[734,8],"def_end_pos":[734,15]},{"full_name":"inv_lt_inv_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[246,8],"def_end_pos":[246,22]}]}]}
{"url":"Mathlib/MeasureTheory/Group/LIntegral.lean","commit":"","full_name":"MeasureTheory.lintegral_mul_left_eq_self","start":[29,0],"end":[37,33],"file_path":"Mathlib/MeasureTheory/Group/LIntegral.lean","tactics":[{"state_before":"G : Type u_1\ninst✝³ : MeasurableSpace G\nμ : Measure G\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulLeftInvariant\nf : G → ℝ≥0∞\ng : G\n⊢ ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ","state_after":"case h.e'_3.h.e'_3\nG : Type u_1\ninst✝³ : MeasurableSpace G\nμ : Measure G\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulLeftInvariant\nf : G → ℝ≥0∞\ng : G\n⊢ μ = map (⇑(MeasurableEquiv.mulLeft g)) μ","tactic":"convert (lintegral_map_equiv f <| MeasurableEquiv.mulLeft g).symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"MeasurableEquiv.mulLeft","def_path":"Mathlib/MeasureTheory/Group/MeasurableEquiv.lean","def_pos":[85,4],"def_end_pos":[85,11]},{"full_name":"MeasureTheory.lintegral_map_equiv","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[1420,8],"def_end_pos":[1420,27]}]},{"state_before":"case h.e'_3.h.e'_3\nG : Type u_1\ninst✝³ : MeasurableSpace G\nμ : Measure G\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulLeftInvariant\nf : G → ℝ≥0∞\ng : G\n⊢ μ = map (⇑(MeasurableEquiv.mulLeft g)) μ","state_after":"no goals","tactic":"simp [map_mul_left_eq_self μ g]","premises":[{"full_name":"MeasureTheory.map_mul_left_eq_self","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[74,8],"def_end_pos":[74,28]}]}]}
{"url":"Mathlib/Analysis/Normed/Lp/ProdLp.lean","commit":"","full_name":"WithLp.prod_aux_uniformity_eq","start":[425,0],"end":[433,43],"file_path":"Mathlib/Analysis/Normed/Lp/ProdLp.lean","tactics":[{"state_before":"p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\n⊢ 𝓤 (WithLp p (α × β)) = 𝓤 (α × β)","state_after":"p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nA : UniformInducing ⇑(WithLp.equiv p (α × β))\n⊢ 𝓤 (WithLp p (α × β)) = 𝓤 (α × β)","tactic":"have A : UniformInducing (WithLp.equiv p (α × β)) :=\n    (prod_antilipschitzWith_equiv_aux p α β).uniformInducing\n      (prod_lipschitzWith_equiv_aux p α β).uniformContinuous","premises":[{"full_name":"AntilipschitzWith.uniformInducing","def_path":"Mathlib/Topology/MetricSpace/Antilipschitz.lean","def_pos":[146,18],"def_end_pos":[146,33]},{"full_name":"LipschitzWith.uniformContinuous","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[178,18],"def_end_pos":[178,35]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"UniformInducing","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[32,10],"def_end_pos":[32,25]},{"full_name":"WithLp.equiv","def_path":"Mathlib/Analysis/Normed/Lp/WithLp.lean","def_pos":[55,14],"def_end_pos":[55,19]},{"full_name":"WithLp.prod_antilipschitzWith_equiv_aux","def_path":"Mathlib/Analysis/Normed/Lp/ProdLp.lean","def_pos":[406,8],"def_end_pos":[406,40]},{"full_name":"WithLp.prod_lipschitzWith_equiv_aux","def_path":"Mathlib/Analysis/Normed/Lp/ProdLp.lean","def_pos":[384,8],"def_end_pos":[384,36]}]},{"state_before":"p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nA : UniformInducing ⇑(WithLp.equiv p (α × β))\n⊢ 𝓤 (WithLp p (α × β)) = 𝓤 (α × β)","state_after":"p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nA : UniformInducing ⇑(WithLp.equiv p (α × β))\nthis : (fun x => ((WithLp.equiv p (α × β)) x.1, (WithLp.equiv p (α × β)) x.2)) = id\n⊢ 𝓤 (WithLp p (α × β)) = 𝓤 (α × β)","tactic":"have : (fun x : WithLp p (α × β) × WithLp p (α × β) =>\n    ((WithLp.equiv p (α × β)) x.fst, (WithLp.equiv p (α × β)) x.snd)) = id := by\n    ext i <;> rfl","premises":[{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"WithLp","def_path":"Mathlib/Analysis/Normed/Lp/WithLp.lean","def_pos":[47,4],"def_end_pos":[47,10]},{"full_name":"WithLp.equiv","def_path":"Mathlib/Analysis/Normed/Lp/WithLp.lean","def_pos":[55,14],"def_end_pos":[55,19]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]},{"state_before":"p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nA : UniformInducing ⇑(WithLp.equiv p (α × β))\nthis : (fun x => ((WithLp.equiv p (α × β)) x.1, (WithLp.equiv p (α × β)) x.2)) = id\n⊢ 𝓤 (WithLp p (α × β)) = 𝓤 (α × β)","state_after":"no goals","tactic":"rw [← A.comap_uniformity, this, comap_id]","premises":[{"full_name":"Filter.comap_id","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1919,8],"def_end_pos":[1919,16]},{"full_name":"UniformInducing.comap_uniformity","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[35,2],"def_end_pos":[35,18]}]}]}
{"url":"Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean","commit":"","full_name":"Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub","start":[28,0],"end":[47,89],"file_path":"Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhm : m ∈ (f.genEigenspace μ) k\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\n⊢ g m = μ • m","tactic":"set p := f.genEigenspace μ k","premises":[{"full_name":"Module.End.genEigenspace","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[156,4],"def_end_pos":[156,17]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\n⊢ g m = μ • m","tactic":"have h₁ : MapsTo g p p := mapsTo_genEigenspace_of_comm hfg μ k","premises":[{"full_name":"Module.End.mapsTo_genEigenspace_of_comm","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[273,6],"def_end_pos":[273,34]},{"full_name":"Set.MapsTo","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[232,4],"def_end_pos":[232,10]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\n⊢ g m = μ • m","tactic":"have h₂ : MapsTo (g - algebraMap R (End R M) μ) p p :=\n    mapsTo_genEigenspace_of_comm (hfg.sub_right <| Algebra.commute_algebraMap_right μ f) μ k","premises":[{"full_name":"Algebra.commute_algebraMap_right","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[288,6],"def_end_pos":[288,30]},{"full_name":"Commute.sub_right","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[97,8],"def_end_pos":[97,17]},{"full_name":"Module.End","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[28,7],"def_end_pos":[28,17]},{"full_name":"Module.End.mapsTo_genEigenspace_of_comm","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[273,6],"def_end_pos":[273,34]},{"full_name":"Set.MapsTo","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[232,4],"def_end_pos":[232,10]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\n⊢ g m = μ • m","tactic":"have h₃ : MapsTo (f - g) p p :=\n    mapsTo_genEigenspace_of_comm (Commute.sub_right rfl hfg) μ k","premises":[{"full_name":"Commute.sub_right","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[97,8],"def_end_pos":[97,17]},{"full_name":"Module.End.mapsTo_genEigenspace_of_comm","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[273,6],"def_end_pos":[273,34]},{"full_name":"Set.MapsTo","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[232,4],"def_end_pos":[232,10]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\nh₄ : MapsTo ⇑(f - (algebraMap R (End R M)) μ) ↑p ↑p\n⊢ g m = μ • m","tactic":"have h₄ : MapsTo (f - algebraMap R (End R M) μ) p p :=\n    mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ k","premises":[{"full_name":"Algebra.mul_sub_algebraMap_commutes","def_path":"Mathlib/Algebra/Algebra/Basic.lean","def_pos":[121,8],"def_end_pos":[121,35]},{"full_name":"Module.End","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[28,7],"def_end_pos":[28,17]},{"full_name":"Module.End.mapsTo_genEigenspace_of_comm","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[273,6],"def_end_pos":[273,34]},{"full_name":"Set.MapsTo","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[232,4],"def_end_pos":[232,10]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\nh₄ : MapsTo ⇑(f - (algebraMap R (End R M)) μ) ↑p ↑p\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\nh₄ : MapsTo ⇑(f - (algebraMap R (End R M)) μ) ↑p ↑p\nhfg : Commute (f - (algebraMap R (End R M)) μ) (f - g)\n⊢ g m = μ • m","tactic":"replace hfg : Commute (f - algebraMap R (End R M) μ) (f - g) :=\n    (Commute.sub_right rfl hfg).sub_left <| Algebra.commute_algebraMap_left μ (f - g)","premises":[{"full_name":"Algebra.commute_algebraMap_left","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[285,6],"def_end_pos":[285,29]},{"full_name":"Commute","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[35,4],"def_end_pos":[35,11]},{"full_name":"Commute.sub_left","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[101,8],"def_end_pos":[101,16]},{"full_name":"Commute.sub_right","def_path":"Mathlib/Algebra/Ring/Commute.lean","def_pos":[97,8],"def_end_pos":[97,17]},{"full_name":"Module.End","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[28,7],"def_end_pos":[28,17]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\nh₄ : MapsTo ⇑(f - (algebraMap R (End R M)) μ) ↑p ↑p\nhfg : Commute (f - (algebraMap R (End R M)) μ) (f - g)\n⊢ g m = μ • m","state_after":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\nh₄ : MapsTo ⇑(f - (algebraMap R (End R M)) μ) ↑p ↑p\nhfg : Commute (f - (algebraMap R (End R M)) μ) (f - g)\n⊢ IsNilpotent (LinearMap.restrict (g - (algebraMap R (End R M)) μ) h₂)","tactic":"suffices IsNilpotent ((g - algebraMap R (End R M) μ).restrict h₂) by\n    replace this : g.restrict h₁ - algebraMap R (End R p) μ = 0 :=\n      eq_zero_of_isNilpotent_isSemisimple this (by simpa using hss.restrict)\n    simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this ⟨m, hm⟩","premises":[{"full_name":"IsNilpotent","def_path":"Mathlib/RingTheory/Nilpotent/Defs.lean","def_pos":[38,4],"def_end_pos":[38,15]},{"full_name":"LinearMap.congr_fun","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[322,18],"def_end_pos":[322,27]},{"full_name":"LinearMap.restrict","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[155,4],"def_end_pos":[155,12]},{"full_name":"LinearMap.restrict_apply","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[164,8],"def_end_pos":[164,22]},{"full_name":"Module.End","def_path":"Mathlib/Algebra/Module/LinearMap/End.lean","def_pos":[28,7],"def_end_pos":[28,17]},{"full_name":"Module.End.IsSemisimple.restrict","def_path":"Mathlib/LinearAlgebra/Semisimple.lean","def_pos":[93,6],"def_end_pos":[93,27]},{"full_name":"Module.End.eq_zero_of_isNilpotent_isSemisimple","def_path":"Mathlib/LinearAlgebra/Semisimple.lean","def_pos":[79,6],"def_end_pos":[79,41]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nμ : R\nk : ℕ\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace μ) k\nhm : m ∈ p\nh₁ : MapsTo ⇑g ↑p ↑p\nh₂ : MapsTo ⇑(g - (algebraMap R (End R M)) μ) ↑p ↑p\nh₃ : MapsTo ⇑(f - g) ↑p ↑p\nh₄ : MapsTo ⇑(f - (algebraMap R (End R M)) μ) ↑p ↑p\nhfg : Commute (f - (algebraMap R (End R M)) μ) (f - g)\n⊢ IsNilpotent (LinearMap.restrict (g - (algebraMap R (End R M)) μ) h₂)","state_after":"no goals","tactic":"simpa [LinearMap.restrict_sub h₄ h₃] using (LinearMap.restrict_commute hfg h₄ h₃).isNilpotent_sub\n    (f.isNilpotent_restrict_sub_algebraMap μ k) (Module.End.isNilpotent.restrict h₃ hnil)","premises":[{"full_name":"Commute.isNilpotent_sub","def_path":"Mathlib/RingTheory/Nilpotent/Basic.lean","def_pos":[168,8],"def_end_pos":[168,23]},{"full_name":"LinearMap.restrict_commute","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[184,6],"def_end_pos":[184,22]},{"full_name":"LinearMap.restrict_sub","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[168,6],"def_end_pos":[168,18]},{"full_name":"Module.End.isNilpotent.restrict","def_path":"Mathlib/RingTheory/Nilpotent/Lemmas.lean","def_pos":[102,6],"def_end_pos":[102,26]},{"full_name":"Module.End.isNilpotent_restrict_sub_algebraMap","def_path":"Mathlib/LinearAlgebra/Eigenspace/Basic.lean","def_pos":[289,6],"def_end_pos":[289,41]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/VectorMeasure.lean","commit":"","full_name":"MeasureTheory.VectorMeasure.mapRange_id","start":[531,0],"end":[534,5],"file_path":"Mathlib/MeasureTheory/Measure/VectorMeasure.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm inst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\nv : VectorMeasure α M\nN : Type u_4\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\n⊢ v.mapRange (AddMonoidHom.id M) ⋯ = v","state_after":"case h\nα : Type u_1\nβ : Type u_2\nm inst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\nv : VectorMeasure α M\nN : Type u_4\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\ni✝ : Set α\na✝ : MeasurableSet i✝\n⊢ ↑(v.mapRange (AddMonoidHom.id M) ⋯) i✝ = ↑v i✝","tactic":"ext","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nm inst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\nv : VectorMeasure α M\nN : Type u_4\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\ni✝ : Set α\na✝ : MeasurableSet i✝\n⊢ ↑(v.mapRange (AddMonoidHom.id M) ⋯) i✝ = ↑v i✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Matroid/Map.lean","commit":"","full_name":"Matroid.mapEmbedding_base_iff","start":[549,0],"end":[555,36],"file_path":"Mathlib/Data/Matroid/Map.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set β\n⊢ (M.mapEmbedding f).Base B ↔ M.Base (⇑f ⁻¹' B) ∧ B ⊆ range ⇑f","state_after":"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set β\n⊢ (∃ B₀, M.Base B₀ ∧ B = ⇑f '' B₀) ↔ M.Base (⇑f ⁻¹' B) ∧ B ⊆ range ⇑f","tactic":"rw [mapEmbedding, map_base_iff]","premises":[{"full_name":"Matroid.mapEmbedding","def_path":"Mathlib/Data/Matroid/Map.lean","def_pos":[524,4],"def_end_pos":[524,16]},{"full_name":"Matroid.map_base_iff","def_path":"Mathlib/Data/Matroid/Map.lean","def_pos":[375,14],"def_end_pos":[375,26]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set β\n⊢ (∃ B₀, M.Base B₀ ∧ B = ⇑f '' B₀) ↔ M.Base (⇑f ⁻¹' B) ∧ B ⊆ range ⇑f","state_after":"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set β\n⊢ (∃ B₀, M.Base B₀ ∧ B = ⇑f '' B₀) → M.Base (⇑f ⁻¹' B) ∧ B ⊆ range ⇑f","tactic":"refine ⟨?_, fun ⟨h,h'⟩ ↦ ⟨f ⁻¹' B, h, by rwa [eq_comm, image_preimage_eq_iff]⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Set.image_preimage_eq_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[672,8],"def_end_pos":[672,29]},{"full_name":"Set.preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[106,4],"def_end_pos":[106,12]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set β\n⊢ (∃ B₀, M.Base B₀ ∧ B = ⇑f '' B₀) → M.Base (⇑f ⁻¹' B) ∧ B ⊆ range ⇑f","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set α\nhB : M.Base B\n⊢ M.Base (⇑f ⁻¹' (⇑f '' B)) ∧ ⇑f '' B ⊆ range ⇑f","tactic":"rintro ⟨B, hB, rfl⟩","premises":[]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set α\nhB : M.Base B\n⊢ M.Base (⇑f ⁻¹' (⇑f '' B)) ∧ ⇑f '' B ⊆ range ⇑f","state_after":"case intro.intro\nα : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set α\nhB : M.Base B\n⊢ M.Base B ∧ ⇑f '' B ⊆ range ⇑f","tactic":"rw [preimage_image_eq _ f.injective]","premises":[{"full_name":"Function.Embedding.injective","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[124,18],"def_end_pos":[124,27]},{"full_name":"Set.preimage_image_eq","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[417,8],"def_end_pos":[417,25]}]},{"state_before":"case intro.intro\nα : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\nf : α ↪ β\nB : Set α\nhB : M.Base B\n⊢ M.Base B ∧ ⇑f '' B ⊆ range ⇑f","state_after":"no goals","tactic":"exact ⟨hB, image_subset_range _ _⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Set.image_subset_range","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[604,8],"def_end_pos":[604,26]}]}]}
{"url":"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean","commit":"","full_name":"norm_lineDeriv_le_of_lipschitzOn","start":[434,0],"end":[440,78],"file_path":"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf✝ f₀ f₁ : E → F\nf' : F\ns✝ t : Set E\nx v : E\nL : E →L[𝕜] F\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖lineDeriv 𝕜 f x₀ v‖ ≤ ↑C * ‖v‖","state_after":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf✝ f₀ f₁ : E → F\nf' : F\ns✝ t : Set E\nx v : E\nL : E →L[𝕜] F\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ ↑C * ‖x - x₀‖","tactic":"refine norm_lineDeriv_le_of_lip' 𝕜 C.coe_nonneg ?_","premises":[{"full_name":"NNReal.coe_nonneg","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[122,8],"def_end_pos":[122,18]},{"full_name":"norm_lineDeriv_le_of_lip'","def_path":"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean","def_pos":[424,8],"def_end_pos":[424,33]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf✝ f₀ f₁ : E → F\nf' : F\ns✝ t : Set E\nx v : E\nL : E →L[𝕜] F\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ ↑C * ‖x - x₀‖","state_after":"no goals","tactic":"filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs)","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"mem_of_mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[737,8],"def_end_pos":[737,23]}]}]}
{"url":"Mathlib/NumberTheory/BernoulliPolynomials.lean","commit":"","full_name":"Polynomial.bernoulli_eval_zero","start":[70,0],"end":[77,13],"file_path":"Mathlib/NumberTheory/BernoulliPolynomials.lean","tactics":[{"state_before":"n : ℕ\n⊢ eval 0 (bernoulli n) = _root_.bernoulli n","state_after":"n : ℕ\n⊢ ∑ x ∈ range n, eval 0 ((monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x))) +\n      eval 0 ((monomial (n - n)) (_root_.bernoulli n * ↑(n.choose n))) =\n    _root_.bernoulli n","tactic":"rw [bernoulli, eval_finset_sum, sum_range_succ]","premises":[{"full_name":"Finset.sum_range_succ","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1296,2],"def_end_pos":[1296,13]},{"full_name":"Polynomial.bernoulli","def_path":"Mathlib/NumberTheory/BernoulliPolynomials.lean","def_pos":[51,4],"def_end_pos":[51,13]},{"full_name":"Polynomial.eval_finset_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[412,8],"def_end_pos":[412,23]}]},{"state_before":"n : ℕ\n⊢ ∑ x ∈ range n, eval 0 ((monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x))) +\n      eval 0 ((monomial (n - n)) (_root_.bernoulli n * ↑(n.choose n))) =\n    _root_.bernoulli n","state_after":"n : ℕ\nthis : ∑ x ∈ range n, _root_.bernoulli x * ↑(n.choose x) * 0 ^ (n - x) = 0\n⊢ ∑ x ∈ range n, eval 0 ((monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x))) +\n      eval 0 ((monomial (n - n)) (_root_.bernoulli n * ↑(n.choose n))) =\n    _root_.bernoulli n","tactic":"have : ∑ x ∈ range n, _root_.bernoulli x * n.choose x * 0 ^ (n - x) = 0 := by\n    apply sum_eq_zero fun x hx => _\n    intros x hx\n    simp [tsub_eq_zero_iff_le, mem_range.1 hx]","premises":[{"full_name":"Finset.mem_range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2450,8],"def_end_pos":[2450,17]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_eq_zero","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[384,2],"def_end_pos":[384,13]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.choose","def_path":"Mathlib/Data/Nat/Choose/Basic.lean","def_pos":[45,4],"def_end_pos":[45,10]},{"full_name":"bernoulli","def_path":"Mathlib/NumberTheory/Bernoulli.lean","def_pos":[184,4],"def_end_pos":[184,13]},{"full_name":"tsub_eq_zero_iff_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[275,8],"def_end_pos":[275,27]}]},{"state_before":"n : ℕ\nthis : ∑ x ∈ range n, _root_.bernoulli x * ↑(n.choose x) * 0 ^ (n - x) = 0\n⊢ ∑ x ∈ range n, eval 0 ((monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x))) +\n      eval 0 ((monomial (n - n)) (_root_.bernoulli n * ↑(n.choose n))) =\n    _root_.bernoulli n","state_after":"no goals","tactic":"simp [this]","premises":[]}]}
{"url":"Mathlib/Data/Sum/Basic.lean","commit":"","full_name":"Sum.getLeft_eq_getLeft?","start":[46,0],"end":[47,72],"file_path":"Mathlib/Data/Sum/Basic.lean","tactics":[{"state_before":"α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nx y : α ⊕ β\nh₁ : x.isLeft = true\nh₂ : x.getLeft?.isSome = true\n⊢ x.getLeft h₁ = x.getLeft?.get h₂","state_after":"no goals","tactic":"simp [← getLeft?_eq_some_iff]","premises":[{"full_name":"Sum.getLeft?_eq_some_iff","def_path":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","def_pos":[65,16],"def_end_pos":[65,36]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","commit":"","full_name":"ContinuousLinearEquiv.comp_fderivWithin","start":[137,0],"end":[143,90],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = (↑iso).comp (fderivWithin 𝕜 f s x)","state_after":"case pos\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt 𝕜 s x\nh : DifferentiableWithinAt 𝕜 f s x\n⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = (↑iso).comp (fderivWithin 𝕜 f s x)\n\ncase neg\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt 𝕜 s x\nh : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = (↑iso).comp (fderivWithin 𝕜 f s x)","tactic":"by_cases h : DifferentiableWithinAt 𝕜 f s x","premises":[{"full_name":"DifferentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[165,4],"def_end_pos":[165,26]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Algebra/Ring/Regular.lean","commit":"","full_name":"isRightRegular_of_non_zero_divisor","start":[23,0],"end":[28,31],"file_path":"Mathlib/Algebra/Ring/Regular.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), x * k = 0 → x = 0\n⊢ IsRightRegular k","state_after":"α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), x * k = 0 → x = 0\nx y : α\nh' : x * k = y * k\n⊢ (x - y) * k = 0","tactic":"refine fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ ?_)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), x * k = 0 → x = 0\nx y : α\nh' : x * k = y * k\n⊢ (x - y) * k = 0","state_after":"no goals","tactic":"rw [sub_mul, sub_eq_zero, h']","premises":[{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","commit":"","full_name":"IsCompact.exists_open_superset_measure_lt_top'","start":[1337,0],"end":[1352,75],"file_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)\n⊢ ∃ U ⊇ s, IsOpen U ∧ μ U < ⊤","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)\n⊢ ∃ U ⊇ ∅, IsOpen U ∧ μ U < ⊤\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)\n⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → (∃ U ⊇ t, IsOpen U ∧ μ U < ⊤) → ∃ U ⊇ s, IsOpen U ∧ μ U < ⊤\n\ncase refine_3\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)\n⊢ ∀ ⦃s t : Set α⦄, (∃ U ⊇ s, IsOpen U ∧ μ U < ⊤) → (∃ U ⊇ t, IsOpen U ∧ μ U < ⊤) → ∃ U ⊇ s ∪ t, IsOpen U ∧ μ U < ⊤\n\ncase refine_4\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∃ U ⊇ t, IsOpen U ∧ μ U < ⊤","tactic":"refine IsCompact.induction_on h ?_ ?_ ?_ ?_","premises":[{"full_name":"IsCompact.induction_on","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[68,8],"def_end_pos":[68,30]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.zero_isRoot_of_coeff_zero_eq_zero","start":[438,0],"end":[439,37],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r : R[X]\nx : R\np : R[X]\nhp : p.coeff 0 = 0\n⊢ p.IsRoot 0","state_after":"no goals","tactic":"rwa [coeff_zero_eq_eval_zero] at hp","premises":[{"full_name":"Polynomial.coeff_zero_eq_eval_zero","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[430,8],"def_end_pos":[430,31]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.add_of_Y_ne'","start":[1221,0],"end":[1230,44],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ W.add P Q =\n    W.dblZ P •\n      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)\n          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),\n        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)\n          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),\n        1]","state_after":"no goals","tactic":"rw [add, if_pos <| equiv_of_X_eq_of_Y_eq hPz hQz hx <| Y_eq_of_Y_ne' hP hQ hx hy,\n    dblXYZ_of_Z_ne_zero hP hQ hPz hQz hx hy]","premises":[{"full_name":"WeierstrassCurve.Jacobian.Y_eq_of_Y_ne'","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[507,6],"def_end_pos":[507,19]},{"full_name":"WeierstrassCurve.Jacobian.add","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[1159,18],"def_end_pos":[1159,21]},{"full_name":"WeierstrassCurve.Jacobian.dblXYZ_of_Z_ne_zero","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[754,6],"def_end_pos":[754,25]},{"full_name":"WeierstrassCurve.Jacobian.equiv_of_X_eq_of_Y_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[204,6],"def_end_pos":[204,27]},{"full_name":"if_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[932,8],"def_end_pos":[932,14]}]}]}
{"url":"Mathlib/Order/Filter/Prod.lean","commit":"","full_name":"Filter.prod_neBot","start":[390,0],"end":[391,48],"file_path":"Mathlib/Order/Filter/Prod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ns : Set α\nt : Set β\nf : Filter α\ng : Filter β\n⊢ (f ×ˢ g).NeBot ↔ f.NeBot ∧ g.NeBot","state_after":"no goals","tactic":"simp only [neBot_iff, Ne, prod_eq_bot, not_or]","premises":[{"full_name":"Filter.neBot_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[462,8],"def_end_pos":[462,17]},{"full_name":"Filter.prod_eq_bot","def_path":"Mathlib/Order/Filter/Prod.lean","def_pos":[382,8],"def_end_pos":[382,19]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","commit":"","full_name":"Complex.arg_lt_pi_div_two_iff","start":[358,0],"end":[365,31],"file_path":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","tactics":[{"state_before":"a x z✝ z : ℂ\n⊢ z.arg < π / 2 ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0","state_after":"a x z✝ z : ℂ\n⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0","tactic":"rw [lt_iff_le_and_ne, arg_le_pi_div_two_iff, Ne, arg_eq_pi_div_two_iff]","premises":[{"full_name":"Complex.arg_eq_pi_div_two_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","def_pos":[240,8],"def_end_pos":[240,29]},{"full_name":"Complex.arg_le_pi_div_two_iff","def_path":"Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean","def_pos":[323,8],"def_end_pos":[323,29]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"lt_iff_le_and_ne","def_path":"Mathlib/Order/Basic.lean","def_pos":[309,8],"def_end_pos":[309,24]}]},{"state_before":"a x z✝ z : ℂ\n⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0","state_after":"case inl\na x z✝ z : ℂ\nhre : z.re < 0\n⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0\n\ncase inr.inl\na x z✝ z : ℂ\nhre : z.re = 0\n⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0\n\ncase inr.inr\na x z✝ z : ℂ\nhre : 0 < z.re\n⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0","tactic":"rcases lt_trichotomy z.re 0 with hre | hre | hre","premises":[{"full_name":"Complex.re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[29,2],"def_end_pos":[29,4]},{"full_name":"lt_trichotomy","def_path":"Mathlib/Order/Defs.lean","def_pos":[266,8],"def_end_pos":[266,21]}]}]}
{"url":"Mathlib/Algebra/Order/UpperLower.lean","commit":"","full_name":"IsUpperSet.div_right","start":[77,0],"end":[80,20],"file_path":"Mathlib/Algebra/Order/UpperLower.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nhs : IsUpperSet s\n⊢ IsUpperSet (s / t)","state_after":"α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nhs : IsUpperSet s\n⊢ IsUpperSet (s * t⁻¹)","tactic":"rw [div_eq_mul_inv]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]},{"state_before":"α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nhs : IsUpperSet s\n⊢ IsUpperSet (s * t⁻¹)","state_after":"no goals","tactic":"exact hs.mul_right","premises":[{"full_name":"IsUpperSet.mul_right","def_path":"Mathlib/Algebra/Order/UpperLower.lean","def_pos":[56,8],"def_end_pos":[56,28]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"one_add_zsmul","start":[822,0],"end":[823,90],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na✝ b c d : G\nn✝ : ℤ\na : G\nn : ℤ\n⊢ a ^ (1 + n) = a * a ^ n","state_after":"no goals","tactic":"rw [zpow_add, zpow_one]","premises":[{"full_name":"zpow_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[816,6],"def_end_pos":[816,14]},{"full_name":"zpow_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[913,6],"def_end_pos":[913,14]}]}]}
{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.sqrt_lt","start":[1396,0],"end":[1396,74],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\n⊢ m.sqrt < n ↔ m < n * n","state_after":"no goals","tactic":"simp only [← not_le, le_sqrt]","premises":[{"full_name":"Nat.le_sqrt","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[1390,6],"def_end_pos":[1390,13]},{"full_name":"not_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[316,8],"def_end_pos":[316,14]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean","commit":"","full_name":"SimpleGraph.degMatrix_mulVec_apply","start":[48,0],"end":[50,33],"file_path":"Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean","tactics":[{"state_before":"V : Type u_1\nR : Type u_2\ninst✝³ : Fintype V\ninst✝² : DecidableEq V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : NonAssocSemiring R\nv : V\nvec : V → R\n⊢ (degMatrix R G *ᵥ vec) v = ↑(G.degree v) * vec v","state_after":"no goals","tactic":"rw [degMatrix, mulVec_diagonal]","premises":[{"full_name":"Matrix.mulVec_diagonal","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1508,8],"def_end_pos":[1508,23]},{"full_name":"SimpleGraph.degMatrix","def_path":"Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean","def_pos":[34,4],"def_end_pos":[34,13]}]}]}
{"url":"Mathlib/Logic/Hydra.lean","commit":"","full_name":"Relation.cutExpand_le_invImage_lex","start":[59,0],"end":[71,33],"file_path":"Mathlib/Logic/Hydra.lean","tactics":[{"state_before":"α : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : IsIrrefl α r\n⊢ CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) ⇑toFinsupp","state_after":"case intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : IsIrrefl α r\ns t u : Multiset α\na : α\nhr : ∀ a' ∈ u, r a' a\nhe : s + {a} = t + u\n⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t","tactic":"rintro s t ⟨u, a, hr, he⟩","premises":[]},{"state_before":"case intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : IsIrrefl α r\ns t u : Multiset α\na : α\nhr : ∀ a' ∈ u, r a' a\nhe : s + {a} = t + u\n⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t","state_after":"case intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : IsIrrefl α r\ns t u : Multiset α\na : α\nhe : s + {a} = t + u\nhr : ∀ (a' : α), ¬r a' a → a' ∉ u\n⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t","tactic":"replace hr := fun a' ↦ mt (hr a')","premises":[{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","commit":"","full_name":"contDiffOn_succ_iff_deriv_of_isOpen","start":[1826,0],"end":[1831,72],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nn : ℕ\nhs : IsOpen s₂\n⊢ ContDiffOn 𝕜 (↑(n + 1)) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂","state_after":"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nn : ℕ\nhs : IsOpen s₂\n⊢ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (derivWithin f₂ s₂) s₂ ↔\n    DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂","tactic":"rw [contDiffOn_succ_iff_derivWithin hs.uniqueDiffOn]","premises":[{"full_name":"IsOpen.uniqueDiffOn","def_path":"Mathlib/Analysis/Calculus/TangentCone.lean","def_pos":[274,8],"def_end_pos":[274,27]},{"full_name":"contDiffOn_succ_iff_derivWithin","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[1806,8],"def_end_pos":[1806,39]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nn : ℕ\nhs : IsOpen s₂\n⊢ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (derivWithin f₂ s₂) s₂ ↔\n    DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂","state_after":"no goals","tactic":"exact Iff.rfl.and (contDiffOn_congr fun _ => derivWithin_of_isOpen hs)","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"contDiffOn_congr","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[659,8],"def_end_pos":[659,24]},{"full_name":"derivWithin_of_isOpen","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[455,8],"def_end_pos":[455,29]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.support_C_mul_X_pow","start":[742,0],"end":[744,52],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nc : R\nh : c ≠ 0\n⊢ (C c * X ^ n).support = {n}","state_after":"no goals","tactic":"rw [C_mul_X_pow_eq_monomial, support_monomial n h]","premises":[{"full_name":"Polynomial.C_mul_X_pow_eq_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[646,8],"def_end_pos":[646,31]},{"full_name":"Polynomial.support_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[729,8],"def_end_pos":[729,24]}]}]}
{"url":"Mathlib/Analysis/Normed/Module/FiniteDimension.lean","commit":"","full_name":"FiniteDimensional.proper","start":[582,0],"end":[589,76],"file_path":"Mathlib/Analysis/Normed/Module/FiniteDimension.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : LocallyCompactSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\n⊢ ProperSpace E","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : LocallyCompactSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nthis : ProperSpace 𝕜\n⊢ ProperSpace E","tactic":"have : ProperSpace 𝕜 := .of_locallyCompactSpace 𝕜","premises":[{"full_name":"ProperSpace","def_path":"Mathlib/Topology/MetricSpace/ProperSpace.lean","def_pos":[35,6],"def_end_pos":[35,17]},{"full_name":"ProperSpace.of_locallyCompactSpace","def_path":"Mathlib/Analysis/Normed/Module/FiniteDimension.lean","def_pos":[501,6],"def_end_pos":[501,40]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : LocallyCompactSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nthis : ProperSpace 𝕜\n⊢ ProperSpace E","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : LocallyCompactSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nthis : ProperSpace 𝕜\ne : E ≃L[𝕜] Fin (finrank 𝕜 E) → 𝕜 := ContinuousLinearEquiv.ofFinrankEq ⋯\n⊢ ProperSpace E","tactic":"set e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm","premises":[{"full_name":"ContinuousLinearEquiv.ofFinrankEq","def_path":"Mathlib/Topology/Algebra/Module/FiniteDimension.lean","def_pos":[406,4],"def_end_pos":[406,37]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]},{"full_name":"FiniteDimensional.finrank_fin_fun","def_path":"Mathlib/LinearAlgebra/Dimension/Constructions.lean","def_pos":[303,8],"def_end_pos":[303,41]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : LocallyCompactSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nthis : ProperSpace 𝕜\ne : E ≃L[𝕜] Fin (finrank 𝕜 E) → 𝕜 := ContinuousLinearEquiv.ofFinrankEq ⋯\n⊢ ProperSpace E","state_after":"no goals","tactic":"exact e.symm.antilipschitz.properSpace e.symm.continuous e.symm.surjective","premises":[{"full_name":"AntilipschitzWith.properSpace","def_path":"Mathlib/Topology/MetricSpace/Antilipschitz.lean","def_pos":[199,18],"def_end_pos":[199,29]},{"full_name":"ContinuousLinearEquiv.antilipschitz","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean","def_pos":[293,18],"def_end_pos":[293,31]},{"full_name":"ContinuousLinearEquiv.continuous","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1725,18],"def_end_pos":[1725,28]},{"full_name":"ContinuousLinearEquiv.surjective","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1855,18],"def_end_pos":[1855,28]},{"full_name":"ContinuousLinearEquiv.symm","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1773,14],"def_end_pos":[1773,18]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean","commit":"","full_name":"Polynomial.hermite_eq_deriv_gaussian'","start":[63,0],"end":[66,31],"file_path":"Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean","tactics":[{"state_before":"n : ℕ\nx : ℝ\n⊢ (aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2)","state_after":"n : ℕ\nx : ℝ\n⊢ (-1) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x / (Real.exp (x ^ 2 / 2))⁻¹ =\n    (-1) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2)","tactic":"rw [hermite_eq_deriv_gaussian, Real.exp_neg]","premises":[{"full_name":"Polynomial.hermite_eq_deriv_gaussian","def_path":"Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean","def_pos":[55,8],"def_end_pos":[55,33]},{"full_name":"Real.exp_neg","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[716,15],"def_end_pos":[716,22]}]},{"state_before":"n : ℕ\nx : ℝ\n⊢ (-1) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x / (Real.exp (x ^ 2 / 2))⁻¹ =\n    (-1) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2)","state_after":"no goals","tactic":"field_simp [Real.exp_ne_zero]","premises":[{"full_name":"Real.exp_ne_zero","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[713,15],"def_end_pos":[713,26]}]}]}
{"url":"Mathlib/RingTheory/SimpleModule.lean","commit":"","full_name":"IsSimpleModule.nontrivial","start":[64,0],"end":[69,56],"file_path":"Mathlib/RingTheory/SimpleModule.lean","tactics":[{"state_before":"ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\n⊢ ∃ y, 0 ≠ y","state_after":"ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\nh : ⊥ ≠ ⊤\n⊢ ∃ y, 0 ≠ y","tactic":"have h : (⊥ : Submodule R M) ≠ ⊤ := bot_ne_top","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Submodule","def_path":"Mathlib/Algebra/Module/Submodule/Basic.lean","def_pos":[36,10],"def_end_pos":[36,19]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"bot_ne_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[758,8],"def_end_pos":[758,18]}]},{"state_before":"ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\nh : ⊥ ≠ ⊤\n⊢ ∃ y, 0 ≠ y","state_after":"ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\nh : ∀ (y : M), 0 = y\n⊢ ⊥ = ⊤","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\nh : ∀ (y : M), 0 = y\n⊢ ⊥ = ⊤","state_after":"case h\nι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\nh : ∀ (y : M), 0 = y\nx : M\n⊢ x ∈ ⊥ ↔ x ∈ ⊤","tactic":"ext x","premises":[]},{"state_before":"case h\nι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Ring S\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSimpleModule R M\nh : ∀ (y : M), 0 = y\nx : M\n⊢ x ∈ ⊥ ↔ x ∈ ⊤","state_after":"no goals","tactic":"simp [Submodule.mem_bot, Submodule.mem_top, h x]","premises":[{"full_name":"Submodule.mem_bot","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[68,8],"def_end_pos":[68,15]},{"full_name":"Submodule.mem_top","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[144,8],"def_end_pos":[144,15]}]}]}
{"url":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","commit":"","full_name":"CategoryTheory.Limits.biproduct.reindex_hom","start":[265,0],"end":[283,57],"file_path":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\n⊢ ((desc fun b => ι f (ε b)) ≫ lift fun b => π f (ε b)) = 𝟙 (⨁ f ∘ ⇑ε)","state_after":"case w.w\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nb b' : β\n⊢ ι (f ∘ ⇑ε) b' ≫ ((desc fun b => ι f (ε b)) ≫ lift fun b => π f (ε b)) ≫ π (f ∘ ⇑ε) b =\n    ι (f ∘ ⇑ε) b' ≫ 𝟙 (⨁ f ∘ ⇑ε) ≫ π (f ∘ ⇑ε) b","tactic":"ext b b'","premises":[]},{"state_before":"case w.w\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nb b' : β\n⊢ ι (f ∘ ⇑ε) b' ≫ ((desc fun b => ι f (ε b)) ≫ lift fun b => π f (ε b)) ≫ π (f ∘ ⇑ε) b =\n    ι (f ∘ ⇑ε) b' ≫ 𝟙 (⨁ f ∘ ⇑ε) ≫ π (f ∘ ⇑ε) b","state_after":"case pos\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nb b' : β\nh : b' = b\n⊢ ι (f ∘ ⇑ε) b' ≫ ((desc fun b => ι f (ε b)) ≫ lift fun b => π f (ε b)) ≫ π (f ∘ ⇑ε) b =\n    ι (f ∘ ⇑ε) b' ≫ 𝟙 (⨁ f ∘ ⇑ε) ≫ π (f ∘ ⇑ε) b\n\ncase neg\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nb b' : β\nh : ¬b' = b\n⊢ ι (f ∘ ⇑ε) b' ≫ ((desc fun b => ι f (ε b)) ≫ lift fun b => π f (ε b)) ≫ π (f ∘ ⇑ε) b =\n    ι (f ∘ ⇑ε) b' ≫ 𝟙 (⨁ f ∘ ⇑ε) ≫ π (f ∘ ⇑ε) b","tactic":"by_cases h : b' = b","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\n⊢ ((lift fun b => π f (ε b)) ≫ desc fun b => ι f (ε b)) = 𝟙 (⨁ f)","state_after":"case intro\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nval✝ : Fintype β\n⊢ ((lift fun b => π f (ε b)) ≫ desc fun b => ι f (ε b)) = 𝟙 (⨁ f)","tactic":"cases nonempty_fintype β","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nval✝ : Fintype β\n⊢ ((lift fun b => π f (ε b)) ≫ desc fun b => ι f (ε b)) = 𝟙 (⨁ f)","state_after":"case intro.w.w\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nval✝ : Fintype β\ng g' : γ\n⊢ ι f g' ≫ ((lift fun b => π f (ε b)) ≫ desc fun b => ι f (ε b)) ≫ π f g = ι f g' ≫ 𝟙 (⨁ f) ≫ π f g","tactic":"ext g g'","premises":[]},{"state_before":"case intro.w.w\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nβ γ : Type\ninst✝² : Finite β\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct (f ∘ ⇑ε)\nval✝ : Fintype β\ng g' : γ\n⊢ ι f g' ≫ ((lift fun b => π f (ε b)) ≫ desc fun b => ι f (ε b)) ≫ π f g = ι f g' ≫ 𝟙 (⨁ f) ≫ π f g","state_after":"no goals","tactic":"by_cases h : g' = g <;>\n      simp [Preadditive.sum_comp, Preadditive.comp_sum, biproduct.lift_desc,\n        biproduct.ι_π, biproduct.ι_π_assoc, comp_dite, Equiv.apply_eq_iff_eq_symm_apply,\n        Finset.sum_dite_eq' Finset.univ (ε.symm g') _, h]","premises":[{"full_name":"CategoryTheory.Limits.biproduct.lift_desc","def_path":"Mathlib/CategoryTheory/Preadditive/Biproducts.lean","def_pos":[217,8],"def_end_pos":[217,27]},{"full_name":"CategoryTheory.Limits.biproduct.ι_π","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean","def_pos":[450,8],"def_end_pos":[450,21]},{"full_name":"CategoryTheory.Preadditive.comp_sum","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[162,8],"def_end_pos":[162,16]},{"full_name":"CategoryTheory.Preadditive.sum_comp","def_path":"Mathlib/CategoryTheory/Preadditive/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]},{"full_name":"CategoryTheory.comp_dite","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[228,8],"def_end_pos":[228,17]},{"full_name":"Equiv.apply_eq_iff_eq_symm_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[272,8],"def_end_pos":[272,34]},{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"Finset.sum_dite_eq'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1116,2],"def_end_pos":[1116,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/FunctorCategory.lean","commit":"","full_name":"CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_π_app","start":[196,0],"end":[203,6],"file_path":"Mathlib/CategoryTheory/Limits/FunctorCategory.lean","tactics":[{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasLimitsOfShape J C\nF : J ⥤ K ⥤ C\nj : J\nk : K\n⊢ (limitObjIsoLimitCompEvaluation F k).inv ≫ (limit.π F j).app k = limit.π (F ⋙ (evaluation K C).obj k) j","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasLimitsOfShape J C\nF : J ⥤ K ⥤ C\nj : J\nk : K\n⊢ (preservesLimitIso ((evaluation K C).obj k) F).inv ≫ (limit.π F j).app k = limit.π (F ⋙ (evaluation K C).obj k) j","tactic":"dsimp [limitObjIsoLimitCompEvaluation]","premises":[{"full_name":"CategoryTheory.Limits.limitObjIsoLimitCompEvaluation","def_path":"Mathlib/CategoryTheory/Limits/FunctorCategory.lean","def_pos":[184,4],"def_end_pos":[184,34]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasLimitsOfShape J C\nF : J ⥤ K ⥤ C\nj : J\nk : K\n⊢ (preservesLimitIso ((evaluation K C).obj k) F).inv ≫ (limit.π F j).app k = limit.π (F ⋙ (evaluation K C).obj k) j","state_after":"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasLimitsOfShape J C\nF : J ⥤ K ⥤ C\nj : J\nk : K\n⊢ (limit.π F j).app k = (preservesLimitIso ((evaluation K C).obj k) F).hom ≫ limit.π (F ⋙ (evaluation K C).obj k) j","tactic":"rw [Iso.inv_comp_eq]","premises":[{"full_name":"CategoryTheory.Iso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[178,8],"def_end_pos":[178,19]}]},{"state_before":"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasLimitsOfShape J C\nF : J ⥤ K ⥤ C\nj : J\nk : K\n⊢ (limit.π F j).app k = (preservesLimitIso ((evaluation K C).obj k) F).hom ≫ limit.π (F ⋙ (evaluation K C).obj k) j","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Path.lean","commit":"","full_name":"SimpleGraph.Walk.IsTrail.of_append_right","start":[168,0],"end":[171,15],"file_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu✝ v✝ w✝ u v w : V\np : G.Walk u v\nq : G.Walk v w\nh : (p.append q).IsTrail\n⊢ q.IsTrail","state_after":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu✝ v✝ w✝ u v w : V\np : G.Walk u v\nq : G.Walk v w\nh : p.edges.Nodup ∧ q.edges.Nodup ∧ p.edges.Disjoint q.edges\n⊢ q.IsTrail","tactic":"rw [isTrail_def, edges_append, List.nodup_append] at h","premises":[{"full_name":"List.nodup_append","def_path":"Mathlib/Data/List/Nodup.lean","def_pos":[199,8],"def_end_pos":[199,20]},{"full_name":"SimpleGraph.Walk.edges_append","def_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","def_pos":[657,8],"def_end_pos":[657,20]},{"full_name":"SimpleGraph.Walk.isTrail_def","def_path":"Mathlib/Combinatorics/SimpleGraph/Path.lean","def_pos":[76,9],"def_end_pos":[76,20]}]},{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu✝ v✝ w✝ u v w : V\np : G.Walk u v\nq : G.Walk v w\nh : p.edges.Nodup ∧ q.edges.Nodup ∧ p.edges.Disjoint q.edges\n⊢ q.IsTrail","state_after":"no goals","tactic":"exact ⟨h.2.1⟩","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Holder.lean","commit":"","full_name":"HolderWith.nndist_le_of_le","start":[203,0],"end":[208,40],"file_path":"Mathlib/Topology/MetricSpace/Holder.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ nndist (f x) (f y) ≤ C * d ^ ↑r","state_after":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist (f x) (f y) ≤ ↑C * ↑d ^ ↑r","tactic":"rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul, ←\n    ENNReal.coe_rpow_of_nonneg _ r.coe_nonneg]","premises":[{"full_name":"ENNReal.coe_le_coe","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[314,27],"def_end_pos":[314,37]},{"full_name":"ENNReal.coe_mul","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[346,25],"def_end_pos":[346,32]},{"full_name":"ENNReal.coe_rpow_of_nonneg","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[416,8],"def_end_pos":[416,26]},{"full_name":"NNReal.coe_nonneg","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[122,8],"def_end_pos":[122,18]},{"full_name":"edist_nndist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[275,8],"def_end_pos":[275,20]}]},{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist (f x) (f y) ≤ ↑C * ↑d ^ ↑r","state_after":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist x y ≤ ↑d","tactic":"apply hf.edist_le_of_le","premises":[{"full_name":"HolderWith.edist_le_of_le","def_path":"Mathlib/Topology/MetricSpace/Holder.lean","def_pos":[168,8],"def_end_pos":[168,22]}]},{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist x y ≤ ↑d","state_after":"no goals","tactic":"rwa [edist_nndist, ENNReal.coe_le_coe]","premises":[{"full_name":"ENNReal.coe_le_coe","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[314,27],"def_end_pos":[314,37]},{"full_name":"edist_nndist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[275,8],"def_end_pos":[275,20]}]}]}
{"url":"Mathlib/Data/Matroid/Basic.lean","commit":"","full_name":"Matroid.Base.card_eq_card_of_base","start":[394,0],"end":[396,45],"file_path":"Mathlib/Data/Matroid/Basic.lean","tactics":[{"state_before":"α : Type u_1\nM : Matroid α\nB B₁ B₂ : Set α\nhB₁ : M.Base B₁\nhB₂ : M.Base B₂\n⊢ B₁.encard = B₂.encard","state_after":"no goals","tactic":"rw [M.base_exchange.encard_base_eq hB₁ hB₂]","premises":[{"full_name":"Matroid.ExchangeProperty.encard_base_eq","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[296,8],"def_end_pos":[296,22]},{"full_name":"Matroid.base_exchange","def_path":"Mathlib/Data/Matroid/Basic.lean","def_pos":[199,3],"def_end_pos":[199,16]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean","commit":"","full_name":"CategoryTheory.Limits.mulIsInitial_hom","start":[88,0],"end":[92,22],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct X I\nhI : IsInitial I\n⊢ X ⨯ I ≅ I","state_after":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct X I\nhI : IsInitial I\nthis : IsIso prod.snd\n⊢ X ⨯ I ≅ I","tactic":"have := hI.isIso_to (prod.snd : X ⨯ I ⟶ I)","premises":[{"full_name":"CategoryTheory.Limits.IsInitial.isIso_to","def_path":"Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean","def_pos":[67,8],"def_end_pos":[67,26]},{"full_name":"CategoryTheory.Limits.prod","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[464,7],"def_end_pos":[464,11]},{"full_name":"CategoryTheory.Limits.prod.snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[483,7],"def_end_pos":[483,15]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct X I\nhI : IsInitial I\nthis : IsIso prod.snd\n⊢ X ⨯ I ≅ I","state_after":"no goals","tactic":"exact asIso prod.snd","premises":[{"full_name":"CategoryTheory.Limits.prod.snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean","def_pos":[483,7],"def_end_pos":[483,15]},{"full_name":"CategoryTheory.asIso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[278,18],"def_end_pos":[278,23]}]}]}
{"url":"Mathlib/Order/SupClosed.lean","commit":"","full_name":"InfClosed.image","start":[131,0],"end":[135,42],"file_path":"Mathlib/Order/SupClosed.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\n⊢ InfClosed (⇑f '' s)","state_after":"case intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na✝ : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ f a ⊓ f b ∈ ⇑f '' s","tactic":"rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩","premises":[]},{"state_before":"case intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na✝ : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ f a ⊓ f b ∈ ⇑f '' s","state_after":"case intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na✝ : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ f (a ⊓ b) ∈ ⇑f '' s","tactic":"rw [← map_inf]","premises":[{"full_name":"InfHomClass.map_inf","def_path":"Mathlib/Order/Hom/Lattice.lean","def_pos":[103,2],"def_end_pos":[103,9]}]},{"state_before":"case intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na✝ : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ f (a ⊓ b) ∈ ⇑f '' s","state_after":"no goals","tactic":"exact Set.mem_image_of_mem _ <| hs ha hb","premises":[{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]}]}]}
{"url":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","commit":"","full_name":"stereographic'_symm_apply","start":[366,0],"end":[373,77],"file_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","tactics":[{"state_before":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\nv : ↑(sphere 0 1)\nx : EuclideanSpace ℝ (Fin n)\n⊢ ↑(↑(stereographic' n v).symm x) =\n    let U := (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr;\n    (‖↑(U.symm x)‖ ^ 2 + 4)⁻¹ • 4 • ↑(U.symm x) + (‖↑(U.symm x)‖ ^ 2 + 4)⁻¹ • (‖↑(U.symm x)‖ ^ 2 - 4) • ↑v","state_after":"no goals","tactic":"simp [real_inner_comm, stereographic, stereographic', ← Submodule.coe_norm]","premises":[{"full_name":"Submodule.coe_norm","def_path":"Mathlib/Analysis/Normed/Group/Submodule.lean","def_pos":[25,8],"def_end_pos":[25,16]},{"full_name":"real_inner_comm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[392,8],"def_end_pos":[392,23]},{"full_name":"stereographic","def_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","def_pos":[255,4],"def_end_pos":[255,17]},{"full_name":"stereographic'","def_path":"Mathlib/Geometry/Manifold/Instances/Sphere.lean","def_pos":[331,4],"def_end_pos":[331,18]}]}]}
{"url":"Mathlib/Data/Ordmap/Ordset.lean","commit":"","full_name":"Ordnode.all_iff_forall","start":[418,0],"end":[420,80],"file_path":"Mathlib/Data/Ordmap/Ordset.lean","tactics":[{"state_before":"α : Type u_1\nP : α → Prop\n⊢ ∀ (x : α), Emem x nil → P x","state_after":"no goals","tactic":"rintro _ ⟨⟩","premises":[]},{"state_before":"α : Type u_1\nP : α → Prop\nsize✝ : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ All P (node size✝ l x r) ↔ ∀ (x_1 : α), Emem x_1 (node size✝ l x r) → P x_1","state_after":"no goals","tactic":"simp [All, Emem, all_iff_forall, Any, or_imp, forall_and]","premises":[{"full_name":"Ordnode.All","def_path":"Mathlib/Data/Ordmap/Ordnode.lean","def_pos":[311,4],"def_end_pos":[311,7]},{"full_name":"Ordnode.Any","def_path":"Mathlib/Data/Ordmap/Ordnode.lean","def_pos":[326,4],"def_end_pos":[326,7]},{"full_name":"Ordnode.Emem","def_path":"Mathlib/Data/Ordmap/Ordnode.lean","def_pos":[343,4],"def_end_pos":[343,8]},{"full_name":"forall_and","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[256,8],"def_end_pos":[256,18]},{"full_name":"or_imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[120,8],"def_end_pos":[120,14]}]}]}
{"url":"Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean","commit":"","full_name":"GenContFract.le_of_succ_get?_den","start":[291,0],"end":[297,56],"file_path":"Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean","tactics":[{"state_before":"K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDenom_eq : (of v).partDens.get? n = some b\n⊢ b * (of v).dens n ≤ (of v).dens (n + 1)","state_after":"K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDenom_eq : (of v).partDens.get? n = some b\n⊢ b * ((of v).contsAux (n + 1)).b ≤ (of v).dens (n + 1)","tactic":"rw [den_eq_conts_b, nth_cont_eq_succ_nth_contAux]","premises":[{"full_name":"GenContFract.den_eq_conts_b","def_path":"Mathlib/Algebra/ContinuedFractions/Translations.lean","def_pos":[84,8],"def_end_pos":[84,22]},{"full_name":"GenContFract.nth_cont_eq_succ_nth_contAux","def_path":"Mathlib/Algebra/ContinuedFractions/Translations.lean","def_pos":[78,8],"def_end_pos":[78,36]}]},{"state_before":"K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDenom_eq : (of v).partDens.get? n = some b\n⊢ b * ((of v).contsAux (n + 1)).b ≤ (of v).dens (n + 1)","state_after":"no goals","tactic":"exact le_of_succ_succ_get?_contsAux_b nth_partDenom_eq","premises":[{"full_name":"GenContFract.le_of_succ_succ_get?_contsAux_b","def_path":"Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean","def_pos":[283,8],"def_end_pos":[283,39]}]}]}
{"url":"Mathlib/NumberTheory/MulChar/Basic.lean","commit":"","full_name":"MulChar.injective_ringHomComp","start":[454,0],"end":[458,33],"file_path":"Mathlib/NumberTheory/MulChar/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nf : R' →+* R''\nhf : Function.Injective ⇑f\n⊢ Function.Injective fun x => x.ringHomComp f","state_after":"no goals","tactic":"simpa\n    only [Function.Injective, MulChar.ext_iff, ringHomComp, coe_mk, MonoidHom.coe_mk, OneHom.coe_mk]\n    using fun χ χ' h a ↦ hf (h a)","premises":[{"full_name":"Function.Injective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[101,4],"def_end_pos":[101,13]},{"full_name":"MonoidHom.coe_mk","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[497,8],"def_end_pos":[497,24]},{"full_name":"MulChar.coe_mk","def_path":"Mathlib/NumberTheory/MulChar/Basic.lean","def_pos":[109,8],"def_end_pos":[109,14]},{"full_name":"MulChar.ext_iff","def_path":"Mathlib/NumberTheory/MulChar/Basic.lean","def_pos":[137,8],"def_end_pos":[137,15]},{"full_name":"MulChar.ringHomComp","def_path":"Mathlib/NumberTheory/MulChar/Basic.lean","def_pos":[428,4],"def_end_pos":[428,15]},{"full_name":"OneHom.coe_mk","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[485,8],"def_end_pos":[485,21]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_eq_sum_range","start":[205,0],"end":[208,69],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ (fun i => eval₂ f x ((monomial i) (p.coeff i))) = fun i => f (p.coeff i) * x ^ i","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Int.ceil_sub_self_eq","start":[1187,0],"end":[1189,6],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\nha : fract a ≠ 0\n⊢ ↑⌈a⌉ - a = 1 - fract a","state_after":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\nha : fract a ≠ 0\n⊢ ↑⌈a⌉ - a = 1 - (a + 1 - ↑⌈a⌉)","tactic":"rw [(or_iff_right ha).mp (fract_eq_zero_or_add_one_sub_ceil a)]","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Int.fract_eq_zero_or_add_one_sub_ceil","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[1170,8],"def_end_pos":[1170,41]},{"full_name":"or_iff_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[92,8],"def_end_pos":[92,20]}]},{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\nha : fract a ≠ 0\n⊢ ↑⌈a⌉ - a = 1 - (a + 1 - ↑⌈a⌉)","state_after":"no goals","tactic":"abel","premises":[]}]}
{"url":"Mathlib/FieldTheory/IntermediateField.lean","commit":"","full_name":"IntermediateField.toSubalgebra_injective","start":[512,0],"end":[516,48],"file_path":"Mathlib/FieldTheory/IntermediateField.lean","tactics":[{"state_before":"K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS : IntermediateField K L\n⊢ Function.Injective toSubalgebra","state_after":"K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS✝ S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\n⊢ S = S'","tactic":"intro S S' h","premises":[]},{"state_before":"K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS✝ S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\n⊢ S = S'","state_after":"case h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS✝ S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\nx✝ : L\n⊢ x✝ ∈ S ↔ x✝ ∈ S'","tactic":"ext","premises":[]},{"state_before":"case h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS✝ S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\nx✝ : L\n⊢ x✝ ∈ S ↔ x✝ ∈ S'","state_after":"no goals","tactic":"rw [← mem_toSubalgebra, ← mem_toSubalgebra, h]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"IntermediateField.mem_toSubalgebra","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[103,8],"def_end_pos":[103,24]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.mem_sum","start":[1873,0],"end":[1878,53],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\nf : α → Multiset β\ns : Finset α\nb : β\n⊢ b ∈ ∑ x ∈ s, f x ↔ ∃ a ∈ s, b ∈ f a","state_after":"no goals","tactic":"classical\n    refine s.induction_on (by simp) ?_\n    intro a t hi ih\n    simp [sum_insert hi, ih, or_and_right, exists_or]","premises":[{"full_name":"Finset.induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1076,18],"def_end_pos":[1076,30]},{"full_name":"Finset.sum_insert","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[328,2],"def_end_pos":[328,13]},{"full_name":"exists_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"or_and_right","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[110,8],"def_end_pos":[110,20]}]}]}
{"url":"Mathlib/Data/PNat/Prime.lean","commit":"","full_name":"PNat.Coprime.symm","start":[217,0],"end":[221,6],"file_path":"Mathlib/Data/PNat/Prime.lean","tactics":[{"state_before":"m n : ℕ+\n⊢ m.Coprime n → n.Coprime m","state_after":"m n : ℕ+\n⊢ m.gcd n = 1 → n.gcd m = 1","tactic":"unfold Coprime","premises":[{"full_name":"PNat.Coprime","def_path":"Mathlib/Data/PNat/Prime.lean","def_pos":[155,4],"def_end_pos":[155,11]}]},{"state_before":"m n : ℕ+\n⊢ m.gcd n = 1 → n.gcd m = 1","state_after":"m n : ℕ+\n⊢ n.gcd m = 1 → n.gcd m = 1","tactic":"rw [gcd_comm]","premises":[{"full_name":"PNat.gcd_comm","def_path":"Mathlib/Data/PNat/Prime.lean","def_pos":[174,8],"def_end_pos":[174,16]}]},{"state_before":"m n : ℕ+\n⊢ n.gcd m = 1 → n.gcd m = 1","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/Convex/Slope.lean","commit":"","full_name":"ConvexOn.secant_mono","start":[251,0],"end":[263,43],"file_path":"Mathlib/Analysis/Convex/Slope.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy : x ≤ y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)","state_after":"case inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhxa : x ≠ a\nhy : x ∈ s\nhya : x ≠ a\nhxy : x ≤ x\n⊢ (f x - f a) / (x - a) ≤ (f x - f a) / (x - a)\n\ncase inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)","tactic":"rcases eq_or_lt_of_le hxy with (rfl | hxy)","premises":[{"full_name":"eq_or_lt_of_le","def_path":"Mathlib/Order/Basic.lean","def_pos":[325,8],"def_end_pos":[325,22]}]},{"state_before":"case inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)","state_after":"case inr.inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x < a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)\n\ncase inr.inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa✝ : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\nhxa : x > a\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)","tactic":"cases' lt_or_gt_of_ne hxa with hxa hxa","premises":[{"full_name":"lt_or_gt_of_ne","def_path":"Mathlib/Order/Defs.lean","def_pos":[299,8],"def_end_pos":[299,22]}]}]}
{"url":"Mathlib/RingTheory/ChainOfDivisors.lean","commit":"","full_name":"multiplicity_prime_eq_multiplicity_image_by_factor_orderIso","start":[320,0],"end":[331,85],"file_path":"Mathlib/RingTheory/ChainOfDivisors.lean","tactics":[{"state_before":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n","state_after":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity p m","tactic":"refine le_antisymm (multiplicity_prime_le_multiplicity_image_by_factor_orderIso hp d) ?_","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"multiplicity_prime_le_multiplicity_image_by_factor_orderIso","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[307,8],"def_end_pos":[307,67]}]},{"state_before":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity p m","state_after":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m","tactic":"suffices multiplicity (↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩)) n ≤\n      multiplicity (↑(d.symm (d ⟨p, dvd_of_mem_normalizedFactors hp⟩))) m by\n    rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalizedFactors hp⟩, Subtype.coe_mk] at this\n    exact this","premises":[{"full_name":"OrderIso.symm","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[754,4],"def_end_pos":[754,8]},{"full_name":"OrderIso.symm_apply_apply","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[761,8],"def_end_pos":[761,24]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors","def_path":"Mathlib/RingTheory/UniqueFactorizationDomain.lean","def_pos":[724,8],"def_end_pos":[724,36]},{"full_name":"multiplicity","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[34,4],"def_end_pos":[34,16]}]},{"state_before":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m","state_after":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nthis : DecidableEq (Associates N) := Classical.decEq (Associates N)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m","tactic":"letI := Classical.decEq (Associates N)","premises":[{"full_name":"Associates","def_path":"Mathlib/Algebra/Associated/Basic.lean","def_pos":[729,7],"def_end_pos":[729,17]},{"full_name":"Classical.decEq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[737,18],"def_end_pos":[737,23]}]},{"state_before":"M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nthis : DecidableEq (Associates N) := Classical.decEq (Associates N)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m","state_after":"no goals","tactic":"simpa only [Subtype.coe_eta] using\n    multiplicity_prime_le_multiplicity_image_by_factor_orderIso\n      (mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors hn hp d) d.symm","premises":[{"full_name":"OrderIso.symm","def_path":"Mathlib/Order/Hom/Basic.lean","def_pos":[754,4],"def_end_pos":[754,8]},{"full_name":"Subtype.coe_eta","def_path":"Mathlib/Data/Subtype.lean","def_pos":[83,8],"def_end_pos":[83,15]},{"full_name":"mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[296,8],"def_end_pos":[296,70]},{"full_name":"multiplicity_prime_le_multiplicity_image_by_factor_orderIso","def_path":"Mathlib/RingTheory/ChainOfDivisors.lean","def_pos":[307,8],"def_end_pos":[307,67]}]}]}
{"url":"Mathlib/Analysis/Convex/Deriv.lean","commit":"","full_name":"ConcaveOn.le_slope_of_hasDerivWithinAt_Iio","start":[699,0],"end":[703,75],"file_path":"Mathlib/Analysis/Convex/Deriv.lean","tactics":[{"state_before":"S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : ConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\n⊢ f' ≤ slope f x y","state_after":"no goals","tactic":"simpa only [neg_neg, Pi.neg_def, slope_neg] using\n    neg_le_neg (hfc.neg.slope_le_of_hasDerivWithinAt_Iio hx hy hxy hf'.neg)","premises":[{"full_name":"ConvexOn.slope_le_of_hasDerivWithinAt_Iio","def_path":"Mathlib/Analysis/Convex/Deriv.lean","def_pos":[444,6],"def_end_pos":[444,38]},{"full_name":"HasDerivWithinAt.neg","def_path":"Mathlib/Analysis/Calculus/Deriv/Add.lean","def_pos":[166,15],"def_end_pos":[166,35]},{"full_name":"Pi.neg_def","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[129,2],"def_end_pos":[129,13]},{"full_name":"neg_le_neg","def_path":"Mathlib/Algebra/Order/Group/Defs.lean","def_pos":[172,31],"def_end_pos":[172,41]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"slope_neg","def_path":"Mathlib/LinearAlgebra/AffineSpace/Slope.lean","def_pos":[80,14],"def_end_pos":[80,23]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_at_apply","start":[281,0],"end":[285,34],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ eval₂ f (f r) p = f (eval r p)","state_after":"R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ ∑ n ∈ p.support, f (p.coeff n) * f r ^ n = ∑ x ∈ p.support, f (p.coeff x * r ^ x)","tactic":"rw [eval₂_eq_sum, eval_eq_sum, sum, sum, map_sum f]","premises":[{"full_name":"Polynomial.eval_eq_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[269,8],"def_end_pos":[269,19]},{"full_name":"Polynomial.eval₂_eq_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[44,8],"def_end_pos":[44,20]},{"full_name":"Polynomial.sum","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[812,4],"def_end_pos":[812,7]},{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]}]},{"state_before":"R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ ∑ n ∈ p.support, f (p.coeff n) * f r ^ n = ∑ x ∈ p.support, f (p.coeff x * r ^ x)","state_after":"no goals","tactic":"simp only [f.map_mul, f.map_pow]","premises":[{"full_name":"RingHom.map_mul","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[484,18],"def_end_pos":[484,25]},{"full_name":"RingHom.map_pow","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[622,16],"def_end_pos":[622,31]}]}]}
{"url":"Mathlib/Geometry/Manifold/Algebra/Monoid.lean","commit":"","full_name":"ContMDiffAt.sum","start":[304,0],"end":[308,32],"file_path":"Mathlib/Geometry/Manifold/Algebra/Monoid.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\ninst✝⁵ : SmoothMul I G\nE' : Type u_6\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝² : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H' M\ns : Set M\nx x₀ : M\nt : Finset ι\nf : ι → M → G\nn : ℕ∞\np : ι → Prop\nh : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀\n⊢ ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x₀","state_after":"ι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\ninst✝⁵ : SmoothMul I G\nE' : Type u_6\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝² : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H' M\ns : Set M\nx x₀ : M\nt : Finset ι\nf : ι → M → G\nn : ℕ∞\np : ι → Prop\nh : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) Set.univ x₀\n⊢ ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) Set.univ x₀","tactic":"simp only [← contMDiffWithinAt_univ] at *","premises":[{"full_name":"contMDiffWithinAt_univ","def_path":"Mathlib/Geometry/Manifold/ContMDiff/Defs.lean","def_pos":[269,8],"def_end_pos":[269,30]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\ninst✝⁵ : SmoothMul I G\nE' : Type u_6\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝² : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H' M\ns : Set M\nx x₀ : M\nt : Finset ι\nf : ι → M → G\nn : ℕ∞\np : ι → Prop\nh : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) Set.univ x₀\n⊢ ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) Set.univ x₀","state_after":"no goals","tactic":"exact ContMDiffWithinAt.prod h","premises":[{"full_name":"ContMDiffWithinAt.prod","def_path":"Mathlib/Geometry/Manifold/Algebra/Monoid.lean","def_pos":[276,8],"def_end_pos":[276,30]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","commit":"","full_name":"MeasureTheory.Measure.measure_isAddHaarMeasure_eq_smul_of_isEverywherePos","start":[694,0],"end":[794,25],"file_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","tactics":[{"state_before":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\n⊢ μ' s = μ'.haarScalarFactor μ • μ s","state_after":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = μ'.haarScalarFactor μ • μ s","tactic":"let ν := haarScalarFactor μ' μ • μ","premises":[{"full_name":"MeasureTheory.Measure.haarScalarFactor","def_path":"Mathlib/MeasureTheory/Measure/Haar/Unique.lean","def_pos":[294,18],"def_end_pos":[294,34]}]},{"state_before":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = μ'.haarScalarFactor μ • μ s","state_after":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = ν s","tactic":"change μ' s = ν s","premises":[]},{"state_before":"G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = ν s","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\n⊢ μ' s = ν s","tactic":"obtain ⟨k, k_comp, k_closed, k_mem⟩ : ∃ k, IsCompact k ∧ IsClosed k ∧ k ∈ 𝓝 (1 : G) := by\n    rcases exists_compact_mem_nhds (1 : G) with ⟨k, hk, hmem⟩\n    exact ⟨closure k, hk.closure, isClosed_closure, mem_of_superset hmem subset_closure⟩","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"IsClosed","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[93,6],"def_end_pos":[93,14]},{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.closure","def_path":"Mathlib/Topology/Separation.lean","def_pos":[1089,18],"def_end_pos":[1089,35]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"WeaklyLocallyCompactSpace.exists_compact_mem_nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[274,2],"def_end_pos":[274,25]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]},{"full_name":"isClosed_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[344,8],"def_end_pos":[344,24]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"subset_closure","def_path":"Mathlib/Topology/Basic.lean","def_pos":[347,8],"def_end_pos":[347,22]}]},{"state_before":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\n⊢ μ' s = ν s","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\n⊢ μ' s = ν s","tactic":"have one_k : 1 ∈ k := mem_of_mem_nhds k_mem","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"mem_of_mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[737,8],"def_end_pos":[737,23]}]},{"state_before":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\n⊢ μ' s = ν s","state_after":"case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\n⊢ μ' s = ν s","tactic":"let A : Set (Set G) := {t | t ⊆ s ∧ PairwiseDisjoint t (fun x ↦ x • k)}","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.PairwiseDisjoint","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[216,4],"def_end_pos":[216,20]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nmA : m ∈ A\nm_max : ∀ a ∈ A, m ⊆ a → a = m\n⊢ μ' s = ν s","state_after":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\n⊢ μ' s = ν s","tactic":"change m ⊆ s ∧ PairwiseDisjoint m (fun x ↦ x • k) at mA","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"Set.PairwiseDisjoint","def_path":"Mathlib/Data/Set/Pairwise/Basic.lean","def_pos":[216,4],"def_end_pos":[216,20]}]},{"state_before":"case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\n⊢ μ' s = ν s","state_after":"case intro.intro.intro.intro.intro.inl\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm_max : ∀ a ∈ A, ∅ ⊆ a → a = ∅\nmA : ∅ ⊆ s ∧ ∅.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ ∅, x • (k * k⁻¹)\n⊢ μ' s = ν s\n\ncase intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\n⊢ μ' s = ν s","tactic":"rcases eq_empty_or_nonempty m with rfl|hm","premises":[{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]}]},{"state_before":"case intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\n⊢ μ' s = ν s","state_after":"case pos\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\nh'm : m.Countable\n⊢ μ' s = ν s\n\ncase neg\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\nhm : m.Nonempty\nh'm : ¬m.Countable\n⊢ μ' s = ν s","tactic":"by_cases h'm : Set.Countable m","premises":[{"full_name":"Set.Countable","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[43,14],"def_end_pos":[43,23]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Topology/VectorBundle/Constructions.lean","commit":"","full_name":"Trivialization.continuousLinearEquivAt_prod","start":[141,0],"end":[150,63],"file_path":"Mathlib/Topology/VectorBundle/Constructions.lean","tactics":[{"state_before":"𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\n⊢ continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx =\n    (continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)","state_after":"case h.h\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv : E₁ x × E₂ x\n⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) v =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) v","tactic":"ext v : 2","premises":[]},{"state_before":"case h.h\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv : E₁ x × E₂ x\n⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) v =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) v","state_after":"case h.h.mk\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) (v₁, v₂) =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)","tactic":"obtain ⟨v₁, v₂⟩ := v","premises":[]},{"state_before":"case h.h.mk\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) (v₁, v₂) =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)","state_after":"case h.h.mk\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (fun y =>\n        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯,\n                continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,\n                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }\n            { proj := x, snd := y }).2)\n      (v₁, v₂) =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)","tactic":"rw [(e₁.prod e₂).continuousLinearEquivAt_apply 𝕜, Trivialization.prod]","premises":[{"full_name":"Trivialization.continuousLinearEquivAt_apply","def_path":"Mathlib/Topology/VectorBundle/Basic.lean","def_pos":[411,26],"def_end_pos":[411,31]},{"full_name":"Trivialization.prod","def_path":"Mathlib/Topology/FiberBundle/Constructions.lean","def_pos":[194,18],"def_end_pos":[194,22]}]},{"state_before":"case h.h.mk\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (fun y =>\n        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯,\n                continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,\n                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }\n            { proj := x, snd := y }).2)\n      (v₁, v₂) =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)","state_after":"no goals","tactic":"exact (congr_arg Prod.snd (prod_apply 𝕜 hx.1 hx.2 v₁ v₂) : _)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"Trivialization.prod_apply","def_path":"Mathlib/Topology/VectorBundle/Constructions.lean","def_pos":[104,8],"def_end_pos":[104,18]}]}]}
{"url":"Mathlib/Data/Sym/Sym2.lean","commit":"","full_name":"Sym2.diag_injective","start":[398,0],"end":[399,28],"file_path":"Mathlib/Data/Sym/Sym2.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\nx y : α\nh : diag x = diag y\n⊢ x = y","state_after":"no goals","tactic":"cases Sym2.exact h <;> rfl","premises":[{"full_name":"Sym2.exact","def_path":"Mathlib/Data/Sym/Sym2.lean","def_pos":[110,18],"def_end_pos":[110,23]}]}]}
{"url":"Mathlib/Algebra/Order/Ring/Rat.lean","commit":"","full_name":"Rat.lt_one_iff_num_lt_denom","start":[231,0],"end":[231,87],"file_path":"Mathlib/Algebra/Order/Ring/Rat.lean","tactics":[{"state_before":"a b c p q✝ q : ℚ\n⊢ q < 1 ↔ q.num < ↑q.den","state_after":"no goals","tactic":"simp [Rat.lt_def]","premises":[{"full_name":"Rat.lt_def","def_path":"Mathlib/Algebra/Order/Ring/Rat.lean","def_pos":[180,16],"def_end_pos":[180,22]}]}]}
{"url":"Mathlib/ModelTheory/Substructures.lean","commit":"","full_name":"FirstOrder.Language.Substructure.realize_boundedFormula_top","start":[576,0],"end":[581,6],"file_path":"Mathlib/ModelTheory/Substructures.lean","tactics":[{"state_before":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\ns : Set M\nα : Type u_3\nn : ℕ\nφ : L.BoundedFormula α n\nv : α → ↥⊤\nxs : Fin n → ↥⊤\n⊢ φ.Realize v xs ↔ φ.Realize (Subtype.val ∘ v) (Subtype.val ∘ xs)","state_after":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\ns : Set M\nα : Type u_3\nn : ℕ\nφ : L.BoundedFormula α n\nv : α → ↥⊤\nxs : Fin n → ↥⊤\n⊢ φ.Realize (⇑topEquiv ∘ v) (⇑topEquiv ∘ xs) ↔ φ.Realize (Subtype.val ∘ v) (Subtype.val ∘ xs)","tactic":"rw [← Substructure.topEquiv.realize_boundedFormula φ]","premises":[{"full_name":"FirstOrder.Language.Equiv.realize_boundedFormula","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[844,8],"def_end_pos":[844,30]},{"full_name":"FirstOrder.Language.Substructure.topEquiv","def_path":"Mathlib/ModelTheory/Substructures.lean","def_pos":[565,4],"def_end_pos":[565,12]}]},{"state_before":"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\ns : Set M\nα : Type u_3\nn : ℕ\nφ : L.BoundedFormula α n\nv : α → ↥⊤\nxs : Fin n → ↥⊤\n⊢ φ.Realize (⇑topEquiv ∘ v) (⇑topEquiv ∘ xs) ↔ φ.Realize (Subtype.val ∘ v) (Subtype.val ∘ xs)","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","commit":"","full_name":"Equiv.Perm.pow_apply_mem_toList_iff_mem_support","start":[315,0],"end":[318,25],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\n⊢ (p ^ n) x ∈ p.toList x ↔ x ∈ p.support","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\n⊢ x ∈ p.support → p.SameCycle x ((p ^ n) x)","tactic":"rw [mem_toList_iff, and_iff_right_iff_imp]","premises":[{"full_name":"Equiv.Perm.mem_toList_iff","def_path":"Mathlib/GroupTheory/Perm/Cycle/Concrete.lean","def_pos":[239,8],"def_end_pos":[239,22]},{"full_name":"and_iff_right_iff_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[165,16],"def_end_pos":[165,37]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\n⊢ x ∈ p.support → p.SameCycle x ((p ^ n) x)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\nx✝ : x ∈ p.support\n⊢ p.SameCycle ((p ^ n) x) x","tactic":"refine fun _ => SameCycle.symm ?_","premises":[{"full_name":"Equiv.Perm.SameCycle.symm","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[62,8],"def_end_pos":[62,22]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\nx✝ : x ∈ p.support\n⊢ p.SameCycle ((p ^ n) x) x","state_after":"no goals","tactic":"rw [sameCycle_pow_left]","premises":[{"full_name":"Equiv.Perm.sameCycle_pow_left","def_path":"Mathlib/GroupTheory/Perm/Cycle/Basic.lean","def_pos":[133,8],"def_end_pos":[133,26]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/IsSheafFor.lean","commit":"","full_name":"CategoryTheory.Presieve.isSheafFor_arrows_iff","start":[708,0],"end":[718,85],"file_path":"Mathlib/CategoryTheory/Sites/IsSheafFor.lean","tactics":[{"state_before":"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP Q U : Cᵒᵖ ⥤ Type w\nX✝ Y : C\nS : Sieve X✝\nR : Presieve X✝\nB : C\nI : Type u_1\nX : I → C\nπ : (i : I) → X i ⟶ B\n⊢ IsSheafFor P (ofArrows X π) ↔\n    ∀ (x : (i : I) → P.obj (op (X i))), Arrows.Compatible P π x → ∃! t, ∀ (i : I), P.map (π i).op t = x i","state_after":"case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP Q U : Cᵒᵖ ⥤ Type w\nX✝ Y : C\nS : Sieve X✝\nR : Presieve X✝\nB : C\nI : Type u_1\nX : I → C\nπ : (i : I) → X i ⟶ B\nh : IsSheafFor P (ofArrows X π)\nx : (i : I) → P.obj (op (X i))\nhx : Arrows.Compatible P π x\n⊢ ∃! t, ∀ (i : I), P.map (π i).op t = x i\n\ncase refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP Q U : Cᵒᵖ ⥤ Type w\nX✝ Y : C\nS : Sieve X✝\nR : Presieve X✝\nB : C\nI : Type u_1\nX : I → C\nπ : (i : I) → X i ⟶ B\nh : ∀ (x : (i : I) → P.obj (op (X i))), Arrows.Compatible P π x → ∃! t, ∀ (i : I), P.map (π i).op t = x i\nx : FamilyOfElements P (ofArrows X π)\nhx : x.Compatible\n⊢ ∃! t, x.IsAmalgamation t","tactic":"refine ⟨fun h x hx ↦ ?_, fun h x hx ↦ ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]}]}
{"url":"Mathlib/Analysis/Analytic/Composition.lean","commit":"","full_name":"FormalMultilinearSeries.comp_assoc","start":[1114,0],"end":[1152,51],"file_path":"Mathlib/Analysis/Analytic/Composition.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\n⊢ (r.comp q).comp p = r.comp (q.comp p)","state_after":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\n⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v","tactic":"ext n v","premises":[]},{"state_before":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\n⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v","state_after":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\n⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v","tactic":"let f : (Σ a : Composition n, Composition a.length) → H := fun c =>\n    r c.2.length (applyComposition q c.2 (applyComposition p c.1 v))","premises":[{"full_name":"Composition","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[96,10],"def_end_pos":[96,21]},{"full_name":"Composition.length","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[137,7],"def_end_pos":[137,13]},{"full_name":"FormalMultilinearSeries.applyComposition","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[100,4],"def_end_pos":[100,20]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.fst","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[175,2],"def_end_pos":[175,5]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]}]},{"state_before":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\n⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v","state_after":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v","tactic":"let g : (Σ c : Composition n, ∀ i : Fin c.length, Composition (c.blocksFun i)) → H := fun c =>\n    r c.1.length fun i : Fin c.1.length =>\n      q (c.2 i).length (applyComposition p (c.2 i) (v ∘ c.1.embedding i))","premises":[{"full_name":"Composition","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[96,10],"def_end_pos":[96,21]},{"full_name":"Composition.blocksFun","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[145,4],"def_end_pos":[145,13]},{"full_name":"Composition.embedding","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[258,4],"def_end_pos":[258,13]},{"full_name":"Composition.length","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[137,7],"def_end_pos":[137,13]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"FormalMultilinearSeries.applyComposition","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[100,4],"def_end_pos":[100,20]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.fst","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[175,2],"def_end_pos":[175,5]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]}]},{"state_before":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v","state_after":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ∑ c : (a : Composition n) × Composition a.length, f c =\n    ∑ c : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)), g c","tactic":"suffices ∑ c, f c = ∑ c, g c by\n    simpa (config := { unfoldPartialApp := true }) only [FormalMultilinearSeries.comp,\n      ContinuousMultilinearMap.sum_apply, compAlongComposition_apply, Finset.sum_sigma',\n      applyComposition, ContinuousMultilinearMap.map_sum]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"ContinuousMultilinearMap.map_sum","def_path":"Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean","def_pos":[395,8],"def_end_pos":[395,15]},{"full_name":"ContinuousMultilinearMap.sum_apply","def_path":"Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean","def_pos":[199,8],"def_end_pos":[199,17]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_sigma'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[549,2],"def_end_pos":[549,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"FormalMultilinearSeries.applyComposition","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[100,4],"def_end_pos":[100,20]},{"full_name":"FormalMultilinearSeries.comp","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[229,14],"def_end_pos":[229,18]},{"full_name":"FormalMultilinearSeries.compAlongComposition_apply","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[215,8],"def_end_pos":[215,34]}]},{"state_before":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ∑ c : (a : Composition n) × Composition a.length, f c =\n    ∑ c : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)), g c","state_after":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ∑ c : (a : Composition n) × Composition a.length, f c =\n    ∑ i : (a : Composition n) × Composition a.length, g ((sigmaEquivSigmaPi n) i)","tactic":"rw [← (sigmaEquivSigmaPi n).sum_comp]","premises":[{"full_name":"Composition.sigmaEquivSigmaPi","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[1048,4],"def_end_pos":[1048,21]},{"full_name":"Equiv.sum_comp","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1940,2],"def_end_pos":[1940,13]}]},{"state_before":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ∑ c : (a : Composition n) × Composition a.length, f c =\n    ∑ i : (a : Composition n) × Composition a.length, g ((sigmaEquivSigmaPi n) i)","state_after":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ∀ x ∈ Finset.univ, f x = g ((sigmaEquivSigmaPi n) x)","tactic":"apply Finset.sum_congr rfl","premises":[{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case h.H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\n⊢ ∀ x ∈ Finset.univ, f x = g ((sigmaEquivSigmaPi n) x)","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\n⊢ f ⟨a, b⟩ = g ((sigmaEquivSigmaPi n) ⟨a, b⟩)","tactic":"rintro ⟨a, b⟩ _","premises":[]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\n⊢ f ⟨a, b⟩ = g ((sigmaEquivSigmaPi n) ⟨a, b⟩)","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\n⊢ f ⟨a, b⟩ = g ⟨a.gather b, a.sigmaCompositionAux b⟩","tactic":"dsimp [sigmaEquivSigmaPi]","premises":[{"full_name":"Composition.sigmaEquivSigmaPi","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[1048,4],"def_end_pos":[1048,21]}]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\n⊢ f ⟨a, b⟩ = g ⟨a.gather b, a.sigmaCompositionAux b⟩","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\n⊢ ∀ (i : ℕ) (him : i < b.length) (hin : i < (a.gather b).length),\n    q.applyComposition ⟨a, b⟩.snd (p.applyComposition ⟨a, b⟩.fst v) ⟨i, him⟩ =\n      (q (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩).length)\n        (p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩)\n          (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hin⟩)))","tactic":"apply r.congr (Composition.length_gather a b).symm","premises":[{"full_name":"Composition.length_gather","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[948,8],"def_end_pos":[948,21]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FormalMultilinearSeries.congr","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[115,8],"def_end_pos":[115,13]}]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\n⊢ ∀ (i : ℕ) (him : i < b.length) (hin : i < (a.gather b).length),\n    q.applyComposition ⟨a, b⟩.snd (p.applyComposition ⟨a, b⟩.fst v) ⟨i, him⟩ =\n      (q (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩).length)\n        (p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩)\n          (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hin⟩)))","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\n⊢ q.applyComposition ⟨a, b⟩.snd (p.applyComposition ⟨a, b⟩.fst v) ⟨i, hi1⟩ =\n    (q (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).length)\n      (p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)\n        (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)))","tactic":"intro i hi1 hi2","premises":[]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\n⊢ q.applyComposition ⟨a, b⟩.snd (p.applyComposition ⟨a, b⟩.fst v) ⟨i, hi1⟩ =\n    (q (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).length)\n      (p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)\n        (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)))","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\n⊢ ∀ (i_1 : ℕ) (him : i_1 < b.blocksFun ⟨i, hi1⟩) (hin : i_1 < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length),\n    (p.applyComposition ⟨a, b⟩.fst v ∘ ⇑(⟨a, b⟩.snd.embedding ⟨i, hi1⟩)) ⟨i_1, him⟩ =\n      p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)\n        (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ⟨i_1, hin⟩","tactic":"apply q.congr (length_sigmaCompositionAux a b _).symm","premises":[{"full_name":"Composition.length_sigmaCompositionAux","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[969,8],"def_end_pos":[969,34]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FormalMultilinearSeries.congr","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[115,8],"def_end_pos":[115,13]}]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\n⊢ ∀ (i_1 : ℕ) (him : i_1 < b.blocksFun ⟨i, hi1⟩) (hin : i_1 < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length),\n    (p.applyComposition ⟨a, b⟩.fst v ∘ ⇑(⟨a, b⟩.snd.embedding ⟨i, hi1⟩)) ⟨i_1, him⟩ =\n      p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)\n        (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ⟨i_1, hin⟩","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\n⊢ (p.applyComposition ⟨a, b⟩.fst v ∘ ⇑(⟨a, b⟩.snd.embedding ⟨i, hi1⟩)) ⟨j, hj1⟩ =\n    p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)\n      (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ⟨j, hj2⟩","tactic":"intro j hj1 hj2","premises":[]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\n⊢ (p.applyComposition ⟨a, b⟩.fst v ∘ ⇑(⟨a, b⟩.snd.embedding ⟨i, hi1⟩)) ⟨j, hj1⟩ =\n    p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩)\n      (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ⟨j, hj2⟩","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\n⊢ ∀ (i_1 : ℕ) (him : i_1 < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩))\n    (hin : i_1 < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩),\n    (v ∘ ⇑(⟨a, b⟩.fst.embedding ((⟨a, b⟩.snd.embedding ⟨i, hi1⟩) ⟨j, hj1⟩))) ⟨i_1, him⟩ =\n      ((v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ∘\n          ⇑((⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).embedding ⟨j, hj2⟩))\n        ⟨i_1, hin⟩","tactic":"apply p.congr (blocksFun_sigmaCompositionAux a b _ _).symm","premises":[{"full_name":"Composition.blocksFun_sigmaCompositionAux","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[977,8],"def_end_pos":[977,37]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"FormalMultilinearSeries.congr","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[115,8],"def_end_pos":[115,13]}]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\n⊢ ∀ (i_1 : ℕ) (him : i_1 < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩))\n    (hin : i_1 < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩),\n    (v ∘ ⇑(⟨a, b⟩.fst.embedding ((⟨a, b⟩.snd.embedding ⟨i, hi1⟩) ⟨j, hj1⟩))) ⟨i_1, him⟩ =\n      ((v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ∘\n          ⇑((⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).embedding ⟨j, hj2⟩))\n        ⟨i_1, hin⟩","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\nk : ℕ\nhk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)\nhk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩\n⊢ (v ∘ ⇑(⟨a, b⟩.fst.embedding ((⟨a, b⟩.snd.embedding ⟨i, hi1⟩) ⟨j, hj1⟩))) ⟨k, hk1⟩ =\n    ((v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ∘\n        ⇑((⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).embedding ⟨j, hj2⟩))\n      ⟨k, hk2⟩","tactic":"intro k hk1 hk2","premises":[]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\nk : ℕ\nhk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)\nhk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩\n⊢ (v ∘ ⇑(⟨a, b⟩.fst.embedding ((⟨a, b⟩.snd.embedding ⟨i, hi1⟩) ⟨j, hj1⟩))) ⟨k, hk1⟩ =\n    ((v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)) ∘\n        ⇑((⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).embedding ⟨j, hj2⟩))\n      ⟨k, hk2⟩","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\nk : ℕ\nhk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)\nhk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩\n⊢ ↑((⟨a, b⟩.fst.embedding ((⟨a, b⟩.snd.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)) ⟨k, hk1⟩) =\n    ↑((⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)\n        (((⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).embedding ⟨j, hj2⟩) ⟨k, hk2⟩))","tactic":"refine congr_arg v (Fin.ext ?_)","premises":[{"full_name":"Fin.ext","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[40,15],"def_end_pos":[40,18]}]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\nk : ℕ\nhk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)\nhk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩\n⊢ ↑((⟨a, b⟩.fst.embedding ((⟨a, b⟩.snd.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)) ⟨k, hk1⟩) =\n    ↑((⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hi2⟩)\n        (((⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hi2⟩).embedding ⟨j, hj2⟩) ⟨k, hk2⟩))","state_after":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\nk : ℕ\nhk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)\nhk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩\n⊢ a.sizeUpTo (b.sizeUpTo i + j) + k = (a.gather b).sizeUpTo i + ((a.sigmaCompositionAux b ⟨i, hi2⟩).sizeUpTo j + k)","tactic":"dsimp [Composition.embedding]","premises":[{"full_name":"Composition.embedding","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[258,4],"def_end_pos":[258,13]}]},{"state_before":"case h.H.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nr : FormalMultilinearSeries 𝕜 G H\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nv : Fin n → E\nf : (a : Composition n) × Composition a.length → H :=\n  fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))\ng : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=\n  fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))\na : Composition n\nb : Composition a.length\na✝ : ⟨a, b⟩ ∈ Finset.univ\ni : ℕ\nhi1 : i < b.length\nhi2 : i < (a.gather b).length\nj : ℕ\nhj1 : j < b.blocksFun ⟨i, hi1⟩\nhj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length\nk : ℕ\nhk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)\nhk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩\n⊢ a.sizeUpTo (b.sizeUpTo i + j) + k = (a.gather b).sizeUpTo i + ((a.sigmaCompositionAux b ⟨i, hi2⟩).sizeUpTo j + k)","state_after":"no goals","tactic":"rw [sizeUpTo_sizeUpTo_add _ _ hi1 hj1, add_assoc]","premises":[{"full_name":"Composition.sizeUpTo_sizeUpTo_add","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[994,8],"def_end_pos":[994,29]},{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"Subgroup.map_normalClosure","start":[2620,0],"end":[2626,75],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\n⊢ map f (normalClosure s) = normalClosure (⇑f '' s)","state_after":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ map f (normalClosure s) = normalClosure (⇑f '' s)","tactic":"have : Normal (map f (normalClosure s)) := Normal.map inferInstance f hf","premises":[{"full_name":"Subgroup.Normal","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1470,10],"def_end_pos":[1470,16]},{"full_name":"Subgroup.Normal.map","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[2253,8],"def_end_pos":[2253,18]},{"full_name":"Subgroup.map","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1091,4],"def_end_pos":[1091,7]},{"full_name":"Subgroup.normalClosure","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[1837,4],"def_end_pos":[1837,17]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ map f (normalClosure s) = normalClosure (⇑f '' s)","state_after":"case a\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ map f (normalClosure s) ≤ normalClosure (⇑f '' s)\n\ncase a\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ normalClosure (⇑f '' s) ≤ map f (normalClosure s)","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
{"url":"Mathlib/Analysis/ConstantSpeed.lean","commit":"","full_name":"hasConstantSpeedOnWith_zero_iff","start":[147,0],"end":[164,47],"file_path":"Mathlib/Analysis/ConstantSpeed.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ HasConstantSpeedOnWith f s 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0","state_after":"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (0 * (y - x))) ↔\n    ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0","tactic":"dsimp [HasConstantSpeedOnWith]","premises":[{"full_name":"HasConstantSpeedOnWith","def_path":"Mathlib/Analysis/ConstantSpeed.lean","def_pos":[51,4],"def_end_pos":[51,26]}]},{"state_before":"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (0 * (y - x))) ↔\n    ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0","state_after":"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) ↔ eVariationOn f s = 0","tactic":"simp only [zero_mul, ENNReal.ofReal_zero, ← eVariationOn.eq_zero_iff]","premises":[{"full_name":"ENNReal.ofReal_zero","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[240,16],"def_end_pos":[240,27]},{"full_name":"MulZeroClass.zero_mul","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[35,2],"def_end_pos":[35,10]},{"full_name":"eVariationOn.eq_zero_iff","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[149,8],"def_end_pos":[149,19]}]},{"state_before":"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) ↔ eVariationOn f s = 0","state_after":"case mp\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) → eVariationOn f s = 0\n\ncase mpr\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ eVariationOn f s = 0 → ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","commit":"","full_name":"Matrix.invOf_submatrix_equiv_eq","start":[669,0],"end":[673,43],"file_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommRing α\nA✝ B : Matrix n n α\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\nA : Matrix m m α\ne₁ e₂ : n ≃ m\ninst✝¹ : Invertible A\ninst✝ : Invertible (A.submatrix ⇑e₁ ⇑e₂)\n⊢ ⅟(A.submatrix ⇑e₁ ⇑e₂) = (⅟A).submatrix ⇑e₂ ⇑e₁","state_after":"l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommRing α\nA✝ B : Matrix n n α\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\nA : Matrix m m α\ne₁ e₂ : n ≃ m\ninst✝¹ : Invertible A\ninst✝ : Invertible (A.submatrix ⇑e₁ ⇑e₂)\nthis : Invertible (A.submatrix ⇑e₁ ⇑e₂) := A.submatrixEquivInvertible e₁ e₂\n⊢ ⅟(A.submatrix ⇑e₁ ⇑e₂) = (⅟A).submatrix ⇑e₂ ⇑e₁","tactic":"letI := submatrixEquivInvertible A e₁ e₂","premises":[{"full_name":"Matrix.submatrixEquivInvertible","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[652,4],"def_end_pos":[652,28]}]},{"state_before":"l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommRing α\nA✝ B : Matrix n n α\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\nA : Matrix m m α\ne₁ e₂ : n ≃ m\ninst✝¹ : Invertible A\ninst✝ : Invertible (A.submatrix ⇑e₁ ⇑e₂)\nthis : Invertible (A.submatrix ⇑e₁ ⇑e₂) := A.submatrixEquivInvertible e₁ e₂\n⊢ ⅟(A.submatrix ⇑e₁ ⇑e₂) = (⅟A).submatrix ⇑e₂ ⇑e₁","state_after":"no goals","tactic":"convert (rfl : ⅟ (A.submatrix e₁ e₂) = _)","premises":[{"full_name":"Invertible.invOf","def_path":"Mathlib/Algebra/Group/Invertible/Defs.lean","def_pos":[86,2],"def_end_pos":[86,7]},{"full_name":"Matrix.submatrix","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[2281,4],"def_end_pos":[2281,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/CategoryTheory/GradedObject.lean","commit":"","full_name":"CategoryTheory.GradedObject.ιMapObjOrZero_mapMap","start":[507,0],"end":[512,69],"file_path":"Mathlib/CategoryTheory/GradedObject.lean","tactics":[{"state_before":"I : Type u_1\nJ : Type u_2\nK : Type u_3\nC : Type u_4\ninst✝⁵ : Category.{u_5, u_4} C\nX Y Z : GradedObject I C\nφ : X ⟶ Y\ne : X ≅ Y\nψ : Y ⟶ Z\np : I → J\nj✝ : J\ninst✝⁴ : X.HasMap p\ninst✝³ : Y.HasMap p\ninst✝² : Z.HasMap p\nq : J → K\nr : I → K\nhpqr : ∀ (i : I), q (p i) = r i\ninst✝¹ : HasZeroMorphisms C\ninst✝ : DecidableEq J\ni : I\nj : J\n⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j","state_after":"case pos\nI : Type u_1\nJ : Type u_2\nK : Type u_3\nC : Type u_4\ninst✝⁵ : Category.{u_5, u_4} C\nX Y Z : GradedObject I C\nφ : X ⟶ Y\ne : X ≅ Y\nψ : Y ⟶ Z\np : I → J\nj✝ : J\ninst✝⁴ : X.HasMap p\ninst✝³ : Y.HasMap p\ninst✝² : Z.HasMap p\nq : J → K\nr : I → K\nhpqr : ∀ (i : I), q (p i) = r i\ninst✝¹ : HasZeroMorphisms C\ninst✝ : DecidableEq J\ni : I\nj : J\nh : p i = j\n⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j\n\ncase neg\nI : Type u_1\nJ : Type u_2\nK : Type u_3\nC : Type u_4\ninst✝⁵ : Category.{u_5, u_4} C\nX Y Z : GradedObject I C\nφ : X ⟶ Y\ne : X ≅ Y\nψ : Y ⟶ Z\np : I → J\nj✝ : J\ninst✝⁴ : X.HasMap p\ninst✝³ : Y.HasMap p\ninst✝² : Z.HasMap p\nq : J → K\nr : I → K\nhpqr : ∀ (i : I), q (p i) = r i\ninst✝¹ : HasZeroMorphisms C\ninst✝ : DecidableEq J\ni : I\nj : J\nh : ¬p i = j\n⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j","tactic":"by_cases h : p i = j","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/SetTheory/Cardinal/Basic.lean","commit":"","full_name":"Cardinal.aleph0_le","start":[1337,0],"end":[1341,67],"file_path":"Mathlib/SetTheory/Cardinal/Basic.lean","tactics":[{"state_before":"α β : Type u\nc : Cardinal.{u_1}\nh : ∀ (n : ℕ), ↑n ≤ c\nhn : c < ℵ₀\n⊢ False","state_after":"case intro\nα β : Type u\nn : ℕ\nh : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n\nhn : ↑n < ℵ₀\n⊢ False","tactic":"rcases lt_aleph0.1 hn with ⟨n, rfl⟩","premises":[{"full_name":"Cardinal.lt_aleph0","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1322,8],"def_end_pos":[1322,17]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]}]},{"state_before":"case intro\nα β : Type u\nn : ℕ\nh : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n\nhn : ↑n < ℵ₀\n⊢ False","state_after":"no goals","tactic":"exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))","premises":[{"full_name":"Cardinal.natCast_le","def_path":"Mathlib/SetTheory/Cardinal/Basic.lean","def_pos":[1240,8],"def_end_pos":[1240,18]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.lt_succ_self","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[358,16],"def_end_pos":[358,28]}]}]}
{"url":"Mathlib/Data/Matroid/Constructions.lean","commit":"","full_name":"Matroid.uniqueBaseOn_base_iff","start":[195,0],"end":[196,93],"file_path":"Mathlib/Data/Matroid/Constructions.lean","tactics":[{"state_before":"α : Type u_1\nM : Matroid α\nE B I X R J : Set α\nhIE : I ⊆ E\n⊢ (uniqueBaseOn I E).Base B ↔ B = I","state_after":"no goals","tactic":"rw [uniqueBaseOn, base_restrict_iff', freeOn_basis'_iff, inter_eq_self_of_subset_right hIE]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Matroid.base_restrict_iff'","def_path":"Mathlib/Data/Matroid/Restrict.lean","def_pos":[156,8],"def_end_pos":[156,26]},{"full_name":"Matroid.freeOn_basis'_iff","def_path":"Mathlib/Data/Matroid/Constructions.lean","def_pos":[161,16],"def_end_pos":[161,33]},{"full_name":"Matroid.uniqueBaseOn","def_path":"Mathlib/Data/Matroid/Constructions.lean","def_pos":[190,4],"def_end_pos":[190,16]},{"full_name":"Set.inter_eq_self_of_subset_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[784,8],"def_end_pos":[784,37]}]}]}
{"url":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","commit":"","full_name":"Submonoid.LocalizationMap.lift_comp","start":[774,0],"end":[775,80],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","tactics":[{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\n⊢ (f.lift hg).comp f.toMap = g","state_after":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nx✝ : M\n⊢ ((f.lift hg).comp f.toMap) x✝ = g x✝","tactic":"ext","premises":[]},{"state_before":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nx✝ : M\n⊢ ((f.lift hg).comp f.toMap) x✝ = g x✝","state_after":"no goals","tactic":"exact f.lift_eq hg _","premises":[{"full_name":"Submonoid.LocalizationMap.lift_eq","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[766,8],"def_end_pos":[766,15]}]}]}
{"url":"Mathlib/Data/Real/Pi/Bounds.lean","commit":"","full_name":"Real.pi_gt_3141592","start":[181,0],"end":[184,100],"file_path":"Mathlib/Data/Real/Pi/Bounds.lean","tactics":[{"state_before":"⊢ 3.141592 < π","state_after":"no goals","tactic":"pi_lower_bound\n        [11482 / 8119, 7792 / 4217, 54055 / 27557, 949247 / 476920, 3310126 / 1657059,\n          2635492 / 1318143, 1580265 / 790192, 1221775 / 610899, 3612247 / 1806132, 849943 / 424972]","premises":[{"full_name":"Real.pi_lower_bound_start","def_path":"Mathlib/Data/Real/Pi/Bounds.lean","def_pos":[73,8],"def_end_pos":[73,28]},{"full_name":"Real.sqrtTwoAddSeries","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean","def_pos":[615,18],"def_end_pos":[615,34]},{"full_name":"Real.sqrtTwoAddSeries_step_up","def_path":"Mathlib/Data/Real/Pi/Bounds.lean","def_pos":[80,8],"def_end_pos":[80,32]}]}]}
{"url":"Mathlib/LinearAlgebra/Determinant.lean","commit":"","full_name":"LinearMap.det_zero","start":[251,0],"end":[257,81],"file_path":"Mathlib/LinearAlgebra/Determinant.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹² : CommRing R\nM✝ : Type u_2\ninst✝¹¹ : AddCommGroup M✝\ninst✝¹⁰ : Module R M✝\nM' : Type u_3\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nι : Type u_4\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ne : Basis ι R M✝\nA : Type u_5\ninst✝⁵ : CommRing A\ninst✝⁴ : Module A M✝\nκ : Type u_6\ninst✝³ : Fintype κ\n𝕜 : Type u_7\ninst✝² : Field 𝕜\nM : Type u_8\ninst✝¹ : AddCommGroup M\ninst✝ : Module 𝕜 M\n⊢ LinearMap.det 0 = 0 ^ FiniteDimensional.finrank 𝕜 M","state_after":"no goals","tactic":"simp only [← zero_smul 𝕜 (1 : M →ₗ[𝕜] M), det_smul, mul_one, MonoidHom.map_one]","premises":[{"full_name":"LinearMap","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[83,10],"def_end_pos":[83,19]},{"full_name":"LinearMap.det_smul","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[232,8],"def_end_pos":[232,16]},{"full_name":"MonoidHom.map_one","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[621,18],"def_end_pos":[621,35]},{"full_name":"RingHom.id","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[538,4],"def_end_pos":[538,6]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"zero_smul","def_path":"Mathlib/Algebra/SMulWithZero.lean","def_pos":[67,8],"def_end_pos":[67,17]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","commit":"","full_name":"Submodule.orthogonal_eq_inter","start":[101,0],"end":[110,20],"file_path":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ Kᗮ = ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)","state_after":"case a\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ Kᗮ ≤ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)\n\ncase a\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v) ≤ Kᗮ","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
{"url":"Mathlib/Algebra/Periodic.lean","commit":"","full_name":"Function.Antiperiodic.const_add","start":[433,0],"end":[434,88],"file_path":"Mathlib/Algebra/Periodic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝¹ : AddSemigroup α\ninst✝ : Neg β\nh : Antiperiodic f c\na x : α\n⊢ (fun x => f (a + x)) (x + c) = -(fun x => f (a + x)) x","state_after":"no goals","tactic":"simpa [add_assoc] using h (a + x)","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]}]}]}
{"url":"Mathlib/Data/Part.lean","commit":"","full_name":"Part.toOption_eq_some_iff","start":[249,0],"end":[252,37],"file_path":"Mathlib/Data/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\no : Part α\ninst✝ : Decidable o.Dom\na : α\n⊢ o.toOption = Option.some a ↔ a ∈ o","state_after":"no goals","tactic":"rw [← Option.mem_def, mem_toOption]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Option.mem_def","def_path":".lake/packages/lean4/src/lean/Init/Data/Option/Instances.lean","def_pos":[24,16],"def_end_pos":[24,23]},{"full_name":"Part.mem_toOption","def_path":"Mathlib/Data/Part.lean","def_pos":[243,8],"def_end_pos":[243,20]}]}]}
{"url":"Mathlib/Computability/Primrec.lean","commit":"","full_name":"Nat.Primrec'.head","start":[1290,0],"end":[1291,43],"file_path":"Mathlib/Computability/Primrec.lean","tactics":[{"state_before":"n : ℕ\nv : Vector ℕ n.succ\n⊢ v.get 0 = v.head","state_after":"no goals","tactic":"simp [get_zero]","premises":[{"full_name":"Mathlib.Vector.get_zero","def_path":"Mathlib/Data/Vector/Basic.lean","def_pos":[220,8],"def_end_pos":[220,16]}]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","commit":"","full_name":"ProbabilityTheory.lintegral_stieltjesOfMeasurableRat","start":[184,0],"end":[188,91],"file_path":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\n⊢ ∫⁻ (b : β), ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (univ ×ˢ Iic x)","state_after":"no goals","tactic":"rw [← setLIntegral_univ, setLIntegral_stieltjesOfMeasurableRat hf _ _ MeasurableSet.univ]","premises":[{"full_name":"MeasurableSet.univ","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[95,18],"def_end_pos":[95,36]},{"full_name":"MeasureTheory.setLIntegral_univ","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[639,8],"def_end_pos":[639,25]},{"full_name":"ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat","def_path":"Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean","def_pos":[119,6],"def_end_pos":[119,43]}]}]}
{"url":"Mathlib/Algebra/DualNumber.lean","commit":"","full_name":"DualNumber.algHom_ext","start":[108,0],"end":[113,26],"file_path":"Mathlib/Algebra/DualNumber.lean","tactics":[{"state_before":"R : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf g : R[ε] →ₐ[R] A\nhε : f ε = g ε\n⊢ f = g","state_after":"case hinr.h\nR : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf g : R[ε] →ₐ[R] A\nhε : f ε = g ε\n⊢ (f.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1 =\n    (g.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1","tactic":"ext","premises":[]},{"state_before":"case hinr.h\nR : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf g : R[ε] →ₐ[R] A\nhε : f ε = g ε\n⊢ (f.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1 =\n    (g.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1","state_after":"case hinr.h\nR : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf g : R[ε] →ₐ[R] A\nhε : f ε = g ε\n⊢ f (1 • ε) = g (1 • ε)","tactic":"dsimp","premises":[]},{"state_before":"case hinr.h\nR : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf g : R[ε] →ₐ[R] A\nhε : f ε = g ε\n⊢ f (1 • ε) = g (1 • ε)","state_after":"no goals","tactic":"simp only [one_smul, hε]","premises":[{"full_name":"one_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[379,6],"def_end_pos":[379,14]}]}]}
{"url":"Mathlib/Algebra/Order/ToIntervalMod.lean","commit":"","full_name":"toIcoDiv_neg","start":[317,0],"end":[326,36],"file_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1)","state_after":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a (-b) = -toIocDiv hp (-(a + p)) b","tactic":"suffices toIcoDiv hp a (-b) = -toIocDiv hp (-(a + p)) b by\n    rwa [neg_add, ← sub_eq_add_neg, toIocDiv_sub_eq_toIocDiv_add', toIocDiv_add_right] at this","premises":[{"full_name":"neg_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[515,14],"def_end_pos":[515,21]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]},{"full_name":"toIcoDiv","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[43,4],"def_end_pos":[43,12]},{"full_name":"toIocDiv","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[55,4],"def_end_pos":[55,12]},{"full_name":"toIocDiv_add_right","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[258,8],"def_end_pos":[258,26]},{"full_name":"toIocDiv_sub_eq_toIocDiv_add'","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[313,8],"def_end_pos":[313,37]}]},{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a (-b) = -toIocDiv hp (-(a + p)) b","state_after":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp (-(a + p)) b = -toIcoDiv hp a (-b)","tactic":"rw [← neg_eq_iff_eq_neg, eq_comm]","premises":[{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"neg_eq_iff_eq_neg","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[298,2],"def_end_pos":[298,13]}]},{"state_before":"α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp (-(a + p)) b = -toIcoDiv hp a (-b)","state_after":"case h\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","tactic":"apply toIocDiv_eq_of_sub_zsmul_mem_Ioc","premises":[{"full_name":"toIocDiv_eq_of_sub_zsmul_mem_Ioc","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[61,8],"def_end_pos":[61,40]}]},{"state_before":"case h\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","state_after":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : a ≤ -b - toIcoDiv hp a (-b) • p\nho : -b - toIcoDiv hp a (-b) • p < a + p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","tactic":"obtain ⟨hc, ho⟩ := sub_toIcoDiv_zsmul_mem_Ico hp a (-b)","premises":[{"full_name":"sub_toIcoDiv_zsmul_mem_Ico","def_path":"Mathlib/Algebra/Order/ToIntervalMod.lean","def_pos":[46,8],"def_end_pos":[46,34]}]},{"state_before":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : a ≤ -b - toIcoDiv hp a (-b) • p\nho : -b - toIcoDiv hp a (-b) • p < a + p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","state_after":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : a ≤ -b - toIcoDiv hp a (-b) • p\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","tactic":"rw [← neg_lt_neg_iff, neg_sub' (-b), neg_neg, ← neg_smul] at ho","premises":[{"full_name":"neg_lt_neg_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[245,2],"def_end_pos":[245,13]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]},{"full_name":"neg_sub'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[518,2],"def_end_pos":[518,13]}]},{"state_before":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : a ≤ -b - toIcoDiv hp a (-b) • p\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","state_after":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : b - -toIcoDiv hp a (-b) • p ≤ -a\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","tactic":"rw [← neg_le_neg_iff, neg_sub' (-b), neg_neg, ← neg_smul] at hc","premises":[{"full_name":"neg_le_neg_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[216,2],"def_end_pos":[216,13]},{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"neg_smul","def_path":"Mathlib/Algebra/Module/Defs.lean","def_pos":[228,8],"def_end_pos":[228,16]},{"full_name":"neg_sub'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[518,2],"def_end_pos":[518,13]}]},{"state_before":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : b - -toIcoDiv hp a (-b) • p ≤ -a\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)","state_after":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : b - -toIcoDiv hp a (-b) • p ≤ -a\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ -a = -(a + p) + p","tactic":"refine ⟨ho, hc.trans_eq ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]}]},{"state_before":"case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : b - -toIcoDiv hp a (-b) • p ≤ -a\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ -a = -(a + p) + p","state_after":"no goals","tactic":"rw [neg_add, neg_add_cancel_right]","premises":[{"full_name":"neg_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[515,14],"def_end_pos":[515,21]},{"full_name":"neg_add_cancel_right","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1070,2],"def_end_pos":[1070,13]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","commit":"","full_name":"SimpleGraph.Walk.mem_support_iff","start":[514,0],"end":[515,69],"file_path":"Mathlib/Combinatorics/SimpleGraph/Walk.lean","tactics":[{"state_before":"V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : G.Walk u v\n⊢ w ∈ p.support ↔ w = u ∨ w ∈ p.support.tail","state_after":"no goals","tactic":"cases p <;> simp","premises":[]}]}
{"url":"Mathlib/Order/Heyting/Basic.lean","commit":"","full_name":"sdiff_bot","start":[448,0],"end":[450,63],"file_path":"Mathlib/Order/Heyting/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedCoheytingAlgebra α\na b✝ c d b : α\n⊢ a \\ ⊥ ≤ b ↔ a ≤ b","state_after":"no goals","tactic":"rw [sdiff_le_iff, bot_sup_eq]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"bot_sup_eq","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[357,8],"def_end_pos":[357,18]},{"full_name":"sdiff_le_iff","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[377,8],"def_end_pos":[377,20]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Basic.lean","commit":"","full_name":"Finset.Icc_diff_both","start":[472,0],"end":[473,83],"file_path":"Mathlib/Order/Interval/Finset/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ c : α\ninst✝ : DecidableEq α\na b : α\n⊢ Icc a b \\ {a, b} = Ioo a b","state_after":"no goals","tactic":"simp [← coe_inj]","premises":[{"full_name":"Finset.coe_inj","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[224,8],"def_end_pos":[224,15]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.prod_piecewise","start":[1560,0],"end":[1563,56],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns t : Finset α\nf g : α → β\n⊢ ∏ x ∈ s, t.piecewise f g x = (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \\ t, g x","state_after":"no goals","tactic":"erw [prod_ite, filter_mem_eq_inter, ← sdiff_eq_filter]","premises":[{"full_name":"Finset.filter_mem_eq_inter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2303,8],"def_end_pos":[2303,27]},{"full_name":"Finset.prod_ite","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1052,8],"def_end_pos":[1052,16]},{"full_name":"Finset.sdiff_eq_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2342,8],"def_end_pos":[2342,23]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","commit":"","full_name":"CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_whiskerRight","start":[388,0],"end":[393,6],"file_path":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasRightDual X\ninst✝ : HasRightDual Y\nf : X ⟶ Y\n⊢ (tensorLeftHomEquiv (𝟙_ C) Y Yᘁ Xᘁ).symm (η_ X Xᘁ ≫ f ▷ Xᘁ) = (ρ_ Yᘁ).hom ≫ fᘁ","state_after":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasRightDual X\ninst✝ : HasRightDual Y\nf : X ⟶ Y\n⊢ Yᘁ ◁ (η_ X Xᘁ ≫ f ▷ Xᘁ) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom =\n    (ρ_ Yᘁ).hom ≫ (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom","tactic":"dsimp [tensorLeftHomEquiv, rightAdjointMate]","premises":[{"full_name":"CategoryTheory.rightAdjointMate","def_path":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","def_pos":[175,4],"def_end_pos":[175,20]},{"full_name":"CategoryTheory.tensorLeftHomEquiv","def_path":"Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean","def_pos":[266,4],"def_end_pos":[266,22]}]},{"state_before":"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nX Y : C\ninst✝¹ : HasRightDual X\ninst✝ : HasRightDual Y\nf : X ⟶ Y\n⊢ Yᘁ ◁ (η_ X Xᘁ ≫ f ▷ Xᘁ) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom =\n    (ρ_ Yᘁ).hom ≫ (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Comma/Presheaf.lean","commit":"","full_name":"CategoryTheory.OverPresheafAux.YonedaCollection.map₁_yonedaEquivFst","start":[307,0],"end":[311,58],"file_path":"Mathlib/CategoryTheory/Comma/Presheaf.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX : C\nG : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nη : F ⟶ G\np : YonedaCollection F X\n⊢ (map₁ η p).yonedaEquivFst = p.yonedaEquivFst","state_after":"no goals","tactic":"simp only [YonedaCollection.yonedaEquivFst_eq, map₁_fst]","premises":[{"full_name":"CategoryTheory.OverPresheafAux.YonedaCollection.map₁_fst","def_path":"Mathlib/CategoryTheory/Comma/Presheaf.lean","def_pos":[303,6],"def_end_pos":[303,14]},{"full_name":"CategoryTheory.OverPresheafAux.YonedaCollection.yonedaEquivFst_eq","def_path":"Mathlib/CategoryTheory/Comma/Presheaf.lean","def_pos":[277,6],"def_end_pos":[277,23]}]}]}
{"url":"Mathlib/Algebra/Ring/Parity.lean","commit":"","full_name":"Nat.even_or_odd'","start":[226,0],"end":[227,66],"file_path":"Mathlib/Algebra/Ring/Parity.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nm n✝ n : ℕ\n⊢ ∃ k, n = 2 * k ∨ n = 2 * k + 1","state_after":"no goals","tactic":"simpa only [← two_mul, exists_or, Odd, Even] using even_or_odd n","premises":[{"full_name":"Even","def_path":"Mathlib/Algebra/Group/Even.lean","def_pos":[44,2],"def_end_pos":[44,13]},{"full_name":"Nat.even_or_odd","def_path":"Mathlib/Algebra/Ring/Parity.lean","def_pos":[224,6],"def_end_pos":[224,17]},{"full_name":"Odd","def_path":"Mathlib/Algebra/Ring/Parity.lean","def_pos":[89,4],"def_end_pos":[89,7]},{"full_name":"exists_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"two_mul","def_path":"Mathlib/Algebra/Ring/Defs.lean","def_pos":[155,8],"def_end_pos":[155,15]}]}]}
{"url":"Mathlib/Geometry/RingedSpace/Stalks.lean","commit":"","full_name":"AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap","start":[190,0],"end":[207,5],"file_path":"Mathlib/Geometry/RingedSpace/Stalks.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\n⊢ Y.presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap f x = Hom.stalkMap f y ≫ X.presheaf.stalkSpecializes h","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\n⊢ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n        { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n          ι :=\n            {\n              app := fun U =>\n                colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n              naturality := ⋯ } } ≫\n      colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n          colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n      colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n        { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n          ι :=\n            {\n              app := fun U =>\n                colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n              naturality := ⋯ } }","tactic":"dsimp [stalkSpecializes, Hom.stalkMap, stalkFunctor, stalkPushforward]","premises":[{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.stalkMap","def_path":"Mathlib/Geometry/RingedSpace/Stalks.lean","def_pos":[40,4],"def_end_pos":[40,16]},{"full_name":"TopCat.Presheaf.stalkFunctor","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[73,4],"def_end_pos":[73,16]},{"full_name":"TopCat.Presheaf.stalkPushforward","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[137,4],"def_end_pos":[137,20]},{"full_name":"TopCat.Presheaf.stalkSpecializes","def_path":"Mathlib/Topology/Sheaves/Stalks.lean","def_pos":[320,18],"def_end_pos":[320,34]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\n⊢ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n        { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n          ι :=\n            {\n              app := fun U =>\n                colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n              naturality := ⋯ } } ≫\n      colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n          colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n      colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n        { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n          ι :=\n            {\n              app := fun U =>\n                colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n              naturality := ⋯ } }","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : (OpenNhds (f.base y))ᵒᵖ\n⊢ colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        j ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        j ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }","tactic":"refine colimit.hom_ext fun j => ?_","premises":[{"full_name":"CategoryTheory.Limits.colimit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[757,8],"def_end_pos":[757,23]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : (OpenNhds (f.base y))ᵒᵖ\n⊢ colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        j ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        j ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }","state_after":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        (op j) ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        (op j) ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }","tactic":"induction j with | h j => ?_","premises":[]},{"state_before":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        (op j) ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι\n        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj\n          Y.presheaf)\n        (op j) ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }","state_after":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }","tactic":"dsimp","premises":[]},{"state_before":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }","state_after":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫\n      X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫\n        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n          ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =\n    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫\n      X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫\n        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n          (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })","tactic":"simp only [colimit.ι_desc_assoc, ι_colimMap_assoc, whiskerLeft_app,\n    whiskerRight_app, NatTrans.id_app, map_id, colimit.ι_pre, id_comp, assoc,\n    colimit.pre_desc, colimit.map_desc, colimit.ι_desc, Cocones.precompose_obj_ι,\n    Cocone.whisker_ι, NatTrans.comp_app]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Limits.Cocone.whisker_ι","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[245,2],"def_end_pos":[245,7]},{"full_name":"CategoryTheory.Limits.Cocones.precompose_obj_ι","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[536,2],"def_end_pos":[536,7]},{"full_name":"CategoryTheory.Limits.colimit.map_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[990,8],"def_end_pos":[990,24]},{"full_name":"CategoryTheory.Limits.colimit.pre_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[874,8],"def_end_pos":[874,24]},{"full_name":"CategoryTheory.Limits.colimit.ι_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[691,8],"def_end_pos":[691,22]},{"full_name":"CategoryTheory.Limits.colimit.ι_pre","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[869,8],"def_end_pos":[869,21]},{"full_name":"CategoryTheory.NatTrans.comp_app","def_path":"Mathlib/CategoryTheory/Functor/Category.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"CategoryTheory.NatTrans.id_app","def_path":"Mathlib/CategoryTheory/Functor/Category.lean","def_pos":[66,8],"def_end_pos":[66,14]},{"full_name":"CategoryTheory.whiskerLeft_app","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[44,2],"def_end_pos":[44,7]},{"full_name":"CategoryTheory.whiskerRight_app","def_path":"Mathlib/CategoryTheory/Whiskering.lean","def_pos":[53,2],"def_end_pos":[53,7]}]},{"state_before":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫\n      X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫\n        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n          ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =\n    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫\n      X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫\n        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n          (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })","state_after":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫\n      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =\n    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫\n      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ }))","tactic":"erw [X.presheaf.map_id, id_comp]","premises":[{"full_name":"AlgebraicGeometry.PresheafedSpace.presheaf","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[48,12],"def_end_pos":[48,20]},{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫\n      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =\n    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫\n      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ }))","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","commit":"","full_name":"MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict'","start":[1060,0],"end":[1072,85],"file_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ✝ ν✝ ν₁ ν₂ : Measure α\ns t : Set α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nP : α → Prop\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x\n⊢ ∀ᵐ (x : α) ∂μ, P x","state_after":"α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ✝ ν✝ ν₁ ν₂ : Measure α\ns t : Set α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nP : α → Prop\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x\n⊢ ∀ᵐ (x : α) ∂μ, P x","tactic":"have : ∀ n, ∀ᵐ x ∂μ, x ∈ spanningSets (μ + ν) n → P x := by\n    intro n\n    have := h\n      (spanningSets (μ + ν) n) (measurable_spanningSets _ _)\n      ((self_le_add_right _ _).trans_lt (measure_spanningSets_lt_top (μ + ν) _))\n      ((self_le_add_left _ _).trans_lt (measure_spanningSets_lt_top (μ + ν) _))\n    exact (ae_restrict_iff' (measurable_spanningSets _ _)).mp this","premises":[{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"MeasureTheory.ae_restrict_iff'","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[547,8],"def_end_pos":[547,24]},{"full_name":"MeasureTheory.measurable_spanningSets","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[583,8],"def_end_pos":[583,31]},{"full_name":"MeasureTheory.measure_spanningSets_lt_top","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[587,8],"def_end_pos":[587,35]},{"full_name":"MeasureTheory.spanningSets","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[577,4],"def_end_pos":[577,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"self_le_add_left","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[75,2],"def_end_pos":[75,13]},{"full_name":"self_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[71,2],"def_end_pos":[71,13]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ✝ ν✝ ν₁ ν₂ : Measure α\ns t : Set α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nP : α → Prop\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x\n⊢ ∀ᵐ (x : α) ∂μ, P x","state_after":"no goals","tactic":"filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.ae_all_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[93,8],"def_end_pos":[93,18]},{"full_name":"MeasureTheory.mem_spanningSetsIndex","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[621,8],"def_end_pos":[621,29]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]}]}
{"url":"Mathlib/Data/ENNReal/Real.lean","commit":"","full_name":"ENNReal.toReal_sub_of_le","start":[40,0],"end":[44,93],"file_path":"Mathlib/Data/ENNReal/Real.lean","tactics":[{"state_before":"a✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : b ≤ a\nha : a ≠ ⊤\n⊢ (a - b).toReal = a.toReal - b.toReal","state_after":"case intro\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nha : a ≠ ⊤\nb : ℝ≥0\nh : ↑b ≤ a\n⊢ (a - ↑b).toReal = a.toReal - (↑b).toReal","tactic":"lift b to ℝ≥0 using ne_top_of_le_ne_top ha h","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]},{"full_name":"ne_top_of_le_ne_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[138,8],"def_end_pos":[138,27]}]},{"state_before":"case intro\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nha : a ≠ ⊤\nb : ℝ≥0\nh : ↑b ≤ a\n⊢ (a - ↑b).toReal = a.toReal - (↑b).toReal","state_after":"case intro.intro\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑b ≤ ↑a\n⊢ (↑a - ↑b).toReal = (↑a).toReal - (↑b).toReal","tactic":"lift a to ℝ≥0 using ha","premises":[{"full_name":"NNReal","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[60,4],"def_end_pos":[60,10]}]},{"state_before":"case intro.intro\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑b ≤ ↑a\n⊢ (↑a - ↑b).toReal = (↑a).toReal - (↑b).toReal","state_after":"no goals","tactic":"simp only [← ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)]","premises":[{"full_name":"ENNReal.coe_le_coe","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[314,27],"def_end_pos":[314,37]},{"full_name":"ENNReal.coe_sub","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[297,16],"def_end_pos":[297,23]},{"full_name":"ENNReal.coe_toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[234,16],"def_end_pos":[234,26]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"NNReal.coe_sub","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[178,18],"def_end_pos":[178,25]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/List/Perm.lean","commit":"","full_name":"List.Perm.filterMap","start":[154,0],"end":[160,48],"file_path":".lake/packages/batteries/Batteries/Data/List/Perm.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nf : α → Option β\nl₁ l₂ : List α\np : l₁ ~ l₂\n⊢ List.filterMap f l₁ ~ List.filterMap f l₂","state_after":"no goals","tactic":"induction p with\n  | nil => simp\n  | cons x _p IH => cases h : f x <;> simp [h, filterMap_cons, IH, Perm.cons]\n  | swap x y l₂ => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap_cons, swap]\n  | trans _p₁ _p₂ IH₁ IH₂ => exact IH₁.trans IH₂","premises":[{"full_name":"List.Perm.cons","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1354,4],"def_end_pos":[1354,8]},{"full_name":"List.Perm.nil","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1352,4],"def_end_pos":[1352,7]},{"full_name":"List.Perm.swap","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1356,4],"def_end_pos":[1356,8]},{"full_name":"List.Perm.trans","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[1358,4],"def_end_pos":[1358,9]},{"full_name":"List.filterMap_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[404,8],"def_end_pos":[404,22]}]}]}
{"url":"Mathlib/Algebra/Periodic.lean","commit":"","full_name":"Function.Periodic.sub_antiperiod","start":[501,0],"end":[503,60],"file_path":"Mathlib/Algebra/Periodic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh1 : Periodic f c₁\nh2 : Antiperiodic f c₂\n⊢ Antiperiodic f (c₁ - c₂)","state_after":"no goals","tactic":"simpa only [sub_eq_add_neg] using h1.add_antiperiod h2.neg","premises":[{"full_name":"Function.Antiperiodic.neg","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[360,18],"def_end_pos":[360,34]},{"full_name":"Function.Periodic.add_antiperiod","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[498,8],"def_end_pos":[498,31]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Periodic.lean","commit":"","full_name":"Function.Periodic.intervalIntegral_add_eq_of_pos","start":[232,0],"end":[239,99],"file_path":"Mathlib/MeasureTheory/Integral/Periodic.lean","tactics":[{"state_before":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\n⊢ ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in s..s + T, f x","state_after":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\n⊢ ∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume","tactic":"simp only [integral_of_le, hT.le, le_add_iff_nonneg_right]","premises":[{"full_name":"intervalIntegral.integral_of_le","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[430,8],"def_end_pos":[430,22]},{"full_name":"le_add_iff_nonneg_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[381,29],"def_end_pos":[381,52]}]},{"state_before":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\n⊢ ∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume","state_after":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ ∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume","tactic":"haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) ℝ volume :=\n    ⟨fun c s _ => measure_preimage_add _ _ _⟩","premises":[{"full_name":"AddSubgroup.zmultiples","def_path":"Mathlib/Algebra/Group/Subgroup/ZPowers.lean","def_pos":[77,4],"def_end_pos":[77,14]},{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"MeasureTheory.VAddInvariantMeasure","def_path":"Mathlib/MeasureTheory/Group/Action.lean","def_pos":[35,6],"def_end_pos":[35,26]},{"full_name":"MeasureTheory.measure_preimage_add","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[263,2],"def_end_pos":[263,13]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ ∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume","state_after":"case hs\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc t (t + T)) volume\n\ncase ht\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc s (s + T)) volume\n\ncase hf\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ ∀ (g : ↥(zmultiples T)) (x : ℝ), f (g +ᵥ x) = f x","tactic":"apply IsAddFundamentalDomain.setIntegral_eq (G := AddSubgroup.zmultiples T)","premises":[{"full_name":"AddSubgroup.zmultiples","def_path":"Mathlib/Algebra/Group/Subgroup/ZPowers.lean","def_pos":[77,4],"def_end_pos":[77,14]},{"full_name":"MeasureTheory.IsAddFundamentalDomain.setIntegral_eq","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[420,2],"def_end_pos":[420,13]}]},{"state_before":"case hs\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc t (t + T)) volume\n\ncase ht\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc s (s + T)) volume\n\ncase hf\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ ∀ (g : ↥(zmultiples T)) (x : ℝ), f (g +ᵥ x) = f x","state_after":"no goals","tactic":"exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomain_Ioc hT s, hf.map_vadd_zmultiples]","premises":[{"full_name":"Function.Periodic.map_vadd_zmultiples","def_path":"Mathlib/Algebra/Periodic.lean","def_pos":[265,8],"def_end_pos":[265,36]},{"full_name":"isAddFundamentalDomain_Ioc","def_path":"Mathlib/MeasureTheory/Integral/Periodic.lean","def_pos":[35,8],"def_end_pos":[35,34]}]}]}
{"url":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","commit":"","full_name":"FractionalIdeal.mapEquiv_refl","start":[149,0],"end":[151,32],"file_path":"Mathlib/RingTheory/FractionalIdeal/Operations.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type u_4\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\nx : FractionalIdeal S P\n⊢ (mapEquiv AlgEquiv.refl) x = (RingEquiv.refl (FractionalIdeal S P)) x","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","commit":"","full_name":"MeasurableSet.measurableSet_bliminf","start":[1298,0],"end":[1302,68],"file_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (bliminf s atTop p)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (⋃ i, ⋂ j, ⋂ (_ : p j ∧ i ≤ j), s j)","tactic":"simp only [Filter.bliminf_eq_iSup_biInf_of_nat, iInf_eq_iInter, iSup_eq_iUnion]","premises":[{"full_name":"Filter.bliminf_eq_iSup_biInf_of_nat","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[819,8],"def_end_pos":[819,36]},{"full_name":"Set.iInf_eq_iInter","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[276,8],"def_end_pos":[276,22]},{"full_name":"Set.iSup_eq_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[272,8],"def_end_pos":[272,22]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (⋃ i, ⋂ j, ⋂ (_ : p j ∧ i ≤ j), s j)","state_after":"no goals","tactic":"exact .iUnion fun n => .iInter fun m => .iInter fun hm => h m hm.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"MeasurableSet.iInter","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[138,8],"def_end_pos":[138,28]},{"full_name":"MeasurableSet.iUnion","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[106,18],"def_end_pos":[106,38]}]}]}
{"url":"Mathlib/Analysis/Normed/Lp/PiLp.lean","commit":"","full_name":"PiLp.nnnorm_eq_ciSup","start":[550,0],"end":[552,39],"file_path":"Mathlib/Analysis/Normed/Lp/PiLp.lean","tactics":[{"state_before":"p : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp ⊤ β\n⊢ ‖f‖₊ = ⨆ i, ‖f i‖₊","state_after":"case a\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp ⊤ β\n⊢ ↑‖f‖₊ = ↑(⨆ i, ‖f i‖₊)","tactic":"ext","premises":[]},{"state_before":"case a\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp ⊤ β\n⊢ ↑‖f‖₊ = ↑(⨆ i, ‖f i‖₊)","state_after":"no goals","tactic":"simp [NNReal.coe_iSup, norm_eq_ciSup]","premises":[{"full_name":"NNReal.coe_iSup","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[461,8],"def_end_pos":[461,16]},{"full_name":"PiLp.norm_eq_ciSup","def_path":"Mathlib/Analysis/Normed/Lp/PiLp.lean","def_pos":[258,8],"def_end_pos":[258,21]}]}]}
{"url":"Mathlib/LinearAlgebra/StdBasis.lean","commit":"","full_name":"LinearMap.stdBasis_apply'","start":[50,0],"end":[53,35],"file_path":"Mathlib/LinearAlgebra/StdBasis.lean","tactics":[{"state_before":"R : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0","state_after":"R : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (if i' = i then 1 else 0) = if i = i' then 1 else 0","tactic":"rw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]","premises":[{"full_name":"Function.update_apply","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[474,8],"def_end_pos":[474,20]},{"full_name":"LinearMap.stdBasis_apply","def_path":"Mathlib/LinearAlgebra/StdBasis.lean","def_pos":[47,8],"def_end_pos":[47,22]},{"full_name":"Pi.zero_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[59,2],"def_end_pos":[59,13]}]},{"state_before":"R : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (if i' = i then 1 else 0) = if i = i' then 1 else 0","state_after":"case e_c\nR : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (i' = i) = (i = i')","tactic":"congr 1","premises":[]},{"state_before":"case e_c\nR : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (i' = i) = (i = i')","state_after":"no goals","tactic":"rw [eq_iff_iff, eq_comm]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"eq_iff_iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1378,16],"def_end_pos":[1378,26]}]}]}
{"url":"Mathlib/Analysis/Seminorm.lean","commit":"","full_name":"Seminorm.rescale_to_shell_zpow","start":[1172,0],"end":[1198,57],"file_path":"Mathlib/Analysis/Seminorm.lean","tactics":[{"state_before":"R : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","state_after":"R : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","tactic":"have xεpos : 0 < (p x)/ε := by positivity","premises":[]},{"state_before":"R : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","state_after":"case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","tactic":"rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩","premises":[{"full_name":"exists_mem_Ico_zpow","def_path":"Mathlib/Algebra/Order/Archimedean.lean","def_pos":[194,8],"def_end_pos":[194,27]}]},{"state_before":"case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","state_after":"case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","tactic":"have cpos : 0 < ‖c‖ := by positivity","premises":[{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]}]},{"state_before":"case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","state_after":"case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","tactic":"have cnpos : 0 < ‖c^(n+1)‖ := by rw [norm_zpow]; exact xεpos.trans hn.2","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"norm_zpow","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[724,8],"def_end_pos":[724,17]}]},{"state_before":"case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","state_after":"case intro.refine_1\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ c ^ (-(n + 1)) ≠ 0\n\ncase intro.refine_2\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ p (c ^ (-(n + 1)) • x) < ε\n\ncase intro.refine_3\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ p (c ^ (-(n + 1)) • x)\n\ncase intro.refine_4\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x","tactic":"refine ⟨-(n+1), ?_, ?_, ?_, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/LinearAlgebra/Dual.lean","commit":"","full_name":"LinearMap.finrank_range_dualMap_eq_finrank_range","start":[1475,0],"end":[1483,79],"file_path":"Mathlib/LinearAlgebra/Dual.lean","tactics":[{"state_before":"K : Type uK\ninst✝⁴ : Field K\nV₁ : Type uV₁\nV₂ : Type uV₂\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ finrank K ↥(range f.dualMap) = finrank K ↥(range f)","state_after":"K : Type uK\ninst✝⁴ : Field K\nV₁ : Type uV₁\nV₂ : Type uV₂\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ Function.Injective ⇑f.rangeRestrict.dualMap","tactic":"rw [congr_arg dualMap (show f = (range f).subtype.comp f.rangeRestrict by rfl),\n    ← dualMap_comp_dualMap, range_comp,\n    range_eq_top.mpr (dualMap_surjective_of_injective (range f).injective_subtype),\n    Submodule.map_top, finrank_range_of_inj, Subspace.dual_finrank_eq]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"LinearMap.dualMap","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[180,4],"def_end_pos":[180,21]},{"full_name":"LinearMap.dualMap_comp_dualMap","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[203,8],"def_end_pos":[203,38]},{"full_name":"LinearMap.dualMap_surjective_of_injective","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[1365,8],"def_end_pos":[1365,39]},{"full_name":"LinearMap.finrank_range_of_inj","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[117,8],"def_end_pos":[117,38]},{"full_name":"LinearMap.range","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[56,4],"def_end_pos":[56,9]},{"full_name":"LinearMap.rangeRestrict","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[137,7],"def_end_pos":[137,20]},{"full_name":"LinearMap.range_comp","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[81,8],"def_end_pos":[81,18]},{"full_name":"LinearMap.range_eq_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[89,8],"def_end_pos":[89,20]},{"full_name":"Submodule.injective_subtype","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[81,8],"def_end_pos":[81,25]},{"full_name":"Submodule.map_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[259,8],"def_end_pos":[259,15]},{"full_name":"Submodule.subtype","def_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","def_pos":[69,14],"def_end_pos":[69,21]},{"full_name":"Subspace.dual_finrank_eq","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[1063,8],"def_end_pos":[1063,23]}]},{"state_before":"K : Type uK\ninst✝⁴ : Field K\nV₁ : Type uV₁\nV₂ : Type uV₂\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ Function.Injective ⇑f.rangeRestrict.dualMap","state_after":"no goals","tactic":"exact dualMap_injective_of_surjective (range_eq_top.mp f.range_rangeRestrict)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LinearMap.dualMap_injective_of_surjective","def_path":"Mathlib/LinearAlgebra/Dual.lean","def_pos":[208,8],"def_end_pos":[208,49]},{"full_name":"LinearMap.range_eq_top","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[89,8],"def_end_pos":[89,20]},{"full_name":"LinearMap.range_rangeRestrict","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[386,16],"def_end_pos":[386,35]}]}]}
{"url":"Mathlib/Order/Hom/CompleteLattice.lean","commit":"","full_name":"FrameHom.cancel_left","start":[563,0],"end":[566,80],"file_path":"Mathlib/Order/Hom/CompleteLattice.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nι : Sort u_6\nκ : ι → Sort u_7\ninst✝⁴ : FunLike F α β\ninst✝³ : CompleteLattice α\ninst✝² : CompleteLattice β\ninst✝¹ : CompleteLattice γ\ninst✝ : CompleteLattice δ\ng : FrameHom β γ\nf₁ f₂ : FrameHom α β\nhg : Injective ⇑g\nh : g.comp f₁ = g.comp f₂\na : α\n⊢ g (f₁ a) = g (f₂ a)","state_after":"no goals","tactic":"rw [← comp_apply, h, comp_apply]","premises":[{"full_name":"FrameHom.comp_apply","def_path":"Mathlib/Order/Hom/CompleteLattice.lean","def_pos":[542,8],"def_end_pos":[542,18]}]}]}
{"url":"Mathlib/Combinatorics/SetFamily/Shadow.lean","commit":"","full_name":"Finset.mem_upShadow_iff_exists_mem_card_add","start":[273,0],"end":[294,7],"file_path":"Mathlib/Combinatorics/SetFamily/Shadow.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\ns t : Finset α\na : α\nk r : ℕ\n⊢ s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card","state_after":"case zero\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ s ∈ ∂⁺ ^[0] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + 0 = s.card\n\ncase succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ s ∈ ∂⁺ ^[k + 1] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","tactic":"induction' k with k ih generalizing 𝒜 s","premises":[]},{"state_before":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ s ∈ ∂⁺ ^[k + 1] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","state_after":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ s ∈ ∂⁺ ^[k] (∂⁺ 𝒜) ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","tactic":"simp only [exists_prop, Function.comp_apply, Function.iterate_succ]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"Function.iterate_succ","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[57,8],"def_end_pos":[57,20]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ s ∈ ∂⁺ ^[k] (∂⁺ 𝒜) ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","state_after":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ t.card + k = s.card) ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","tactic":"refine ih.trans ?_","premises":[{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]}]},{"state_before":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ t.card + k = s.card) ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","state_after":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ t.card + k = s.card) ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","tactic":"clear ih","premises":[]},{"state_before":"case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ t.card + k = s.card) ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card","state_after":"case succ.mp\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ t.card + k = s.card) → ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card\n\ncase succ.mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card) → ∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ t.card + k = s.card","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Topology/Bases.lean","commit":"","full_name":"exists_countable_dense_bot_top","start":[628,0],"end":[635,62],"file_path":"Mathlib/Topology/Bases.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : PartialOrder α\n⊢ ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∈ s) ∧ ∀ (x : α), IsTop x → x ∈ s","state_after":"no goals","tactic":"simpa using dense_univ.exists_countable_dense_subset_bot_top","premises":[{"full_name":"Dense.exists_countable_dense_subset_bot_top","def_path":"Mathlib/Topology/Bases.lean","def_pos":[613,8],"def_end_pos":[613,51]},{"full_name":"dense_univ","def_path":"Mathlib/Topology/Basic.lean","def_pos":[522,8],"def_end_pos":[522,18]}]}]}
{"url":"Mathlib/Data/Finset/Basic.lean","commit":"","full_name":"Finset.erase_union_of_mem","start":[1956,0],"end":[1957,98],"file_path":"Mathlib/Data/Finset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na b : α\nha : a ∈ t\ns : Finset α\n⊢ s.erase a ∪ t = s ∪ t","state_after":"no goals","tactic":"rw [← insert_erase (mem_union_right s ha), erase_union_distrib, ← union_insert, insert_erase ha]","premises":[{"full_name":"Finset.erase_union_distrib","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1947,8],"def_end_pos":[1947,27]},{"full_name":"Finset.insert_erase","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1646,16],"def_end_pos":[1646,28]},{"full_name":"Finset.mem_union_right","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1201,8],"def_end_pos":[1201,23]},{"full_name":"Finset.union_insert","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1289,8],"def_end_pos":[1289,20]}]}]}
{"url":"Mathlib/CategoryTheory/DifferentialObject.lean","commit":"","full_name":"CategoryTheory.DifferentialObject.shiftZero_hom_app_f","start":[265,0],"end":[272,13],"file_path":"Mathlib/CategoryTheory/DifferentialObject.lean","tactics":[{"state_before":"S : Type u_1\ninst✝³ : AddCommGroupWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\n⊢ shiftFunctor C 0 ≅ 𝟭 (DifferentialObject S C)","state_after":"case refine_1\nS : Type u_1\ninst✝³ : AddCommGroupWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX : DifferentialObject S C\n⊢ ((shiftFunctor C 0).obj X).d ≫ (CategoryTheory.shiftFunctor C 1).map ((shiftFunctorZero C S).app X.obj).hom =\n    ((shiftFunctorZero C S).app X.obj).hom ≫ ((𝟭 (DifferentialObject S C)).obj X).d\n\ncase refine_2\nS : Type u_1\ninst✝³ : AddCommGroupWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX✝ Y✝ : DifferentialObject S C\nf : X✝ ⟶ Y✝\n⊢ (shiftFunctor C 0).map f ≫ ((fun X => mkIso ((shiftFunctorZero C S).app X.obj) ⋯) Y✝).hom =\n    ((fun X => mkIso ((shiftFunctorZero C S).app X.obj) ⋯) X✝).hom ≫ (𝟭 (DifferentialObject S C)).map f","tactic":"refine NatIso.ofComponents (fun X => mkIso ((shiftFunctorZero C S).app X.obj) ?_) (fun f => ?_)","premises":[{"full_name":"CategoryTheory.DifferentialObject.mkIso","def_path":"Mathlib/CategoryTheory/DifferentialObject.lean","def_pos":[141,4],"def_end_pos":[141,9]},{"full_name":"CategoryTheory.DifferentialObject.obj","def_path":"Mathlib/CategoryTheory/DifferentialObject.lean","def_pos":[39,2],"def_end_pos":[39,5]},{"full_name":"CategoryTheory.Iso.app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[51,4],"def_end_pos":[51,7]},{"full_name":"CategoryTheory.NatIso.ofComponents","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[186,4],"def_end_pos":[186,16]},{"full_name":"CategoryTheory.shiftFunctorZero","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[179,4],"def_end_pos":[179,20]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","commit":"","full_name":"CategoryTheory.Limits.MultispanIndex.multicoforkEquivSigmaCofork_functor","start":[650,0],"end":[670,12],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Multicofork I\n⊢ ∀ (j : WalkingMultispan I.fstFrom I.sndFrom),\n    ((𝟭 (Multicofork I)).obj K).ι.app j ≫ (Iso.refl ((𝟭 (Multicofork I)).obj K).pt).hom =\n      ((I.toSigmaCoforkFunctor ⋙ I.ofSigmaCoforkFunctor).obj K).ι.app j","state_after":"no goals","tactic":"rintro (_ | _) <;> simp","premises":[]},{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n      (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","state_after":"case w\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ∀ (j : Discrete I.R),\n    colimit.ι (Discrete.functor I.right) j ≫\n        ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n          (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n      colimit.ι (Discrete.functor I.right) j ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","tactic":"apply Limits.colimit.hom_ext","premises":[{"full_name":"CategoryTheory.Limits.colimit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[757,8],"def_end_pos":[757,23]}]},{"state_before":"case w\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ∀ (j : Discrete I.R),\n    colimit.ι (Discrete.functor I.right) j ≫\n        ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n          (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n      colimit.ι (Discrete.functor I.right) j ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","state_after":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫\n      ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n        (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","tactic":"rintro ⟨j⟩","premises":[]},{"state_before":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫\n      ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n        (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π","state_after":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫ Sigma.desc (Multicofork.ofSigmaCofork I K).π ≫ 𝟙 K.pt =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π","tactic":"dsimp","premises":[]},{"state_before":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫ Sigma.desc (Multicofork.ofSigmaCofork I K).π ≫ 𝟙 K.pt =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π","state_after":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ (Multicofork.ofSigmaCofork I K).π j = colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π","tactic":"simp only [Category.comp_id, colimit.ι_desc, Cofan.mk_ι_app]","premises":[{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Limits.Cofan.mk_ι_app","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Products.lean","def_pos":[63,12],"def_end_pos":[63,17]},{"full_name":"CategoryTheory.Limits.colimit.ι_desc","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[691,8],"def_end_pos":[691,22]}]},{"state_before":"case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ (Multicofork.ofSigmaCofork I K).π j = colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Limits/HasLimits.lean","commit":"","full_name":"CategoryTheory.Limits.limit.pre_pre","start":[363,0],"end":[367,64],"file_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","tactics":[{"state_before":"J : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\n⊢ pre F E ≫ pre (E ⋙ F) D = pre F (D ⋙ E)","state_after":"J : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\nthis : HasLimit ((D ⋙ E) ⋙ F)\n⊢ pre F E ≫ pre (E ⋙ F) D = pre F (D ⋙ E)","tactic":"haveI : HasLimit ((D ⋙ E) ⋙ F) := h","premises":[{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Limits.HasLimit","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[82,6],"def_end_pos":[82,14]}]},{"state_before":"J : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\nthis : HasLimit ((D ⋙ E) ⋙ F)\n⊢ pre F E ≫ pre (E ⋙ F) D = pre F (D ⋙ E)","state_after":"case w\nJ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\nthis : HasLimit ((D ⋙ E) ⋙ F)\nj : L\n⊢ (pre F E ≫ pre (E ⋙ F) D) ≫ π (D ⋙ E ⋙ F) j = pre F (D ⋙ E) ≫ π (D ⋙ E ⋙ F) j","tactic":"ext j","premises":[]},{"state_before":"case w\nJ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\nthis : HasLimit ((D ⋙ E) ⋙ F)\nj : L\n⊢ (pre F E ≫ pre (E ⋙ F) D) ≫ π (D ⋙ E ⋙ F) j = pre F (D ⋙ E) ≫ π (D ⋙ E ⋙ F) j","state_after":"case w\nJ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\nthis : HasLimit ((D ⋙ E) ⋙ F)\nj : L\n⊢ π F (E.obj (D.obj j)) = π F ((D ⋙ E).obj j)","tactic":"erw [assoc, limit.pre_π, limit.pre_π, limit.pre_π]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.limit.pre_π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[352,8],"def_end_pos":[352,19]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case w\nJ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝⁴ : Category.{v₂, u₂} K\nC : Type u\ninst✝³ : Category.{v, u} C\nF : J ⥤ C\ninst✝² : HasLimit F\nE : K ⥤ J\ninst✝¹ : HasLimit (E ⋙ F)\nL : Type u₃\ninst✝ : Category.{v₃, u₃} L\nD : L ⥤ K\nh : HasLimit (D ⋙ E ⋙ F)\nthis : HasLimit ((D ⋙ E) ⋙ F)\nj : L\n⊢ π F (E.obj (D.obj j)) = π F ((D ⋙ E).obj j)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","commit":"","full_name":"Matrix.PosSemidef.sq_sqrt","start":[165,0],"end":[176,50],"file_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","tactics":[{"state_before":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ hA.sqrt ^ 2 = A","state_after":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\n⊢ hA.sqrt ^ 2 = A","tactic":"let C : Matrix n n 𝕜 := hA.1.eigenvectorUnitary","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Matrix","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[55,4],"def_end_pos":[55,10]},{"full_name":"Matrix.IsHermitian.eigenvectorUnitary","def_path":"Mathlib/LinearAlgebra/Matrix/Spectrum.lean","def_pos":[71,18],"def_end_pos":[71,36]}]},{"state_before":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\n⊢ hA.sqrt ^ 2 = A","state_after":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\nE : Matrix n n 𝕜 := diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)\n⊢ hA.sqrt ^ 2 = A","tactic":"let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Matrix.IsHermitian.eigenvalues","def_path":"Mathlib/LinearAlgebra/Matrix/Spectrum.lean","def_pos":[40,18],"def_end_pos":[40,29]},{"full_name":"Matrix.diagonal","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[356,4],"def_end_pos":[356,12]},{"full_name":"Real.sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[109,18],"def_end_pos":[109,22]}]},{"state_before":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\nE : Matrix n n 𝕜 := diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)\n⊢ hA.sqrt ^ 2 = A","state_after":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\nE : Matrix n n 𝕜 := diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)\n⊢ C * (E * (star C * C) * E) * star C = A","tactic":"suffices C * (E * (star C * C) * E) * star C = A by\n    rw [Matrix.PosSemidef.sqrt, pow_two]\n    simpa only [← mul_assoc] using this","premises":[{"full_name":"Matrix.PosSemidef.sqrt","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[138,18],"def_end_pos":[138,22]},{"full_name":"Star.star","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[46,2],"def_end_pos":[46,6]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]}]},{"state_before":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\nE : Matrix n n 𝕜 := diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)\n⊢ C * (E * (star C * C) * E) * star C = A","state_after":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\nE : Matrix n n 𝕜 := diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)\nthis : E * E = diagonal (RCLike.ofReal ∘ ⋯.eigenvalues)\n⊢ C * (E * (star C * C) * E) * star C = A","tactic":"have : E * E = diagonal ((↑) ∘ hA.1.eigenvalues) := by\n    rw [diagonal_mul_diagonal]\n    congr! with v\n    simp [← pow_two, ← RCLike.ofReal_pow, Real.sq_sqrt (hA.eigenvalues_nonneg v)]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Matrix.IsHermitian.eigenvalues","def_path":"Mathlib/LinearAlgebra/Matrix/Spectrum.lean","def_pos":[40,18],"def_end_pos":[40,29]},{"full_name":"Matrix.PosSemidef.eigenvalues_nonneg","def_path":"Mathlib/LinearAlgebra/Matrix/PosDef.lean","def_pos":[129,6],"def_end_pos":[129,24]},{"full_name":"Matrix.diagonal","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[356,4],"def_end_pos":[356,12]},{"full_name":"Matrix.diagonal_mul_diagonal","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[933,8],"def_end_pos":[933,29]},{"full_name":"RCLike.ofReal_pow","def_path":"Mathlib/Analysis/RCLike/Basic.lean","def_pos":[196,8],"def_end_pos":[196,18]},{"full_name":"Real.sq_sqrt","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[170,8],"def_end_pos":[170,15]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]}]},{"state_before":"m : Type u_1\nn : Type u_2\nR : Type u_3\n𝕜 : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : RCLike 𝕜\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nC : Matrix n n 𝕜 := ↑⋯.eigenvectorUnitary\nE : Matrix n n 𝕜 := diagonal (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues)\nthis : E * E = diagonal (RCLike.ofReal ∘ ⋯.eigenvalues)\n⊢ C * (E * (star C * C) * E) * star C = A","state_after":"no goals","tactic":"simpa [C, this] using hA.1.spectral_theorem.symm","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Matrix.IsHermitian.spectral_theorem","def_path":"Mathlib/LinearAlgebra/Matrix/Spectrum.lean","def_pos":[116,8],"def_end_pos":[116,24]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/DominatedConvergence.lean","commit":"","full_name":"intervalIntegral.continuousOn_primitive","start":[449,0],"end":[462,30],"file_path":"Mathlib/MeasureTheory/Integral/DominatedConvergence.lean","tactics":[{"state_before":"E : Type u_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : TopologicalSpace X\na b b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f (Icc a b) μ\n⊢ ContinuousOn (fun x => ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)","state_after":"case pos\nE : Type u_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : TopologicalSpace X\na b b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f (Icc a b) μ\nh : a ≤ b\n⊢ ContinuousOn (fun x => ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)\n\ncase neg\nE : Type u_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : TopologicalSpace X\na b b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntegrableOn f (Icc a b) μ\nh : ¬a ≤ b\n⊢ ContinuousOn (fun x => ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)","tactic":"by_cases h : a ≤ b","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.GoodProducts.spanFin","start":[599,0],"end":[650,48],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => x ∈ s)))","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s)))","tactic":"rw [span_iff_products]","premises":[{"full_name":"Profinite.NobelingProof.GoodProducts.span_iff_products","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[430,8],"def_end_pos":[430,38]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s)))","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ Submodule.span ℤ (Set.range (spanFinBasis C s)) ≤ Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s)))","tactic":"refine le_trans (spanFinBasis.span C s) ?_","premises":[{"full_name":"Profinite.NobelingProof.spanFinBasis.span","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[506,8],"def_end_pos":[506,25]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ Submodule.span ℤ (Set.range (spanFinBasis C s)) ≤ Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s)))","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ Set.range (spanFinBasis C s) ⊆ ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","tactic":"rw [Submodule.span_le]","premises":[{"full_name":"Submodule.span_le","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[74,8],"def_end_pos":[74,15]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\n⊢ Set.range (spanFinBasis C s) ⊆ ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","state_after":"case intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx : ↑(π C fun x => x ∈ s)\n⊢ spanFinBasis C s x ∈ ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","tactic":"rintro _ ⟨x, rfl⟩","premises":[]},{"state_before":"case intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx : ↑(π C fun x => x ∈ s)\n⊢ spanFinBasis C s x ∈ ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","state_after":"case intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx : ↑(π C fun x => x ∈ s)\n⊢ (factors C s x).prod ∈ ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","tactic":"rw [← factors_prod_eq_basis]","premises":[{"full_name":"Profinite.NobelingProof.factors_prod_eq_basis","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[574,8],"def_end_pos":[574,29]}]},{"state_before":"case intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx : ↑(π C fun x => x ∈ s)\nl : List I := Finset.sort (fun x x_1 => x ≥ x_1) s\n⊢ (factors C s x).prod ∈ ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","state_after":"case intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\ns : Finset I\nx : ↑(π C fun x => x ∈ s)\nl : List I := Finset.sort (fun x x_1 => x ≥ x_1) s\n⊢ (List.map (fun i => if ↑x i = true then e (π C fun x => x ∈ s) i else 1 - e (π C fun x => x ∈ s) i)\n        (Finset.sort (fun x x_1 => x ≥ x_1) s)).prod ∈\n    ↑(Submodule.span ℤ (Set.range (Products.eval (π C fun x => x ∈ s))))","tactic":"dsimp [factors]","premises":[{"full_name":"Profinite.NobelingProof.factors","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[521,4],"def_end_pos":[521,11]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.sum_sdiff","start":[509,0],"end":[512,59],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\nh : s₁ ⊆ s₂\n⊢ (∏ x ∈ s₂ \\ s₁, f x) * ∏ x ∈ s₁, f x = ∏ x ∈ s₂, f x","state_after":"no goals","tactic":"rw [← prod_union sdiff_disjoint, sdiff_union_of_subset h]","premises":[{"full_name":"Finset.prod_union","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[411,8],"def_end_pos":[411,18]},{"full_name":"Finset.sdiff_disjoint","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2016,8],"def_end_pos":[2016,22]},{"full_name":"Finset.sdiff_union_of_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1777,8],"def_end_pos":[1777,29]}]}]}
{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.union_iInter₂","start":[827,0],"end":[828,78],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ns : Set α\nt : (i : ι) → κ i → Set α\n⊢ s ∪ ⋂ i, ⋂ j, t i j = ⋂ i, ⋂ j, s ∪ t i j","state_after":"no goals","tactic":"simp_rw [union_iInter]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.union_iInter","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[441,8],"def_end_pos":[441,20]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","commit":"","full_name":"EuclideanSpace.inner_single_right","start":[246,0],"end":[247,95],"file_path":"Mathlib/Analysis/InnerProductSpace/PiL2.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : _root_.RCLike 𝕜\nE : Type u_4\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : InnerProductSpace 𝕜 E\nE' : Type u_5\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : InnerProductSpace 𝕜 E'\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\nF' : Type u_7\ninst✝³ : NormedAddCommGroup F'\ninst✝² : InnerProductSpace ℝ F'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ni : ι\na : 𝕜\nv : EuclideanSpace 𝕜 ι\n⊢ ⟪v, single i a⟫_𝕜 = a * (starRingEnd ((fun x => 𝕜) i)) (v i)","state_after":"no goals","tactic":"simp [apply_ite conj, mul_comm]","premises":[{"full_name":"apply_ite","def_path":".lake/packages/lean4/src/lean/Init/ByCases.lean","def_pos":[36,8],"def_end_pos":[36,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]},{"full_name":"starRingEnd","def_path":"Mathlib/Algebra/Star/Basic.lean","def_pos":[321,4],"def_end_pos":[321,15]}]}]}
{"url":"Mathlib/Tactic/NormNum/DivMod.lean","commit":"","full_name":"Mathlib.Meta.NormNum.isInt_ediv_zero","start":[23,0],"end":[24,54],"file_path":"Mathlib/Tactic/NormNum/DivMod.lean","tactics":[{"state_before":"n✝ : ℤ\n⊢ ↑n✝ / ↑0 = ↑0","state_after":"no goals","tactic":"simp [Int.ediv_zero]","premises":[{"full_name":"Int.ediv_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[139,26],"def_end_pos":[139,35]}]}]}
{"url":"Mathlib/RingTheory/Noetherian.lean","commit":"","full_name":"LinearIndependent.finite_of_isNoetherian","start":[368,0],"end":[377,41],"file_path":"Mathlib/RingTheory/Noetherian.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\n⊢ Finite ι","state_after":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\n⊢ Finite ι","tactic":"have hwf := isNoetherian_iff_wellFounded.mp (by infer_instance : IsNoetherian R M)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]},{"full_name":"isNoetherian_iff_wellFounded","def_path":"Mathlib/RingTheory/Noetherian.lean","def_pos":[291,8],"def_end_pos":[291,36]}]},{"state_before":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\n⊢ Finite ι","state_after":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\ni : ι\ncontra : span R {v i} = ⊥\n⊢ False","tactic":"refine CompleteLattice.WellFounded.finite_of_independent hwf hv.independent_span_singleton\n    fun i contra => ?_","premises":[{"full_name":"CompleteLattice.WellFounded.finite_of_independent","def_path":"Mathlib/Order/CompactlyGenerated/Basic.lean","def_pos":[304,8],"def_end_pos":[304,41]},{"full_name":"LinearIndependent.independent_span_singleton","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[899,8],"def_end_pos":[899,52]}]},{"state_before":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\ni : ι\ncontra : span R {v i} = ⊥\n⊢ False","state_after":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\ni : ι\ncontra : span R {v i} = ⊥\n⊢ v i = 0","tactic":"apply hv.ne_zero i","premises":[{"full_name":"LinearIndependent.ne_zero","def_path":"Mathlib/LinearAlgebra/LinearIndependent.lean","def_pos":[191,8],"def_end_pos":[191,33]}]},{"state_before":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\ni : ι\ncontra : span R {v i} = ⊥\n⊢ v i = 0","state_after":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\ni : ι\ncontra : span R {v i} = ⊥\nthis : v i ∈ span R {v i}\n⊢ v i = 0","tactic":"have : v i ∈ R ∙ v i := Submodule.mem_span_singleton_self (v i)","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Submodule.mem_span_singleton_self","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[418,8],"def_end_pos":[418,31]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]}]},{"state_before":"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_4\nv : ι → M\nhv : LinearIndependent R v\nhwf : WellFounded fun x x_1 => x > x_1\ni : ι\ncontra : span R {v i} = ⊥\nthis : v i ∈ span R {v i}\n⊢ v i = 0","state_after":"no goals","tactic":"rwa [contra, Submodule.mem_bot] at this","premises":[{"full_name":"Submodule.mem_bot","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[68,8],"def_end_pos":[68,15]}]}]}
{"url":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","commit":"","full_name":"finrank_top","start":[127,0],"end":[130,17],"file_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","tactics":[{"state_before":"R : Type u\nM : Type v\nN : Type w\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\n⊢ finrank R ↥⊤ = finrank R M","state_after":"R : Type u\nM : Type v\nN : Type w\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\n⊢ toNat (Module.rank R ↥⊤) = toNat (Module.rank R M)","tactic":"unfold finrank","premises":[{"full_name":"FiniteDimensional.finrank","def_path":"Mathlib/LinearAlgebra/Dimension/Finrank.lean","def_pos":[52,18],"def_end_pos":[52,25]}]},{"state_before":"R : Type u\nM : Type v\nN : Type w\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\n⊢ toNat (Module.rank R ↥⊤) = toNat (Module.rank R M)","state_after":"no goals","tactic":"simp [rank_top]","premises":[{"full_name":"rank_top","def_path":"Mathlib/LinearAlgebra/Dimension/Basic.lean","def_pos":[312,8],"def_end_pos":[312,16]}]}]}
{"url":"Mathlib/Dynamics/OmegaLimit.lean","commit":"","full_name":"mem_omegaLimit_iff_frequently₂","start":[126,0],"end":[131,67],"file_path":"Mathlib/Dynamics/OmegaLimit.lean","tactics":[{"state_before":"τ : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns s₁ s₂ : Set α\ny : β\n⊢ y ∈ ω f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ (t : τ) in f, (ϕ t '' s ∩ n).Nonempty","state_after":"no goals","tactic":"simp_rw [mem_omegaLimit_iff_frequently, image_inter_nonempty_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.image_inter_nonempty_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[450,8],"def_end_pos":[450,32]},{"full_name":"mem_omegaLimit_iff_frequently","def_path":"Mathlib/Dynamics/OmegaLimit.lean","def_pos":[115,8],"def_end_pos":[115,37]}]}]}
{"url":"Mathlib/Algebra/Order/Floor/Div.lean","commit":"","full_name":"smul_floorDiv","start":[140,0],"end":[142,66],"file_path":"Mathlib/Algebra/Order/Floor/Div.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring α\ninst✝⁴ : OrderedAddCommMonoid β\ninst✝³ : MulActionWithZero α β\ninst✝² : FloorDiv α β\na : α\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : 0 < a\nb : β\n⊢ ∀ (c : β), c ≤ a • b ⌊/⌋ a ↔ c ≤ b","state_after":"no goals","tactic":"simp [smul_le_smul_iff_of_pos_left, ha]","premises":[{"full_name":"smul_le_smul_iff_of_pos_left","def_path":"Mathlib/Algebra/Order/Module/Defs.lean","def_pos":[288,6],"def_end_pos":[288,34]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Roots.lean","commit":"","full_name":"Polynomial.aroots_X_sub_C","start":[373,0],"end":[376,66],"file_path":"Mathlib/Algebra/Polynomial/Roots.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\np q : R[X]\ninst✝³ : CommRing T\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra T S\nr : T\n⊢ (X - C r).aroots S = {(algebraMap T S) r}","state_after":"no goals","tactic":"rw [aroots_def, Polynomial.map_sub, map_X, map_C, roots_X_sub_C]","premises":[{"full_name":"Polynomial.aroots_def","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[352,8],"def_end_pos":[352,18]},{"full_name":"Polynomial.map_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[601,8],"def_end_pos":[601,13]},{"full_name":"Polynomial.map_X","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[605,8],"def_end_pos":[605,13]},{"full_name":"Polynomial.map_sub","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[1085,18],"def_end_pos":[1085,25]},{"full_name":"Polynomial.roots_X_sub_C","def_path":"Mathlib/Algebra/Polynomial/Roots.lean","def_pos":[156,8],"def_end_pos":[156,21]}]}]}
{"url":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","commit":"","full_name":"Measurable.map_measurableSpace_eq_borel","start":[542,0],"end":[546,38],"file_path":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","tactics":[{"state_before":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","state_after":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","tactic":"have d := hf.mono le_rfl OpensMeasurableSpace.borel_le","premises":[{"full_name":"Measurable.mono","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Basic.lean","def_pos":[189,8],"def_end_pos":[189,23]},{"full_name":"OpensMeasurableSpace.borel_le","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[115,2],"def_end_pos":[115,10]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","state_after":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\nthis : MeasurableSpace Y := borel Y\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","tactic":"letI := borel Y","premises":[{"full_name":"borel","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[49,4],"def_end_pos":[49,9]}]},{"state_before":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\nthis : MeasurableSpace Y := borel Y\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","state_after":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\nthis✝ : MeasurableSpace Y := borel Y\nthis : BorelSpace Y\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","tactic":"haveI : BorelSpace Y := ⟨rfl⟩","premises":[{"full_name":"BorelSpace","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean","def_pos":[119,6],"def_end_pos":[119,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u_1\nι : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\nβ : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : StandardBorelSpace X\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : T0Space Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace Z\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\nthis✝ : MeasurableSpace Y := borel Y\nthis : BorelSpace Y\n⊢ MeasurableSpace.map f inst✝⁸ = borel Y","state_after":"no goals","tactic":"exact d.map_measurableSpace_eq hsurj","premises":[{"full_name":"Measurable.map_measurableSpace_eq","def_path":"Mathlib/MeasureTheory/Constructions/Polish/Basic.lean","def_pos":[537,8],"def_end_pos":[537,30]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/NormPow.lean","commit":"","full_name":"norm_fderiv_norm_rpow_le","start":[76,0],"end":[86,9],"file_path":"Mathlib/Analysis/InnerProductSpace/NormPow.lean","tactics":[{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖fderiv ℝ (fun x => ‖f x‖ ^ p) x‖ ≤ p * ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","state_after":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖p‖ * ‖‖f x‖ ^ (p - 2)‖ * ‖((innerSL ℝ) (f x)).comp (fderiv ℝ f x)‖ ≤ p * ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","tactic":"rw [hf.fderiv_norm_rpow hp, norm_smul, norm_mul]","premises":[{"full_name":"Differentiable.fderiv_norm_rpow","def_path":"Mathlib/Analysis/InnerProductSpace/NormPow.lean","def_pos":[70,8],"def_end_pos":[70,39]},{"full_name":"norm_mul","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[661,8],"def_end_pos":[661,16]},{"full_name":"norm_smul","def_path":"Mathlib/Analysis/Normed/MulAction.lean","def_pos":[79,8],"def_end_pos":[79,17]}]},{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖p‖ * ‖‖f x‖ ^ (p - 2)‖ * ‖((innerSL ℝ) (f x)).comp (fderiv ℝ f x)‖ ≤ p * ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","state_after":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ p * (‖f x‖ ^ (p - 2) * ‖((innerSL ℝ) (f x)).comp (fderiv ℝ f x)‖) ≤ p * (‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖)","tactic":"simp_rw [norm_rpow_of_nonneg (norm_nonneg _), norm_norm, norm_eq_abs,\n    abs_eq_self.mpr <| zero_le_one.trans hp.le, mul_assoc]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real.norm_eq_abs","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1132,8],"def_end_pos":[1132,19]},{"full_name":"Real.norm_rpow_of_nonneg","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[166,8],"def_end_pos":[166,27]},{"full_name":"abs_eq_self","def_path":"Mathlib/Algebra/Order/Group/Abs.lean","def_pos":[196,8],"def_end_pos":[196,19]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]},{"full_name":"norm_norm","def_path":"Mathlib/Analysis/Normed/Field/Basic.lean","def_pos":[1028,8],"def_end_pos":[1028,17]},{"full_name":"zero_le_one","def_path":"Mathlib/Algebra/Order/ZeroLEOne.lean","def_pos":[23,14],"def_end_pos":[23,25]}]},{"state_before":"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ p * (‖f x‖ ^ (p - 2) * ‖((innerSL ℝ) (f x)).comp (fderiv ℝ f x)‖) ≤ p * (‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖)","state_after":"case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2) * ‖((innerSL ℝ) (f x)).comp (fderiv ℝ f x)‖ ≤ ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","tactic":"gcongr _ * ?_","premises":[]},{"state_before":"case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2) * ‖((innerSL ℝ) (f x)).comp (fderiv ℝ f x)‖ ≤ ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","state_after":"case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2) * (‖(innerSL ℝ) (f x)‖ * ‖fderiv ℝ f x‖) = ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","tactic":"refine mul_le_mul_of_nonneg_left (ContinuousLinearMap.opNorm_comp_le ..) (by positivity)\n    |>.trans_eq ?_","premises":[{"full_name":"ContinuousLinearMap.opNorm_comp_le","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean","def_pos":[349,8],"def_end_pos":[349,22]},{"full_name":"mul_le_mul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[190,8],"def_end_pos":[190,33]}]},{"state_before":"case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2) * (‖(innerSL ℝ) (f x)‖ * ‖fderiv ℝ f x‖) = ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","state_after":"case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2 + 1) * ‖fderiv ℝ f x‖ = ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","tactic":"rw [innerSL_apply_norm, ← mul_assoc, ← Real.rpow_add_one' (by positivity) (by linarith)]","premises":[{"full_name":"Real.rpow_add_one'","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","def_pos":[404,6],"def_end_pos":[404,19]},{"full_name":"innerSL_apply_norm","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1565,8],"def_end_pos":[1565,26]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2 + 1) * ‖fderiv ℝ f x‖ = ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖","state_after":"no goals","tactic":"ring_nf","premises":[]}]}
{"url":"Mathlib/RingTheory/Ideal/Quotient.lean","commit":"","full_name":"Ideal.Quotient.exists_inv","start":[174,0],"end":[182,59],"file_path":"Mathlib/RingTheory/Ideal/Quotient.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\n⊢ ∀ {a : R ⧸ I}, a ≠ 0 → ∃ b, a * b = 1","state_after":"case mk\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\n⊢ ∃ b, Quot.mk Setoid.r a * b = 1","tactic":"rintro ⟨a⟩ h","premises":[]},{"state_before":"case mk\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\n⊢ ∃ b, Quot.mk Setoid.r a * b = 1","state_after":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : b * a + c = 1\n⊢ ∃ b, Quot.mk Setoid.r a * b = 1","tactic":"rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩","premises":[{"full_name":"Ideal.IsMaximal.exists_inv","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[335,8],"def_end_pos":[335,28]},{"full_name":"Ideal.Quotient.eq_zero_iff_mem","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[112,8],"def_end_pos":[112,23]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]},{"state_before":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : b * a + c = 1\n⊢ ∃ b, Quot.mk Setoid.r a * b = 1","state_after":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : a * b + c = 1\n⊢ ∃ b, Quot.mk Setoid.r a * b = 1","tactic":"rw [mul_comm] at abc","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : a * b + c = 1\n⊢ ∃ b, Quot.mk Setoid.r a * b = 1","state_after":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : a * b + c = 1\n⊢ Setoid.r ((fun x x_1 => x * x_1) a b) 1","tactic":"refine ⟨mk _ b, Quot.sound ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Quot.sound","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1471,6],"def_end_pos":[1471,11]}]},{"state_before":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : a * b + c = 1\n⊢ Setoid.r ((fun x x_1 => x * x_1) a b) 1","state_after":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : a * b + c = 1\n⊢ a * b - 1 ∈ I","tactic":"simp only [Submodule.quotientRel_r_def]","premises":[{"full_name":"Submodule.quotientRel_r_def","def_path":"Mathlib/LinearAlgebra/Quotient.lean","def_pos":[35,8],"def_end_pos":[35,25]}]},{"state_before":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : a * b + c = 1\n⊢ a * b - 1 ∈ I","state_after":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : c = 1 - a * b\n⊢ a * b - 1 ∈ I","tactic":"rw [← eq_sub_iff_add_eq'] at abc","premises":[{"full_name":"eq_sub_iff_add_eq'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[917,14],"def_end_pos":[917,32]}]},{"state_before":"case mk.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na✝¹ b✝ : R\nS : Type v\nx y : R\nI : Ideal R\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk Setoid.r a ≠ 0\nb c : R\nhc : c ∈ I\nabc : c = 1 - a * b\n⊢ a * b - 1 ∈ I","state_after":"no goals","tactic":"rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc","premises":[{"full_name":"neg_mem_iff","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[121,2],"def_end_pos":[121,13]},{"full_name":"neg_sub","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[399,2],"def_end_pos":[399,13]}]}]}
{"url":"Mathlib/Algebra/Star/StarAlgHom.lean","commit":"","full_name":"NonUnitalStarAlgHom.prodEquiv_apply","start":[537,0],"end":[544,31],"file_path":"Mathlib/Algebra/Star/StarAlgHom.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : Monoid R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : DistribMulAction R A\ninst✝⁶ : Star A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : DistribMulAction R B\ninst✝³ : Star B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : DistribMulAction R C\ninst✝ : Star C\nf : (A →⋆ₙₐ[R] B) × (A →⋆ₙₐ[R] C)\n⊢ (fun f => ((fst R B C).comp f, (snd R B C).comp f)) ((fun f => f.1.prod f.2) f) = f","state_after":"no goals","tactic":"ext <;> rfl","premises":[]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : Monoid R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : DistribMulAction R A\ninst✝⁶ : Star A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : DistribMulAction R B\ninst✝³ : Star B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : DistribMulAction R C\ninst✝ : Star C\nf : A →⋆ₙₐ[R] B × C\n⊢ (fun f => f.1.prod f.2) ((fun f => ((fst R B C).comp f, (snd R B C).comp f)) f) = f","state_after":"no goals","tactic":"ext <;> rfl","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","commit":"","full_name":"dist_vadd_left","start":[100,0],"end":[103,41],"file_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","tactics":[{"state_before":"α : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv : V\nx : P\n⊢ dist (v +ᵥ x) x = ‖v‖","state_after":"no goals","tactic":"rw [dist_eq_norm_vsub V _ x, vadd_vsub]","premises":[{"full_name":"dist_eq_norm_vsub","def_path":"Mathlib/Analysis/Normed/Group/AddTorsor.lean","def_pos":[68,8],"def_end_pos":[68,25]},{"full_name":"vadd_vsub","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[86,8],"def_end_pos":[86,17]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/List.lean","commit":"","full_name":"List.prod_range_div'","start":[422,0],"end":[428,99],"file_path":"Mathlib/Algebra/BigOperators/Group/List.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : Group G\nn : ℕ\nf : ℕ → G\n⊢ (map (fun k => f k / f (k + 1)) (range n)).prod = f 0 / f n","state_after":"case zero\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : Group G\nf : ℕ → G\n⊢ (map (fun k => f k / f (k + 1)) (range 0)).prod = f 0 / f 0\n\ncase succ\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : Group G\nf : ℕ → G\nn : ℕ\nh : (map (fun k => f k / f (k + 1)) (range n)).prod = f 0 / f n\n⊢ (map (fun k => f k / f (k + 1)) (range (n + 1))).prod = f 0 / f (n + 1)","tactic":"induction' n with n h","premises":[]}]}
{"url":"Mathlib/RingTheory/Ideal/Basic.lean","commit":"","full_name":"Ideal.span_eq_top_iff_finite","start":[182,0],"end":[185,84],"file_path":"Mathlib/RingTheory/Ideal/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\ns : Set α\n⊢ span s = ⊤ ↔ ∃ s', ↑s' ⊆ s ∧ span ↑s' = ⊤","state_after":"α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\ns : Set α\n⊢ 1 ∈ span s ↔ ∃ s', ↑s' ⊆ s ∧ 1 ∈ span ↑s'","tactic":"simp_rw [eq_top_iff_one]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Ideal.eq_top_iff_one","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[87,8],"def_end_pos":[87,22]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]}]},{"state_before":"α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\ns : Set α\n⊢ 1 ∈ span s ↔ ∃ s', ↑s' ⊆ s ∧ 1 ∈ span ↑s'","state_after":"no goals","tactic":"exact ⟨Submodule.mem_span_finite_of_mem_span, fun ⟨s', h₁, h₂⟩ => span_mono h₁ h₂⟩","premises":[{"full_name":"Ideal.span_mono","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[131,8],"def_end_pos":[131,17]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Submodule.mem_span_finite_of_mem_span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[700,8],"def_end_pos":[700,35]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Sieves.lean","commit":"","full_name":"CategoryTheory.Sieve.bind_apply","start":[367,0],"end":[375,41],"file_path":"Mathlib/CategoryTheory/Sites/Sieves.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ : Sieve X\nS : Presieve X\nR : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → Sieve Y\n⊢ ∀ {Y Z : C} {f : Y ⟶ X},\n    S.bind (fun Y f h => (R h).arrows) f → ∀ (g : Z ⟶ Y), S.bind (fun Y f h => (R h).arrows) (g ≫ f)","state_after":"case intro.intro.intro.intro.intro\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf✝ : Y✝ ⟶ X\nS✝ R✝ : Sieve X\nS : Presieve X\nR : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → Sieve Y\nY Z W : C\nf : Y ⟶ W\nh : W ⟶ X\nhh : S h\nhf : (R hh).arrows f\ng : Z ⟶ Y\n⊢ S.bind (fun Y f h => (R h).arrows) (g ≫ f ≫ h)","tactic":"rintro Y Z f ⟨W, f, h, hh, hf, rfl⟩ g","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y✝ Z✝ : C\nf✝ : Y✝ ⟶ X\nS✝ R✝ : Sieve X\nS : Presieve X\nR : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → Sieve Y\nY Z W : C\nf : Y ⟶ W\nh : W ⟶ X\nhh : S h\nhf : (R hh).arrows f\ng : Z ⟶ Y\n⊢ S.bind (fun Y f h => (R h).arrows) (g ≫ f ≫ h)","state_after":"no goals","tactic":"exact ⟨_, g ≫ f, _, hh, by simp [hf]⟩","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/Tactic/NormNum/BigOperators.lean","commit":"","full_name":"Mathlib.Meta.List.range_zero'","start":[98,0],"end":[99,71],"file_path":"Mathlib/Tactic/NormNum/BigOperators.lean","tactics":[{"state_before":"u v : Level\nn : ℕ\npn : NormNum.IsNat n 0\n⊢ List.range n = []","state_after":"no goals","tactic":"rw [pn.out, Nat.cast_zero, List.range_zero]","premises":[{"full_name":"List.range_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[1162,16],"def_end_pos":[1162,26]},{"full_name":"Mathlib.Meta.NormNum.IsNat.out","def_path":"Mathlib/Tactic/NormNum/Result.lean","def_pos":[127,2],"def_end_pos":[127,5]},{"full_name":"Nat.cast_zero","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[110,8],"def_end_pos":[110,17]}]}]}
{"url":"Mathlib/Tactic/NormNum/LegendreSymbol.lean","commit":"","full_name":"Mathlib.Meta.NormNum.jacobiSymNat.even_odd₁","start":[126,0],"end":[133,41],"file_path":"Mathlib/Tactic/NormNum/LegendreSymbol.lean","tactics":[{"state_before":"a b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ jacobiSymNat a b = r","state_after":"a b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ jacobiSym (↑a) b = jacobiSym (↑a / 2) b","tactic":"simp only [jacobiSymNat, ← hr, ← hc, Int.ofNat_ediv, Nat.cast_ofNat]","premises":[{"full_name":"Int.ofNat_ediv","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivMod.lean","def_pos":[161,27],"def_end_pos":[161,37]},{"full_name":"Mathlib.Meta.NormNum.jacobiSymNat","def_path":"Mathlib/Tactic/NormNum/LegendreSymbol.lean","def_pos":[53,4],"def_end_pos":[53,16]},{"full_name":"Nat.cast_ofNat","def_path":"Mathlib/Data/Nat/Cast/Defs.lean","def_pos":[68,27],"def_end_pos":[68,41]}]},{"state_before":"a b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ jacobiSym (↑a) b = jacobiSym (↑a / 2) b","state_after":"a b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ b % 2 = 1","tactic":"rw [← jacobiSym.even_odd (mod_cast ha), if_neg (by simp [hb])]","premises":[{"full_name":"if_neg","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[937,8],"def_end_pos":[937,14]},{"full_name":"jacobiSym.even_odd","def_path":"Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean","def_pos":[342,8],"def_end_pos":[342,16]}]},{"state_before":"a b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ b % 2 = 1","state_after":"case h\na b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ 2 ∣ 8","tactic":"rw [← Nat.mod_mod_of_dvd, hb]","premises":[{"full_name":"Nat.mod_mod_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[94,16],"def_end_pos":[94,30]}]},{"state_before":"case h\na b c : ℕ\nr : ℤ\nha : a % 2 = 0\nhb : b % 8 = 1\nhc : a / 2 = c\nhr : jacobiSymNat c b = r\n⊢ 2 ∣ 8","state_after":"no goals","tactic":"norm_num","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.monomial_sub","start":[1023,0],"end":[1025,4],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\n⊢ (monomial n) (a - b) = (monomial n) a - (monomial n) b","state_after":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\n⊢ (monomial n) a + -(monomial n) b = (monomial n) a - (monomial n) b","tactic":"rw [sub_eq_add_neg, monomial_add, monomial_neg]","premises":[{"full_name":"Polynomial.monomial_add","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[395,8],"def_end_pos":[395,20]},{"full_name":"Polynomial.monomial_neg","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[1020,8],"def_end_pos":[1020,20]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]},{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\n⊢ (monomial n) a + -(monomial n) b = (monomial n) a - (monomial n) b","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","commit":"","full_name":"Set.Finite.summable","start":[111,0],"end":[115,31],"file_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf✝ g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhs : s.Finite\nf : β → α\n⊢ Multipliable (f ∘ Subtype.val)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf✝ g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhs : s.Finite\nf : β → α\nthis : Multipliable (f ∘ Subtype.val)\n⊢ Multipliable (f ∘ Subtype.val)","tactic":"have := hs.toFinset.multipliable f","premises":[{"full_name":"Finset.multipliable","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Basic.lean","def_pos":[107,18],"def_end_pos":[107,37]},{"full_name":"Set.Finite.toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[91,28],"def_end_pos":[91,43]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf✝ g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhs : s.Finite\nf : β → α\nthis : Multipliable (f ∘ Subtype.val)\n⊢ Multipliable (f ∘ Subtype.val)","state_after":"no goals","tactic":"rwa [hs.coe_toFinset] at this","premises":[{"full_name":"Set.Finite.coe_toFinset","def_path":"Mathlib/Data/Set/Finite.lean","def_pos":[150,18],"def_end_pos":[150,30]}]}]}
{"url":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","commit":"","full_name":"CategoryTheory.NonPreadditiveAbelian.add_assoc","start":[381,0],"end":[384,44],"file_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a + b + c = a + (b + c)","state_after":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - -b + c = a + (b + c)","tactic":"conv_lhs =>\n    congr; rw [add_def]","premises":[{"full_name":"CategoryTheory.NonPreadditiveAbelian.add_def","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[322,8],"def_end_pos":[322,15]}]},{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - -b + c = a + (b + c)","state_after":"no goals","tactic":"rw [sub_add, ← add_neg, neg_sub', neg_neg]","premises":[{"full_name":"CategoryTheory.NonPreadditiveAbelian.add_neg","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[365,8],"def_end_pos":[365,15]},{"full_name":"CategoryTheory.NonPreadditiveAbelian.neg_neg","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[349,8],"def_end_pos":[349,15]},{"full_name":"CategoryTheory.NonPreadditiveAbelian.neg_sub'","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[371,8],"def_end_pos":[371,16]},{"full_name":"CategoryTheory.NonPreadditiveAbelian.sub_add","def_path":"Mathlib/CategoryTheory/Abelian/NonPreadditive.lean","def_pos":[378,8],"def_end_pos":[378,15]}]}]}
{"url":"Mathlib/Data/Nat/Cast/Basic.lean","commit":"","full_name":"NeZero.nat_of_neZero","start":[175,0],"end":[178,61],"file_path":"Mathlib/Data/Nat/Cast/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nR✝ : Type u_3\nS✝ : Type u_4\nF✝ : Type u_5\ninst✝⁵ : NonAssocSemiring R✝\ninst✝⁴ : NonAssocSemiring S✝\nR : Type u_6\nS : Type u_7\ninst✝³ : Semiring R\ninst✝² : Semiring S\nF : Type u_8\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nn : ℕ\nhn : NeZero ↑n\n⊢ NeZero (f ↑n)","state_after":"no goals","tactic":"simp only [map_natCast, hn]","premises":[{"full_name":"map_natCast","def_path":"Mathlib/Data/Nat/Cast/Basic.lean","def_pos":[163,8],"def_end_pos":[163,19]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Fin.lean","commit":"","full_name":"Fin.partialSum_right_neg","start":[210,0],"end":[223,67],"file_path":"Mathlib/Algebra/BigOperators/Fin.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : Fin n\n⊢ (partialProd f i.castSucc)⁻¹ * partialProd f i.succ = f i","state_after":"case mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : i < n\n⊢ (partialProd f ⟨i, hn⟩.castSucc)⁻¹ * partialProd f ⟨i, hn⟩.succ = f ⟨i, hn⟩","tactic":"cases' i with i hn","premises":[]},{"state_before":"case mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : i < n\n⊢ (partialProd f ⟨i, hn⟩.castSucc)⁻¹ * partialProd f ⟨i, hn⟩.succ = f ⟨i, hn⟩","state_after":"no goals","tactic":"induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn), ← Fin.succ_mk _ _ hn]\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc, mul_left_eq_self, mul_assoc, hi, mul_left_inv]","premises":[{"full_name":"Fin.castSucc_mk","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[372,16],"def_end_pos":[372,27]},{"full_name":"Fin.coe_eq_castSucc","def_path":"Mathlib/Data/Fin/Basic.lean","def_pos":[726,8],"def_end_pos":[726,23]},{"full_name":"Fin.partialProd_succ","def_path":"Mathlib/Algebra/BigOperators/Fin.lean","def_pos":[193,8],"def_end_pos":[193,24]},{"full_name":"Fin.succ_mk","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[258,16],"def_end_pos":[258,23]},{"full_name":"Nat.lt_succ_self","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[358,16],"def_end_pos":[358,28]},{"full_name":"lt_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[74,8],"def_end_pos":[74,16]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_inv_rev","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[982,8],"def_end_pos":[982,19]},{"full_name":"mul_left_eq_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[254,8],"def_end_pos":[254,24]},{"full_name":"mul_left_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1039,8],"def_end_pos":[1039,20]}]}]}
{"url":"Mathlib/Order/Lattice.lean","commit":"","full_name":"le_iff_exists_sup","start":[170,0],"end":[175,21],"file_path":"Mathlib/Order/Lattice.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na b c d : α\n⊢ a ≤ b ↔ ∃ c, b = a ⊔ c","state_after":"case mp\nα : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na b c d : α\n⊢ a ≤ b → ∃ c, b = a ⊔ c\n\ncase mpr\nα : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na b c d : α\n⊢ (∃ c, b = a ⊔ c) → a ≤ b","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Algebra/GroupWithZero/Basic.lean","commit":"","full_name":"right_eq_mul₀","start":[221,0],"end":[222,91],"file_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","tactics":[{"state_before":"α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝ : CancelMonoidWithZero M₀\na b c : M₀\nhb : b ≠ 0\n⊢ b = a * b ↔ a = 1","state_after":"no goals","tactic":"rw [eq_comm, mul_eq_right₀ hb]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]},{"full_name":"mul_eq_right₀","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[215,8],"def_end_pos":[215,21]}]}]}
{"url":"Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean","commit":"","full_name":"ModuleCat.Free.μ_natural","start":[81,0],"end":[101,63],"file_path":"Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ ((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom = (μ R X X').hom ≫ (free R).map (f ⊗ g)","state_after":"case H\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n      (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) =\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g))","tactic":"apply TensorProduct.ext","premises":[{"full_name":"TensorProduct.ext","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[545,8],"def_end_pos":[545,11]}]},{"state_before":"case H\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n      (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) =\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g))","state_after":"case H.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ ∀ (a : X),\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n          (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n        Finsupp.lsingle a =\n      (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n        Finsupp.lsingle a","tactic":"apply Finsupp.lhom_ext'","premises":[{"full_name":"Finsupp.lhom_ext'","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[150,8],"def_end_pos":[150,17]}]},{"state_before":"case H.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ ∀ (a : X),\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n          (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n        Finsupp.lsingle a =\n      (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n        Finsupp.lsingle a","state_after":"case H.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n        (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n      Finsupp.lsingle x =\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n      Finsupp.lsingle x","tactic":"intro x","premises":[]},{"state_before":"case H.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n        (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n      Finsupp.lsingle x =\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n      Finsupp.lsingle x","state_after":"case H.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n          (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n        Finsupp.lsingle x)\n      1 =\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n        Finsupp.lsingle x)\n      1","tactic":"apply LinearMap.ext_ring","premises":[{"full_name":"LinearMap.ext_ring","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[456,8],"def_end_pos":[456,16]}]},{"state_before":"case H.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n          (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n        Finsupp.lsingle x)\n      1 =\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n        Finsupp.lsingle x)\n      1","state_after":"case H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ ∀ (a : X'),\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n              (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle a =\n      ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle a","tactic":"apply Finsupp.lhom_ext'","premises":[{"full_name":"Finsupp.lhom_ext'","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[150,8],"def_end_pos":[150,17]}]},{"state_before":"case H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ ∀ (a : X'),\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n              (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle a =\n      ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle a","state_after":"case H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n            (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n          Finsupp.lsingle x)\n        1 ∘ₗ\n      Finsupp.lsingle x' =\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n          Finsupp.lsingle x)\n        1 ∘ₗ\n      Finsupp.lsingle x'","tactic":"intro x'","premises":[]},{"state_before":"case H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n            (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n          Finsupp.lsingle x)\n        1 ∘ₗ\n      Finsupp.lsingle x' =\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n          Finsupp.lsingle x)\n        1 ∘ₗ\n      Finsupp.lsingle x'","state_after":"case H.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n              (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle x')\n      1 =\n    (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle x')\n      1","tactic":"apply LinearMap.ext_ring","premises":[{"full_name":"LinearMap.ext_ring","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[456,8],"def_end_pos":[456,16]}]},{"state_before":"case H.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n              (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle x')\n      1 =\n    (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle x')\n      1","state_after":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ ∀ (a : Y ⊗ Y'),\n    ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n                  (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n                Finsupp.lsingle x)\n              1 ∘ₗ\n            Finsupp.lsingle x')\n          1)\n        a =\n      ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n                Finsupp.lsingle x)\n              1 ∘ₗ\n            Finsupp.lsingle x')\n          1)\n        a","tactic":"apply Finsupp.ext","premises":[{"full_name":"Finsupp.ext","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[125,8],"def_end_pos":[125,11]}]},{"state_before":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ ∀ (a : Y ⊗ Y'),\n    ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n                  (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n                Finsupp.lsingle x)\n              1 ∘ₗ\n            Finsupp.lsingle x')\n          1)\n        a =\n      ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n                Finsupp.lsingle x)\n              1 ∘ₗ\n            Finsupp.lsingle x')\n          1)\n        a","state_after":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n                (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n              Finsupp.lsingle x)\n            1 ∘ₗ\n          Finsupp.lsingle x')\n        1)\n      (y, y') =\n    ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n              Finsupp.lsingle x)\n            1 ∘ₗ\n          Finsupp.lsingle x')\n        1)\n      (y, y')","tactic":"intro ⟨y, y'⟩","premises":[]},{"state_before":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n                (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n              Finsupp.lsingle x)\n            1 ∘ₗ\n          Finsupp.lsingle x')\n        1)\n      (y, y') =\n    ((((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n              Finsupp.lsingle x)\n            1 ∘ₗ\n          Finsupp.lsingle x')\n        1)\n      (y, y')","state_after":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ ((finsuppTensorFinsupp' R Y Y')\n        (Finsupp.mapDomain f (Finsupp.single x 1) ⊗ₜ[R] Finsupp.mapDomain g (Finsupp.single x' 1)))\n      (y, y') =\n    (Finsupp.mapDomain (f ⊗ g) ((finsuppTensorFinsupp' R X X') (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single x' 1))) (y, y')","tactic":"change (finsuppTensorFinsupp' R Y Y')\n    (Finsupp.mapDomain f (Finsupp.single x 1) ⊗ₜ[R] Finsupp.mapDomain g (Finsupp.single x' 1)) _\n    = (Finsupp.mapDomain (f ⊗ g) (finsuppTensorFinsupp' R X X'\n    (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single x' 1))) _","premises":[{"full_name":"CategoryTheory.MonoidalCategoryStruct.tensorHom","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[89,2],"def_end_pos":[89,11]},{"full_name":"Finsupp.mapDomain","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[394,4],"def_end_pos":[394,13]},{"full_name":"Finsupp.single","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[228,4],"def_end_pos":[228,10]},{"full_name":"TensorProduct.tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[122,4],"def_end_pos":[122,8]},{"full_name":"finsuppTensorFinsupp'","def_path":"Mathlib/LinearAlgebra/DirectSum/Finsupp.lean","def_pos":[327,4],"def_end_pos":[327,25]}]},{"state_before":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ ((finsuppTensorFinsupp' R Y Y')\n        (Finsupp.mapDomain f (Finsupp.single x 1) ⊗ₜ[R] Finsupp.mapDomain g (Finsupp.single x' 1)))\n      (y, y') =\n    (Finsupp.mapDomain (f ⊗ g) ((finsuppTensorFinsupp' R X X') (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single x' 1))) (y, y')","state_after":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ (Finsupp.single (f x, g x') 1) (y, y') = (Finsupp.single (f x, g x') 1) (y, y')","tactic":"simp_rw [Finsupp.mapDomain_single, finsuppTensorFinsupp'_single_tmul_single, mul_one,\n    Finsupp.mapDomain_single, CategoryTheory.tensor_apply]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"CategoryTheory.tensor_apply","def_path":"Mathlib/CategoryTheory/Monoidal/Types/Basic.lean","def_pos":[29,8],"def_end_pos":[29,20]},{"full_name":"Finsupp.mapDomain_single","def_path":"Mathlib/Data/Finsupp/Basic.lean","def_pos":[425,8],"def_end_pos":[425,24]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"finsuppTensorFinsupp'_single_tmul_single","def_path":"Mathlib/LinearAlgebra/DirectSum/Finsupp.lean","def_pos":[336,8],"def_end_pos":[336,48]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]}]},{"state_before":"case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ (Finsupp.single (f x, g x') 1) (y, y') = (Finsupp.single (f x, g x') 1) (y, y')","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Option/NAry.lean","commit":"","full_name":"Option.map₂_coe_right","start":[63,0],"end":[65,59],"file_path":"Mathlib/Data/Option/NAry.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf✝ : α → β → γ\na✝ : Option α\nb✝ : Option β\nc : Option γ\nf : α → β → γ\na : Option α\nb : β\n⊢ map₂ f a (some b) = Option.map (fun a => f a b) a","state_after":"no goals","tactic":"cases a <;> rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Field/Basic.lean","commit":"","full_name":"div_mul_le_div_mul_of_div_le_div","start":[399,0],"end":[402,70],"file_path":"Mathlib/Algebra/Order/Field/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nh : a / b ≤ c / d\nhe : 0 ≤ e\n⊢ a / (b * e) ≤ c / (d * e)","state_after":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nh : a / b ≤ c / d\nhe : 0 ≤ e\n⊢ a / b * (1 / e) ≤ c / d * (1 / e)","tactic":"rw [div_mul_eq_div_mul_one_div, div_mul_eq_div_mul_one_div]","premises":[{"full_name":"div_mul_eq_div_mul_one_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[570,8],"def_end_pos":[570,34]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nh : a / b ≤ c / d\nhe : 0 ≤ e\n⊢ a / b * (1 / e) ≤ c / d * (1 / e)","state_after":"no goals","tactic":"exact mul_le_mul_of_nonneg_right h (one_div_nonneg (α := α) |>.2 he)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"mul_le_mul_of_nonneg_right","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[194,8],"def_end_pos":[194,34]},{"full_name":"one_div_nonneg","def_path":"Mathlib/Algebra/Order/Field/Unbundled/Basic.lean","def_pos":[41,6],"def_end_pos":[41,20]}]}]}
{"url":"Mathlib/Data/Complex/Exponential.lean","commit":"","full_name":"Real.tan_div_sqrt_one_add_tan_sq","start":[843,0],"end":[845,73],"file_path":"Mathlib/Data/Complex/Exponential.lean","tactics":[{"state_before":"x✝ y x : ℝ\nhx : 0 < cos x\n⊢ tan x / √(1 + tan x ^ 2) = sin x","state_after":"no goals","tactic":"rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Real.inv_sqrt_one_add_tan_sq","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[840,8],"def_end_pos":[840,31]},{"full_name":"Real.tan_mul_cos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[766,8],"def_end_pos":[766,19]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/RingTheory/Prime.lean","commit":"","full_name":"mul_eq_mul_prime_prod","start":[23,0],"end":[44,76],"file_path":"Mathlib/RingTheory/Prime.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nα : Type u_2\ninst✝ : DecidableEq α\nx y a : R\ns : Finset α\np : α → R\nhp : ∀ i ∈ s, Prime (p i)\nhx : x * y = a * ∏ i ∈ s, p i\n⊢ ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i","state_after":"case empty\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nα : Type u_2\ninst✝ : DecidableEq α\np : α → R\nx y a : R\nhp : ∀ i ∈ ∅, Prime (p i)\nhx : x * y = a * ∏ i ∈ ∅, p i\n⊢ ∃ t u b c, t ∪ u = ∅ ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i\n\ncase insert\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nα : Type u_2\ninst✝ : DecidableEq α\np : α → R\ni : α\ns : Finset α\nhis : i ∉ s\nih :\n  ∀ {x y a : R},\n    (∀ i ∈ s, Prime (p i)) →\n      x * y = a * ∏ i ∈ s, p i →\n        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i\nx y a : R\nhp : ∀ i_1 ∈ insert i s, Prime (p i_1)\nhx : x * y = a * ∏ i ∈ insert i s, p i\n⊢ ∃ t u b c, t ∪ u = insert i s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i","tactic":"induction' s using Finset.induction with i s his ih generalizing x y a","premises":[{"full_name":"Finset.induction","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1066,18],"def_end_pos":[1066,27]}]}]}
{"url":"Mathlib/Analysis/Analytic/Within.lean","commit":"","full_name":"hasFPowerSeriesWithinAt_univ","start":[55,0],"end":[57,90],"file_path":"Mathlib/Analysis/Analytic/Within.lean","tactics":[{"state_before":"α : Type u_1\n𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\n⊢ HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x","state_after":"no goals","tactic":"simp only [HasFPowerSeriesWithinAt, hasFPowerSeriesWithinOnBall_univ, HasFPowerSeriesAt]","premises":[{"full_name":"HasFPowerSeriesAt","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[372,4],"def_end_pos":[372,21]},{"full_name":"HasFPowerSeriesWithinAt","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[376,4],"def_end_pos":[376,27]},{"full_name":"hasFPowerSeriesWithinOnBall_univ","def_path":"Mathlib/Analysis/Analytic/Within.lean","def_pos":[45,14],"def_end_pos":[45,46]}]}]}
{"url":"Mathlib/RingTheory/Localization/Basic.lean","commit":"","full_name":"IsLocalization.sec_spec'","start":[189,0],"end":[193,25],"file_path":"Mathlib/RingTheory/Localization/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nz : S\n⊢ (algebraMap R S) (sec M z).1 = (algebraMap R S) ↑(sec M z).2 * z","state_after":"no goals","tactic":"rw [mul_comm, sec_spec]","premises":[{"full_name":"IsLocalization.sec_spec","def_path":"Mathlib/RingTheory/Localization/Basic.lean","def_pos":[185,8],"def_end_pos":[185,16]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","commit":"","full_name":"AkraBazziRecurrence.dist_r_b'","start":[133,0],"end":[136,51],"file_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\n⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ‖↑(r i n) - b i * ↑n‖ ≤ ↑n / log ↑n ^ 2","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\n⊢ ∀ (i : α), ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2","tactic":"rw [Filter.eventually_all]","premises":[{"full_name":"Filter.eventually_all","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1024,8],"def_end_pos":[1024,22]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\n⊢ ∀ (i : α), ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\ni : α\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2","tactic":"intro i","premises":[]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\ni : α\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2","state_after":"no goals","tactic":"simpa using IsLittleO.eventuallyLE (R.dist_r_b i)","premises":[{"full_name":"AkraBazziRecurrence.dist_r_b","def_path":"Mathlib/Computability/AkraBazzi/AkraBazzi.lean","def_pos":[104,2],"def_end_pos":[104,10]},{"full_name":"Asymptotics.IsLittleO.eventuallyLE","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[180,8],"def_end_pos":[180,30]}]}]}
{"url":"Mathlib/NumberTheory/DiophantineApproximation.lean","commit":"","full_name":"Real.exists_rat_abs_sub_le_and_den_le","start":[143,0],"end":[159,41],"file_path":"Mathlib/NumberTheory/DiophantineApproximation.lean","tactics":[{"state_before":"ξ : ℝ\nn : ℕ\nn_pos : 0 < n\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","state_after":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","tactic":"obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos","premises":[{"full_name":"Real.exists_int_int_abs_mul_sub_le","def_path":"Mathlib/NumberTheory/DiophantineApproximation.lean","def_pos":[91,8],"def_end_pos":[91,37]}]},{"state_before":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","state_after":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","tactic":"have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.cast_pos","def_path":"Mathlib/Algebra/Order/Ring/Cast.lean","def_pos":[57,14],"def_end_pos":[57,22]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","state_after":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","tactic":"have hden : ((j / k : ℚ).den : ℤ) ≤ k := by\n    convert le_of_dvd hk₀ (Rat.den_dvd j k)\n    exact Rat.intCast_div_eq_divInt _ _","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.le_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[97,8],"def_end_pos":[97,17]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Rat.den","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[24,2],"def_end_pos":[24,5]},{"full_name":"Rat.den_dvd","def_path":"Mathlib/Data/Rat/Lemmas.lean","def_pos":[28,8],"def_end_pos":[28,15]},{"full_name":"Rat.intCast_div_eq_divInt","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[436,6],"def_end_pos":[436,27]}]},{"state_before":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n","state_after":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| ≤ 1 / ((↑n + 1) * ↑(↑j / ↑k).den)","tactic":"refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Nat.cast_le","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[78,8],"def_end_pos":[78,15]}]},{"state_before":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| ≤ 1 / ((↑n + 1) * ↑(↑j / ↑k).den)","state_after":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| * ↑(↑j / ↑k).den ≤ 1 / (↑n + 1)","tactic":"rw [← div_div, le_div_iff (Nat.cast_pos.mpr <| Rat.pos _ : (0 : ℝ) < _)]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.cast_pos","def_path":"Mathlib/Data/Nat/Cast/Order/Ring.lean","def_pos":[53,8],"def_end_pos":[53,16]},{"full_name":"Rat.pos","def_path":"Mathlib/Data/Rat/Defs.lean","def_pos":[45,8],"def_end_pos":[45,11]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"div_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[543,8],"def_end_pos":[543,15]},{"full_name":"le_div_iff","def_path":"Mathlib/Algebra/Order/Field/Basic.lean","def_pos":[41,8],"def_end_pos":[41,18]}]},{"state_before":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| * ↑(↑j / ↑k).den ≤ 1 / (↑n + 1)","state_after":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| * ↑k ≤ 1 / (↑n + 1)","tactic":"refine (mul_le_mul_of_nonneg_left (Int.cast_le.mpr hden : _ ≤ (k : ℝ)) (abs_nonneg _)).trans ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.cast_le","def_path":"Mathlib/Algebra/Order/Ring/Cast.lean","def_pos":[47,25],"def_end_pos":[47,32]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"abs_nonneg","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[95,29],"def_end_pos":[95,39]},{"full_name":"mul_le_mul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[190,8],"def_end_pos":[190,33]}]},{"state_before":"case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| * ↑k ≤ 1 / (↑n + 1)","state_after":"no goals","tactic":"rwa [← abs_of_pos hk₀', Rat.cast_div, Rat.cast_intCast, Rat.cast_intCast, ← abs_mul, sub_mul,\n    div_mul_cancel₀ _ hk₀'.ne', mul_comm]","premises":[{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Rat.cast_div","def_path":"Mathlib/Data/Rat/Cast/CharZero.lean","def_pos":[80,8],"def_end_pos":[80,16]},{"full_name":"Rat.cast_intCast","def_path":"Mathlib/Data/Rat/Cast/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]},{"full_name":"abs_mul","def_path":"Mathlib/Algebra/Order/Ring/Abs.lean","def_pos":[42,6],"def_end_pos":[42,13]},{"full_name":"abs_of_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[79,2],"def_end_pos":[79,13]},{"full_name":"div_mul_cancel₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[289,14],"def_end_pos":[289,29]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Order/SymmDiff.lean","commit":"","full_name":"bihimp_of_ge","start":[226,0],"end":[227,46],"file_path":"Mathlib/Order/SymmDiff.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c d a b : α\nh : b ≤ a\n⊢ a ⇔ b = a ⇨ b","state_after":"no goals","tactic":"rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"bihimp","def_path":"Mathlib/Order/SymmDiff.lean","def_pos":[63,4],"def_end_pos":[63,10]},{"full_name":"himp_eq_top_iff","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[268,8],"def_end_pos":[268,23]},{"full_name":"top_inf_eq","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[374,6],"def_end_pos":[374,16]}]}]}
{"url":"Mathlib/Algebra/CharP/Defs.lean","commit":"","full_name":"ringChar.Nat.cast_ringChar","start":[185,0],"end":[185,71],"file_path":"Mathlib/Algebra/CharP/Defs.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : NonAssocSemiring R\n⊢ ↑(ringChar R) = 0","state_after":"no goals","tactic":"rw [ringChar.spec]","premises":[{"full_name":"ringChar.spec","def_path":"Mathlib/Algebra/CharP/Defs.lean","def_pos":[159,6],"def_end_pos":[159,10]}]}]}
{"url":"Mathlib/Order/Heyting/Basic.lean","commit":"","full_name":"sdiff_sup_cancel","start":[437,0],"end":[437,98],"file_path":"Mathlib/Order/Heyting/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\nh : b ≤ a\n⊢ a \\ b ⊔ b = a","state_after":"no goals","tactic":"rw [sup_comm, sup_sdiff_cancel_right h]","premises":[{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]},{"full_name":"sup_sdiff_cancel_right","def_path":"Mathlib/Order/Heyting/Basic.lean","def_pos":[434,8],"def_end_pos":[434,30]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","commit":"","full_name":"WeierstrassCurve.Affine.nonsingular_neg_iff","start":[383,0],"end":[386,52],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Nonsingular x (W.negY x y) ↔ W.Nonsingular x y","state_after":"R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ∧ (W.a₁ * W.negY x y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ W.negY x y) ↔\n    W.Equation x y ∧ (W.a₁ * y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ -y - W.a₁ * x - W.a₃)","tactic":"rw [nonsingular_iff, equation_neg_iff, ← negY, negY_negY, ← @ne_comm _ y, nonsingular_iff]","premises":[{"full_name":"WeierstrassCurve.Affine.equation_neg_iff","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","def_pos":[378,6],"def_end_pos":[378,22]},{"full_name":"WeierstrassCurve.Affine.negY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","def_pos":[309,4],"def_end_pos":[309,8]},{"full_name":"WeierstrassCurve.Affine.negY_negY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","def_pos":[312,6],"def_end_pos":[312,15]},{"full_name":"WeierstrassCurve.Affine.nonsingular_iff","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","def_pos":[259,6],"def_end_pos":[259,21]},{"full_name":"ne_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[706,8],"def_end_pos":[706,15]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ∧ (W.a₁ * W.negY x y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ W.negY x y) ↔\n    W.Equation x y ∧ (W.a₁ * y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ -y - W.a₁ * x - W.a₃)","state_after":"no goals","tactic":"exact and_congr_right' <| (iff_congr not_and_or.symm not_and_or.symm).mpr <|\n    not_congr <| and_congr_left fun h => by rw [← h]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"and_congr_left","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[145,8],"def_end_pos":[145,22]},{"full_name":"and_congr_right'","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[46,8],"def_end_pos":[46,24]},{"full_name":"iff_congr","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[37,8],"def_end_pos":[37,17]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]},{"full_name":"not_congr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1349,8],"def_end_pos":[1349,17]}]}]}
{"url":"Mathlib/Topology/Clopen.lean","commit":"","full_name":"continuous_boolIndicator_iff_isClopen","start":[123,0],"end":[125,60],"file_path":"Mathlib/Topology/Clopen.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns t U : Set X\n⊢ Continuous U.boolIndicator ↔ IsClopen U","state_after":"no goals","tactic":"rw [continuous_bool_rng true, preimage_boolIndicator_true]","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Set.preimage_boolIndicator_true","def_path":"Mathlib/Data/Set/BoolIndicator.lean","def_pos":[32,8],"def_end_pos":[32,35]},{"full_name":"continuous_bool_rng","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[802,6],"def_end_pos":[802,25]}]}]}
{"url":"Mathlib/Data/PFunctor/Univariate/M.lean","commit":"","full_name":"PFunctor.M.isubtree_cons","start":[452,0],"end":[455,94],"file_path":"Mathlib/Data/PFunctor/Univariate/M.lean","tactics":[{"state_before":"F : PFunctor.{u}\nX : Type u_1\nf✝ : X → ↑F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited F.M\nps : Path F\na : F.A\nf : F.B a → F.M\ni : F.B a\n⊢ isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = isubtree ps (f i)","state_after":"F : PFunctor.{u}\nX : Type u_1\nf✝ : X → ↑F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited F.M\nps : Path F\na : F.A\nf : F.B a → F.M\ni : F.B a\n⊢ isubtree ps (f (cast ⋯ i)) = isubtree ps (f i)","tactic":"simp only [isubtree, ichildren_mk, PFunctor.Obj.iget, dif_pos, isubtree, M.casesOn_mk']","premises":[{"full_name":"PFunctor.M.casesOn_mk'","def_path":"Mathlib/Data/PFunctor/Univariate/M.lean","def_pos":[373,8],"def_end_pos":[373,19]},{"full_name":"PFunctor.M.ichildren_mk","def_path":"Mathlib/Data/PFunctor/Univariate/M.lean","def_pos":[447,8],"def_end_pos":[447,20]},{"full_name":"PFunctor.M.isubtree","def_path":"Mathlib/Data/PFunctor/Univariate/M.lean","def_pos":[402,4],"def_end_pos":[402,12]},{"full_name":"PFunctor.Obj.iget","def_path":"Mathlib/Data/PFunctor/Univariate/Basic.lean","def_pos":[130,4],"def_end_pos":[130,12]},{"full_name":"dif_pos","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[949,8],"def_end_pos":[949,15]}]},{"state_before":"F : PFunctor.{u}\nX : Type u_1\nf✝ : X → ↑F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited F.M\nps : Path F\na : F.A\nf : F.B a → F.M\ni : F.B a\n⊢ isubtree ps (f (cast ⋯ i)) = isubtree ps (f i)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Ring/Int.lean","commit":"","full_name":"Int.two_mul_ediv_two_add_one_of_odd","start":[206,0],"end":[210,21],"file_path":"Mathlib/Algebra/Ring/Int.lean","tactics":[{"state_before":"m n : ℤ\n⊢ Odd n → 2 * (n / 2) + 1 = n","state_after":"case intro\nm c : ℤ\n⊢ 2 * ((2 * c + 1) / 2) + 1 = 2 * c + 1","tactic":"rintro ⟨c, rfl⟩","premises":[]},{"state_before":"case intro\nm c : ℤ\n⊢ 2 * ((2 * c + 1) / 2) + 1 = 2 * c + 1","state_after":"case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1","tactic":"rw [mul_comm]","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1","state_after":"case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2","tactic":"convert Int.ediv_add_emod' (2 * c + 1) 2","premises":[{"full_name":"Int.ediv_add_emod'","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[221,8],"def_end_pos":[221,22]}]},{"state_before":"case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2","state_after":"no goals","tactic":"simp [Int.add_emod]","premises":[{"full_name":"Int.add_emod","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[476,8],"def_end_pos":[476,16]}]}]}
{"url":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","commit":"","full_name":"AddSubsemigroup.dense_induction","start":[322,0],"end":[333,25],"file_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nmem : ∀ x ∈ s, p x\nmul : ∀ (x y : M), p x → p y → p (x * y)\n⊢ p x","state_after":"M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nmem : ∀ x ∈ s, p x\nmul : ∀ (x y : M), p x → p y → p (x * y)\nthis : ∀ x ∈ closure s, p x\n⊢ p x","tactic":"have : ∀ x ∈ closure s, p x := fun x hx => closure_induction hx mem mul","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Subsemigroup.closure","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[253,4],"def_end_pos":[253,11]},{"full_name":"Subsemigroup.closure_induction","def_path":"Mathlib/Algebra/Group/Subsemigroup/Basic.lean","def_pos":[297,8],"def_end_pos":[297,25]}]},{"state_before":"M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nmem : ∀ x ∈ s, p x\nmul : ∀ (x y : M), p x → p y → p (x * y)\nthis : ∀ x ∈ closure s, p x\n⊢ p x","state_after":"no goals","tactic":"simpa [hs] using this x","premises":[]}]}
{"url":"Mathlib/RingTheory/MvPolynomial/Homogeneous.lean","commit":"","full_name":"_private.Mathlib.RingTheory.MvPolynomial.Homogeneous.0.MvPolynomial.IsHomogeneous.exists_eval_ne_zero_of_coeff_finSuccEquiv_ne_zero_aux","start":[307,0],"end":[338,20],"file_path":"Mathlib/RingTheory/MvPolynomial/Homogeneous.lean","tactics":[{"state_before":"σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\n⊢ ∃ r, (eval r) F ≠ 0","state_after":"σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\n⊢ ∃ r, (eval r) F ≠ 0","tactic":"have hF₀ : F ≠ 0 := by contrapose! hFn; simp [hFn]","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\n⊢ ∃ r, (eval r) F ≠ 0","state_after":"σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\n⊢ ∃ r, (eval r) F ≠ 0","tactic":"have hdeg : natDegree (finSuccEquiv R N F) < n + 1 := by\n    linarith [natDegree_finSuccEquiv F, degreeOf_le_totalDegree F 0, hF.totalDegree hF₀]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"MvPolynomial.IsHomogeneous.totalDegree","def_path":"Mathlib/RingTheory/MvPolynomial/Homogeneous.lean","def_pos":[250,8],"def_end_pos":[250,19]},{"full_name":"MvPolynomial.degreeOf_le_totalDegree","def_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","def_pos":[442,6],"def_end_pos":[442,29]},{"full_name":"MvPolynomial.finSuccEquiv","def_path":"Mathlib/Algebra/MvPolynomial/Equiv.lean","def_pos":[303,4],"def_end_pos":[303,16]},{"full_name":"MvPolynomial.natDegree_finSuccEquiv","def_path":"Mathlib/Algebra/MvPolynomial/Equiv.lean","def_pos":[469,8],"def_end_pos":[469,30]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]}]},{"state_before":"σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\n⊢ ∃ r, (eval r) F ≠ 0","state_after":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\n⊢ (eval (Fin.cons 1 0)) F ≠ 0","tactic":"use Fin.cons 1 0","premises":[{"full_name":"Fin.cons","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[108,4],"def_end_pos":[108,8]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\n⊢ (eval (Fin.cons 1 0)) F ≠ 0","state_after":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ (eval (Fin.cons 1 0)) F ≠ 0","tactic":"have aux : ∀ i ∈ Finset.range n, constantCoeff ((finSuccEquiv R N F).coeff i) = 0 := by\n    intro i hi\n    rw [Finset.mem_range] at hi\n    apply (hF.finSuccEquiv_coeff_isHomogeneous i (n-i) (by omega)).coeff_eq_zero\n    simp only [Finsupp.degree_zero]\n    rw [← Nat.sub_ne_zero_iff_lt] at hi\n    exact hi.symm","premises":[{"full_name":"Finset.mem_range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2450,8],"def_end_pos":[2450,17]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Finsupp.degree_zero","def_path":"Mathlib/Data/Finsupp/Weight.lean","def_pos":[182,8],"def_end_pos":[182,19]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"MvPolynomial.IsHomogeneous.coeff_eq_zero","def_path":"Mathlib/RingTheory/MvPolynomial/Homogeneous.lean","def_pos":[148,8],"def_end_pos":[148,21]},{"full_name":"MvPolynomial.IsHomogeneous.finSuccEquiv_coeff_isHomogeneous","def_path":"Mathlib/RingTheory/MvPolynomial/Homogeneous.lean","def_pos":[276,6],"def_end_pos":[276,38]},{"full_name":"MvPolynomial.constantCoeff","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[822,4],"def_end_pos":[822,17]},{"full_name":"MvPolynomial.finSuccEquiv","def_path":"Mathlib/Algebra/MvPolynomial/Equiv.lean","def_pos":[303,4],"def_end_pos":[303,16]},{"full_name":"Nat.sub_ne_zero_iff_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[1039,18],"def_end_pos":[1039,36]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]}]},{"state_before":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ (eval (Fin.cons 1 0)) F ≠ 0","state_after":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ constantCoeff (((finSuccEquiv R N) F).coeff n) ≠ 0","tactic":"simp_rw [eval_eq_eval_mv_eval', eval_one_map, Polynomial.eval_eq_sum_range' hdeg,\n    eval_zero, one_pow, mul_one, map_sum, Finset.sum_range_succ, Finset.sum_eq_zero aux, zero_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.sum_eq_zero","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[384,2],"def_end_pos":[384,13]},{"full_name":"Finset.sum_range_succ","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1296,2],"def_end_pos":[1296,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MvPolynomial.eval_eq_eval_mv_eval'","def_path":"Mathlib/Algebra/MvPolynomial/Equiv.lean","def_pos":[366,8],"def_end_pos":[366,29]},{"full_name":"MvPolynomial.eval_zero","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[1405,8],"def_end_pos":[1405,17]},{"full_name":"Polynomial.eval_eq_sum_range'","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[277,8],"def_end_pos":[277,26]},{"full_name":"Polynomial.eval_one_map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[828,8],"def_end_pos":[828,20]},{"full_name":"map_sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[286,2],"def_end_pos":[286,13]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ constantCoeff (((finSuccEquiv R N) F).coeff n) ≠ 0","state_after":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\n⊢ ((finSuccEquiv R N) F).coeff n = 0","tactic":"contrapose! hFn","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\n⊢ ((finSuccEquiv R N) F).coeff n = 0","state_after":"case h.a\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\nd : Fin N →₀ ℕ\n⊢ coeff d (((finSuccEquiv R N) F).coeff n) = coeff d 0","tactic":"ext d","premises":[]},{"state_before":"case h.a\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\nd : Fin N →₀ ℕ\n⊢ coeff d (((finSuccEquiv R N) F).coeff n) = coeff d 0","state_after":"case h.a\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\nd : Fin N →₀ ℕ\n⊢ coeff d (((finSuccEquiv R N) F).coeff n) = 0","tactic":"rw [coeff_zero]","premises":[{"full_name":"MvPolynomial.coeff_zero","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[545,8],"def_end_pos":[545,18]}]},{"state_before":"case h.a\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\nd : Fin N →₀ ℕ\n⊢ coeff d (((finSuccEquiv R N) F).coeff n) = 0","state_after":"case h.a.inl\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\n⊢ coeff 0 (((finSuccEquiv R N) F).coeff n) = 0\n\ncase h.a.inr\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) = 0\nd : Fin N →₀ ℕ\nhd : d ≠ 0\n⊢ coeff d (((finSuccEquiv R N) F).coeff n) = 0","tactic":"obtain rfl | hd := eq_or_ne d 0","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]}]}
{"url":"Mathlib/RingTheory/IsTensorProduct.lean","commit":"","full_name":"IsTensorProduct.lift_eq","start":[89,0],"end":[92,6],"file_path":"Mathlib/RingTheory/IsTensorProduct.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nf' : M₁ →ₗ[R] M₂ →ₗ[R] M'\nx₁ : M₁\nx₂ : M₂\n⊢ (h.lift f') ((f x₁) x₂) = (f' x₁) x₂","state_after":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nf' : M₁ →ₗ[R] M₂ →ₗ[R] M'\nx₁ : M₁\nx₂ : M₂\n⊢ (TensorProduct.lift f' ∘ₗ ↑h.equiv.symm) ((f x₁) x₂) = (f' x₁) x₂","tactic":"delta IsTensorProduct.lift","premises":[{"full_name":"IsTensorProduct.lift","def_path":"Mathlib/RingTheory/IsTensorProduct.lean","def_pos":[85,18],"def_end_pos":[85,38]}]},{"state_before":"R : Type u_1\ninst✝¹⁴ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_6\nN₂ : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nf' : M₁ →ₗ[R] M₂ →ₗ[R] M'\nx₁ : M₁\nx₂ : M₂\n⊢ (TensorProduct.lift f' ∘ₗ ↑h.equiv.symm) ((f x₁) x₂) = (f' x₁) x₂","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","commit":"","full_name":"Orientation.oangle_add_oangle_rev_neg_right","start":[241,0],"end":[245,68],"file_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","tactics":[{"state_before":"V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ o.oangle x (-y) + o.oangle y (-x) = 0","state_after":"no goals","tactic":"rw [o.oangle_rev (-x), oangle_neg_left_eq_neg_right, add_neg_self]","premises":[{"full_name":"Orientation.oangle_neg_left_eq_neg_right","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[212,8],"def_end_pos":[212,36]},{"full_name":"Orientation.oangle_rev","def_path":"Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean","def_pos":[165,8],"def_end_pos":[165,18]},{"full_name":"add_neg_self","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1054,2],"def_end_pos":[1054,13]}]}]}
{"url":"Mathlib/Order/Filter/SmallSets.lean","commit":"","full_name":"Filter.bind_smallSets_gc","start":[43,0],"end":[47,5],"file_path":"Mathlib/Order/Filter/SmallSets.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nl l' la : Filter α\nlb : Filter β\n⊢ GaloisConnection (fun L => L.bind 𝓟) smallSets","state_after":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nl✝ l' la : Filter α\nlb : Filter β\nL : Filter (Set α)\nl : Filter α\n⊢ (fun L => L.bind 𝓟) L ≤ l ↔ L ≤ l.smallSets","tactic":"intro L l","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nl✝ l' la : Filter α\nlb : Filter β\nL : Filter (Set α)\nl : Filter α\n⊢ (fun L => L.bind 𝓟) L ≤ l ↔ L ≤ l.smallSets","state_after":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nl✝ l' la : Filter α\nlb : Filter β\nL : Filter (Set α)\nl : Filter α\n⊢ L.bind 𝓟 ≤ l ↔ l.sets ⊆ powerset ⁻¹' L.sets","tactic":"simp_rw [smallSets_eq_generate, le_generate_iff, image_subset_iff]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.le_generate_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[333,8],"def_end_pos":[333,23]},{"full_name":"Filter.smallSets_eq_generate","def_path":"Mathlib/Order/Filter/SmallSets.lean","def_pos":[37,8],"def_end_pos":[37,29]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.image_subset_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[407,8],"def_end_pos":[407,24]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Sort u_3\nl✝ l' la : Filter α\nlb : Filter β\nL : Filter (Set α)\nl : Filter α\n⊢ L.bind 𝓟 ≤ l ↔ l.sets ⊆ powerset ⁻¹' L.sets","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Topology/Bornology/Absorbs.lean","commit":"","full_name":"absorbent_iff_inv_smul","start":[238,0],"end":[240,96],"file_path":"Mathlib/Topology/Bornology/Absorbs.lean","tactics":[{"state_before":"G₀ : Type u_1\nα : Type u_2\nE : Type u_3\ninst✝² : GroupWithZero G₀\ninst✝¹ : Bornology G₀\ninst✝ : MulAction G₀ α\ns : Set α\nx : α\n⊢ Absorbs G₀ s {x} ↔ ∀ᶠ (c : G₀) in cobounded G₀, c⁻¹ • x ∈ s","state_after":"no goals","tactic":"simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_singleton]","premises":[{"full_name":"Set.mapsTo_singleton","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[344,8],"def_end_pos":[344,24]},{"full_name":"absorbs_iff_eventually_cobounded_mapsTo","def_path":"Mathlib/Topology/Bornology/Absorbs.lean","def_pos":[145,6],"def_end_pos":[145,52]}]}]}
{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Int.ceil_add_one","start":[1091,0],"end":[1094,41],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\n⊢ ⌈a + 1⌉ = ⌈a⌉ + 1","state_after":"no goals","tactic":"rw [← ceil_add_int a (1 : ℤ), cast_one]","premises":[{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Int.cast_one","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[70,8],"def_end_pos":[70,16]},{"full_name":"Int.ceil_add_int","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[1085,8],"def_end_pos":[1085,20]}]}]}
{"url":"Mathlib/Analysis/BoundedVariation.lean","commit":"","full_name":"LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_pi","start":[802,0],"end":[811,52],"file_path":"Mathlib/Analysis/BoundedVariation.lean","tactics":[{"state_before":"α : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","state_after":"α : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nA : ∀ (i : ι), LipschitzWith 1 fun x => x i\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","tactic":"have A : ∀ i : ι, LipschitzWith 1 fun x : ι → ℝ => x i := fun i => LipschitzWith.eval i","premises":[{"full_name":"LipschitzWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[54,4],"def_end_pos":[54,17]},{"full_name":"LipschitzWith.eval","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[205,18],"def_end_pos":[205,22]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"α : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nA : ∀ (i : ι), LipschitzWith 1 fun x => x i\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","state_after":"α : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nA : ∀ (i : ι), LipschitzWith 1 fun x => x i\nthis : ∀ (i : ι), ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ (fun x => f x i) s x\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","tactic":"have : ∀ i : ι, ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ (fun x : ℝ => f x i) s x := fun i ↦ by\n    apply ae_differentiableWithinAt_of_mem_real\n    exact LipschitzWith.comp_locallyBoundedVariationOn (A i) h","premises":[{"full_name":"DifferentiableWithinAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Basic.lean","def_pos":[165,4],"def_end_pos":[165,26]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"LipschitzWith.comp_locallyBoundedVariationOn","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[770,8],"def_end_pos":[770,52]},{"full_name":"LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_real","def_path":"Mathlib/Analysis/BoundedVariation.lean","def_pos":[794,8],"def_end_pos":[794,45]},{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"α : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nA : ∀ (i : ι), LipschitzWith 1 fun x => x i\nthis : ∀ (i : ι), ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ (fun x => f x i) s x\n⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","state_after":"case h\nα : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nA : ∀ (i : ι), LipschitzWith 1 fun x => x i\nthis : ∀ (i : ι), ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ (fun x => f x i) s x\nx : ℝ\nhx : ∀ (i : ι), x ∈ s → DifferentiableWithinAt ℝ (fun x => f x i) s x\nxs : x ∈ s\n⊢ DifferentiableWithinAt ℝ f s x","tactic":"filter_upwards [ae_all_iff.2 this] with x hx xs","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.ae_all_iff","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[93,8],"def_end_pos":[93,18]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\nα : Type u_1\ninst✝⁵ : LinearOrder α\nE : Type u_2\ninst✝⁴ : PseudoEMetricSpace E\nV : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : FiniteDimensional ℝ V\nι : Type u_4\ninst✝ : Fintype ι\nf : ℝ → ι → ℝ\ns : Set ℝ\nh : LocallyBoundedVariationOn f s\nA : ∀ (i : ι), LipschitzWith 1 fun x => x i\nthis : ∀ (i : ι), ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ (fun x => f x i) s x\nx : ℝ\nhx : ∀ (i : ι), x ∈ s → DifferentiableWithinAt ℝ (fun x => f x i) s x\nxs : x ∈ s\n⊢ DifferentiableWithinAt ℝ f s x","state_after":"no goals","tactic":"exact differentiableWithinAt_pi.2 fun i => hx i xs","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"differentiableWithinAt_pi","def_path":"Mathlib/Analysis/Calculus/FDeriv/Prod.lean","def_pos":[424,8],"def_end_pos":[424,33]}]}]}
{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.diff_iInter","start":[453,0],"end":[454,47],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ns : Set β\nt : ι → Set β\n⊢ s \\ ⋂ i, t i = ⋃ i, s \\ t i","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ns : Set β\nt : ι → Set β\n⊢ ⋃ i, s ∩ (t i)ᶜ = ⋃ i, s \\ t i","tactic":"rw [diff_eq, compl_iInter, inter_iUnion]","premises":[{"full_name":"Set.compl_iInter","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[398,8],"def_end_pos":[398,20]},{"full_name":"Set.diff_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[86,8],"def_end_pos":[86,15]},{"full_name":"Set.inter_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[414,8],"def_end_pos":[414,20]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ns : Set β\nt : ι → Set β\n⊢ ⋃ i, s ∩ (t i)ᶜ = ⋃ i, s \\ t i","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/Int/GCD.lean","commit":"","full_name":"Int.dvd_of_dvd_mul_left_of_gcd_one","start":[300,0],"end":[309,91],"file_path":"Mathlib/Data/Int/GCD.lean","tactics":[{"state_before":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\n⊢ a ∣ b","state_after":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis : ↑(a.gcd c) = a * a.gcdA c + c * a.gcdB c\n⊢ a ∣ b","tactic":"have := gcd_eq_gcd_ab a c","premises":[{"full_name":"Int.gcd_eq_gcd_ab","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[167,8],"def_end_pos":[167,21]}]},{"state_before":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis : ↑(a.gcd c) = a * a.gcdA c + c * a.gcdB c\n⊢ a ∣ b","state_after":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis : 1 = a * a.gcdA c + c * a.gcdB c\n⊢ a ∣ b","tactic":"simp only [hab, Int.ofNat_zero, Int.ofNat_succ, zero_add] at this","premises":[{"full_name":"Int.ofNat_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[27,8],"def_end_pos":[27,18]},{"full_name":"Int.ofNat_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[73,16],"def_end_pos":[73,26]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]},{"state_before":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis : 1 = a * a.gcdA c + c * a.gcdB c\n⊢ a ∣ b","state_after":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis✝ : 1 = a * a.gcdA c + c * a.gcdB c\nthis : b * a * a.gcdA c + b * c * a.gcdB c = b\n⊢ a ∣ b","tactic":"have : b * a * gcdA a c + b * c * gcdB a c = b := by simp [mul_assoc, ← Int.mul_add, ← this]","premises":[{"full_name":"Int.gcdA","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[157,4],"def_end_pos":[157,8]},{"full_name":"Int.gcdB","def_path":"Mathlib/Data/Int/GCD.lean","def_pos":[162,4],"def_end_pos":[162,8]},{"full_name":"Int.mul_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[425,18],"def_end_pos":[425,25]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis✝ : 1 = a * a.gcdA c + c * a.gcdB c\nthis : b * a * a.gcdA c + b * c * a.gcdB c = b\n⊢ a ∣ b","state_after":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis✝ : 1 = a * a.gcdA c + c * a.gcdB c\nthis : b * a * a.gcdA c + b * c * a.gcdB c = b\n⊢ a ∣ b * a * a.gcdA c + b * c * a.gcdB c","tactic":"rw [← this]","premises":[]},{"state_before":"a b c : ℤ\nhabc : a ∣ b * c\nhab : a.gcd c = 1\nthis✝ : 1 = a * a.gcdA c + c * a.gcdB c\nthis : b * a * a.gcdA c + b * c * a.gcdB c = b\n⊢ a ∣ b * a * a.gcdA c + b * c * a.gcdB c","state_after":"no goals","tactic":"exact Int.dvd_add (dvd_mul_of_dvd_left (dvd_mul_left a b) _) (dvd_mul_of_dvd_left habc _)","premises":[{"full_name":"Int.dvd_add","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[78,18],"def_end_pos":[78,25]},{"full_name":"dvd_mul_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[176,8],"def_end_pos":[176,20]},{"full_name":"dvd_mul_of_dvd_left","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[78,8],"def_end_pos":[78,27]}]}]}
{"url":"Mathlib/CategoryTheory/Grothendieck.lean","commit":"","full_name":"CategoryTheory.Grothendieck.grothendieckTypeToCat_inverse_map_base","start":[265,0],"end":[302,7],"file_path":"Mathlib/CategoryTheory/Grothendieck.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nX : Grothendieck (G ⋙ typeToCat)\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).obj X ≅ (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).obj X","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base✝, fiber := { as := as✝ } } ≅\n    (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).obj { base := base✝, fiber := { as := as✝ } }","tactic":"rcases X with ⟨_, ⟨⟩⟩","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base✝, fiber := { as := as✝ } } ≅\n    (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).obj { base := base✝, fiber := { as := as✝ } }","state_after":"no goals","tactic":"exact Iso.refl _","premises":[{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ {X Y : Grothendieck (G ⋙ typeToCat)} (f : X ⟶ Y),\n    (𝟭 (Grothendieck (G ⋙ typeToCat))).map f ≫\n        ((fun X =>\n              Grothendieck.casesOn X fun base fiber =>\n                Discrete.casesOn fiber fun as =>\n                  Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n            Y).hom =\n      ((fun X =>\n              Grothendieck.casesOn X fun base fiber =>\n                Discrete.casesOn fiber fun as =>\n                  Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n            X).hom ≫\n        (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).map f","state_after":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).map { base := base, fiber := { down := { down := f } } } ≫\n      ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝, fiber := { as := as✝ } }).hom =\n    ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝¹, fiber := { as := as✝¹ } }).hom ≫\n      (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).map\n        { base := base, fiber := { down := { down := f } } }","tactic":"rintro ⟨_, ⟨⟩⟩ ⟨_, ⟨⟩⟩ ⟨base, ⟨⟨f⟩⟩⟩","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ (𝟭 (Grothendieck (G ⋙ typeToCat))).map { base := base, fiber := { down := { down := f } } } ≫\n      ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝, fiber := { as := as✝ } }).hom =\n    ((fun X =>\n            Grothendieck.casesOn X fun base fiber =>\n              Discrete.casesOn fiber fun as =>\n                Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n          { base := base✝¹, fiber := { as := as✝¹ } }).hom ≫\n      (grothendieckTypeToCatFunctor G ⋙ grothendieckTypeToCatInverse G).map\n        { base := base, fiber := { down := { down := f } } }","state_after":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } ≫ 𝟙 { base := base✝, fiber := { as := as✝ } } =\n    𝟙 { base := base✝¹, fiber := { as := as✝¹ } } ≫\n      (grothendieckTypeToCatInverse G).map\n        ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","tactic":"dsimp at *","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } ≫ 𝟙 { base := base✝, fiber := { as := as✝ } } =\n    𝟙 { base := base✝¹, fiber := { as := as✝¹ } } ≫\n      (grothendieckTypeToCatInverse G).map\n        ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","state_after":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } =\n    (grothendieckTypeToCatInverse G).map\n      ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","tactic":"simp","premises":[]},{"state_before":"case mk.mk.mk.mk.mk.up.up\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝¹ : C\nas✝¹ : G.obj base✝¹\nbase✝ : C\nas✝ : G.obj base✝\nbase : { base := base✝¹, fiber := { as := as✝¹ } }.base ⟶ { base := base✝, fiber := { as := as✝ } }.base\nf :\n  (((G ⋙ typeToCat).map base).obj { base := base✝¹, fiber := { as := as✝¹ } }.fiber).as =\n    { base := base✝, fiber := { as := as✝ } }.fiber.as\n⊢ { base := base, fiber := { down := { down := f } } } =\n    (grothendieckTypeToCatInverse G).map\n      ((grothendieckTypeToCatFunctor G).map { base := base, fiber := { down := { down := f } } })","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nX : G.Elements\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X","state_after":"case mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝ : C\nsnd✝ : G.obj fst✝\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst✝, snd✝⟩ ≅ (𝟭 G.Elements).obj ⟨fst✝, snd✝⟩","tactic":"cases X","premises":[]},{"state_before":"case mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝ : C\nsnd✝ : G.obj fst✝\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst✝, snd✝⟩ ≅ (𝟭 G.Elements).obj ⟨fst✝, snd✝⟩","state_after":"no goals","tactic":"exact Iso.refl _","premises":[{"full_name":"CategoryTheory.Iso.refl","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[114,4],"def_end_pos":[114,8]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ {X Y : G.Elements} (f : X ⟶ Y),\n    (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map f ≫\n        ((fun X =>\n              Sigma.casesOn (motive := fun t =>\n                X = t →\n                  ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                X\n                (fun fst snd h =>\n                  ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                ⋯)\n            Y).hom =\n      ((fun X =>\n              Sigma.casesOn (motive := fun t =>\n                X = t →\n                  ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                X\n                (fun fst snd h =>\n                  ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                ⋯)\n            X).hom ≫\n        (𝟭 G.Elements).map f","state_after":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map ⟨f, e⟩ ≫\n      ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝, snd✝⟩).hom =\n    ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝¹, snd✝¹⟩).hom ≫\n      (𝟭 G.Elements).map ⟨f, e⟩","tactic":"rintro ⟨⟩ ⟨⟩ ⟨f, e⟩","premises":[]},{"state_before":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map ⟨f, e⟩ ≫\n      ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝, snd✝⟩).hom =\n    ((fun X =>\n            Sigma.casesOn (motive := fun t =>\n              X = t → ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n              X\n              (fun fst snd h =>\n                ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n              ⋯)\n          ⟨fst✝¹, snd✝¹⟩).hom ≫\n      (𝟭 G.Elements).map ⟨f, e⟩","state_after":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) ≫\n      𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝, snd✝⟩)) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝¹, snd✝¹⟩)) ≫ ⟨f, e⟩","tactic":"dsimp at *","premises":[]},{"state_before":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) ≫\n      𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝, snd✝⟩)) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst✝¹, snd✝¹⟩)) ≫ ⟨f, e⟩","state_after":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) = ⟨f, e⟩","tactic":"simp","premises":[]},{"state_before":"case mk.mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : G.map f ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCatFunctor G).map ((grothendieckTypeToCatInverse G).map ⟨f, e⟩) = ⟨f, e⟩","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ (X : Grothendieck (G ⋙ typeToCat)),\n    (grothendieckTypeToCatFunctor G).map\n          ((NatIso.ofComponents\n                  (fun X =>\n                    Grothendieck.casesOn X fun base fiber =>\n                      Discrete.casesOn fiber fun as =>\n                        Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n                  ⋯).hom.app\n            X) ≫\n        (NatIso.ofComponents\n                (fun X =>\n                  Sigma.casesOn (motive := fun t =>\n                    X = t →\n                      ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                    X\n                    (fun fst snd h =>\n                      ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                    ⋯)\n                ⋯).hom.app\n          ((grothendieckTypeToCatFunctor G).obj X) =\n      𝟙 ((grothendieckTypeToCatFunctor G).obj X)","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map\n        ((NatIso.ofComponents\n                (fun X =>\n                  Grothendieck.casesOn X fun base fiber =>\n                    Discrete.casesOn fiber fun as =>\n                      Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n                ⋯).hom.app\n          { base := base✝, fiber := { as := as✝ } }) ≫\n      (NatIso.ofComponents\n              (fun X =>\n                Sigma.casesOn (motive := fun t =>\n                  X = t →\n                    ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                  X\n                  (fun fst snd h =>\n                    ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                  ⋯)\n              ⋯).hom.app\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","tactic":"rintro ⟨_, ⟨⟩⟩","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map\n        ((NatIso.ofComponents\n                (fun X =>\n                  Grothendieck.casesOn X fun base fiber =>\n                    Discrete.casesOn fiber fun as =>\n                      Iso.refl ((𝟭 (Grothendieck (G ⋙ typeToCat))).obj { base := base, fiber := { as := as } }))\n                ⋯).hom.app\n          { base := base✝, fiber := { as := as✝ } }) ≫\n      (NatIso.ofComponents\n              (fun X =>\n                Sigma.casesOn (motive := fun t =>\n                  X = t →\n                    ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj X ≅ (𝟭 G.Elements).obj X))\n                  X\n                  (fun fst snd h =>\n                    ⋯ ▸ Iso.refl ((grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).obj ⟨fst, snd⟩))\n                  ⋯)\n              ⋯).hom.app\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map (𝟙 { base := base✝, fiber := { as := as✝ } }) ≫\n      (Sigma.rec (motive := fun t =>\n          (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n            ((grothendieckTypeToCatFunctor G).obj\n                ((grothendieckTypeToCatInverse G).obj\n                  ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n              (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n          (fun fst snd h =>\n            ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n          ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","tactic":"dsimp","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (grothendieckTypeToCatFunctor G).map (𝟙 { base := base✝, fiber := { as := as✝ } }) ≫\n      (Sigma.rec (motive := fun t =>\n          (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n            ((grothendieckTypeToCatFunctor G).obj\n                ((grothendieckTypeToCatInverse G).obj\n                  ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n              (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n          (fun fst snd h =>\n            ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n          ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","state_after":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (Sigma.rec (motive := fun t =>\n        (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n          ((grothendieckTypeToCatFunctor G).obj\n              ((grothendieckTypeToCatInverse G).obj\n                ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n            (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n        (fun fst snd h =>\n          ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","tactic":"simp","premises":[]},{"state_before":"case mk.mk\nC : Type u_1\nD : Type u_2\ninst✝¹ : Category.{?u.62170, u_1} C\ninst✝ : Category.{?u.62174, u_2} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nbase✝ : C\nas✝ : G.obj base✝\n⊢ (Sigma.rec (motive := fun t =>\n        (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } } = t →\n          ((grothendieckTypeToCatFunctor G).obj\n              ((grothendieckTypeToCatInverse G).obj\n                ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })) ≅\n            (grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }))\n        (fun fst snd h =>\n          ⋯ ▸ Iso.refl ((grothendieckTypeToCatFunctor G).obj ((grothendieckTypeToCatInverse G).obj ⟨fst, snd⟩)))\n        ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } }) ⋯).hom =\n    𝟙 ((grothendieckTypeToCatFunctor G).obj { base := base✝, fiber := { as := as✝ } })","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subset","start":[207,0],"end":[222,91],"file_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","state_after":"case intro.mk.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","tactic":"obtain ⟨⟨V, hxV⟩, R, ⟨e⟩⟩ := X.2 x","premises":[{"full_name":"AlgebraicGeometry.Scheme.local_affine","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[45,2],"def_end_pos":[45,14]}]},{"state_before":"case intro.mk.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","state_after":"case intro.mk.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","tactic":"have : e.hom.1.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.1.base ≫ V.inclusion)).obj U :=\n    show ((e.hom ≫ e.inv).1.base ⟨x, hxV⟩).1 ∈ U from e.hom_inv_id ▸ hxU","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Prefunctor.obj","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[55,2],"def_end_pos":[55,5]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]},{"full_name":"TopologicalSpace.Opens.inclusion","def_path":"Mathlib/Topology/Category/TopCat/Opens.lean","def_pos":[114,4],"def_end_pos":[114,13]},{"full_name":"TopologicalSpace.Opens.map","def_path":"Mathlib/Topology/Category/TopCat/Opens.lean","def_pos":[133,4],"def_end_pos":[133,7]}]},{"state_before":"case intro.mk.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","tactic":"obtain ⟨_, ⟨_, ⟨r : R, rfl⟩, rfl⟩, hr, hr'⟩ :=\n    PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open this (Opens.is_open' _)","premises":[{"full_name":"PrimeSpectrum.isBasis_basic_opens","def_path":"Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean","def_pos":[476,8],"def_end_pos":[476,27]},{"full_name":"TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open","def_path":"Mathlib/Topology/Bases.lean","def_pos":[177,8],"def_end_pos":[177,52]},{"full_name":"TopologicalSpace.Opens.is_open'","def_path":"Mathlib/Topology/Sets/Opens.lean","def_pos":[65,2],"def_end_pos":[65,10]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","tactic":"let f : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n    Spec.map (CommRingCat.ofHom (algebraMap R (Localization.Away r))) ≫ (e.inv ≫ X.ofRestrict _ : _)","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.ofRestrict","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[205,4],"def_end_pos":[205,14]},{"full_name":"AlgebraicGeometry.Spec","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[306,4],"def_end_pos":[306,8]},{"full_name":"AlgebraicGeometry.Spec.map","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[316,4],"def_end_pos":[316,12]},{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CommRingCat.of","def_path":"Mathlib/Algebra/Category/Ring/Basic.lean","def_pos":[470,4],"def_end_pos":[470,6]},{"full_name":"CommRingCat.ofHom","def_path":"Mathlib/Algebra/Category/Ring/Basic.lean","def_pos":[489,4],"def_end_pos":[489,9]},{"full_name":"Localization.Away","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[1398,4],"def_end_pos":[1398,8]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]},{"full_name":"algebraMap","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[108,4],"def_end_pos":[108,14]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ ∃ R f, IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","tactic":"refine ⟨.of (Localization.Away r), f, inferInstance, ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"CommRingCat.of","def_path":"Mathlib/Algebra/Category/Ring/Basic.lean","def_pos":[470,4],"def_end_pos":[470,6]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Localization.Away","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[1398,4],"def_end_pos":[1398,8]},{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈\n      ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) ''\n        Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ∧\n    ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) ''\n        Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ⊆\n      ↑U","tactic":"rw [Scheme.comp_val_base, LocallyRingedSpace.comp_val, SheafedSpace.comp_base, TopCat.coe_comp,\n    Set.range_comp]","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.comp_val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[137,8],"def_end_pos":[137,16]},{"full_name":"AlgebraicGeometry.Scheme.comp_val_base","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[225,8],"def_end_pos":[225,21]},{"full_name":"AlgebraicGeometry.SheafedSpace.comp_base","def_path":"Mathlib/Geometry/RingedSpace/SheafedSpace.lean","def_pos":[130,8],"def_end_pos":[130,17]},{"full_name":"Set.range_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[621,8],"def_end_pos":[621,18]},{"full_name":"TopCat.coe_comp","def_path":"Mathlib/Topology/Category/TopCat/Basic.lean","def_pos":[71,16],"def_end_pos":[71,24]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈\n      ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) ''\n        Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ∧\n    ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) ''\n        Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ⊆\n      ↑U","state_after":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ∧\n    ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ⊆ ↑U","tactic":"erw [PrimeSpectrum.localization_away_comap_range (Localization.Away r) r]","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Localization.Away","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[1398,4],"def_end_pos":[1398,8]},{"full_name":"PrimeSpectrum.localization_away_comap_range","def_path":"Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean","def_pos":[489,8],"def_end_pos":[489,37]}]},{"state_before":"case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ∧\n    ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ⊆ ↑U","state_after":"no goals","tactic":"exact ⟨⟨_, hr, congr(($(e.hom_inv_id).1.base ⟨x, hxV⟩).1)⟩, Set.image_subset_iff.mpr hr'⟩","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.image_subset_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[407,8],"def_end_pos":[407,24]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]}]}
{"url":"Mathlib/Data/Rat/Cast/Order.lean","commit":"","full_name":"Rat.cast_lt_zero","start":[55,0],"end":[55,64],"file_path":"Mathlib/Data/Rat/Cast/Order.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\np q : ℚ\nK : Type u_5\ninst✝ : LinearOrderedField K\n⊢ ↑q < 0 ↔ q < 0","state_after":"no goals","tactic":"norm_cast","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/Probability/Kernel/Composition.lean","commit":"","full_name":"ProbabilityTheory.Kernel.compProd_add_right","start":[550,0],"end":[556,44],"file_path":"Mathlib/Probability/Kernel/Composition.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\n⊢ μ ⊗ₖ (κ + η) = μ ⊗ₖ κ + μ ⊗ₖ η","state_after":"case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ ((μ ⊗ₖ (κ + η)) a) s = ((μ ⊗ₖ κ + μ ⊗ₖ η) a) s","tactic":"ext a s hs","premises":[]},{"state_before":"case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ ((μ ⊗ₖ (κ + η)) a) s = ((μ ⊗ₖ κ + μ ⊗ₖ η) a) s","state_after":"case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} + (η (a, b)) {c | (b, c) ∈ s} ∂μ a =\n    ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} ∂μ a + ∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂μ a","tactic":"simp only [compProd_apply _ _ _ hs, coe_add, Pi.add_apply, Measure.coe_add]","premises":[{"full_name":"MeasureTheory.Measure.coe_add","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[781,8],"def_end_pos":[781,15]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"ProbabilityTheory.Kernel.coe_add","def_path":"Mathlib/Probability/Kernel/Basic.lean","def_pos":[86,25],"def_end_pos":[86,32]},{"full_name":"ProbabilityTheory.Kernel.compProd_apply","def_path":"Mathlib/Probability/Kernel/Composition.lean","def_pos":[230,8],"def_end_pos":[230,22]}]},{"state_before":"case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} + (η (a, b)) {c | (b, c) ∈ s} ∂μ a =\n    ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} ∂μ a + ∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂μ a","state_after":"case h.h.hf\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ Measurable fun b => (κ (a, b)) {c | (b, c) ∈ s}","tactic":"rw [lintegral_add_left]","premises":[{"full_name":"MeasureTheory.lintegral_add_left","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[552,8],"def_end_pos":[552,26]}]},{"state_before":"case h.h.hf\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ Measurable fun b => (κ (a, b)) {c | (b, c) ∈ s}","state_after":"no goals","tactic":"exact measurable_kernel_prod_mk_left' hs a","premises":[{"full_name":"ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left'","def_path":"Mathlib/Probability/Kernel/MeasurableIntegral.lean","def_pos":[109,8],"def_end_pos":[109,39]}]}]}
{"url":"Mathlib/Probability/Martingale/Upcrossing.lean","commit":"","full_name":"MeasureTheory.crossing_pos_eq","start":[614,0],"end":[661,9],"file_path":"Mathlib/Probability/Martingale/Upcrossing.lean","tactics":[{"state_before":"Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n","state_after":"Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n","tactic":"have hab' : 0 < b - a := sub_pos.2 hab","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"sub_pos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[603,29],"def_end_pos":[603,36]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n","state_after":"Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n","tactic":"have hf' (ω i) : (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by rw [posPart_nonpos, sub_nonpos]","premises":[{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"posPart","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[55,2],"def_end_pos":[55,13]},{"full_name":"posPart_nonpos","def_path":"Mathlib/Algebra/Order/Group/PosPart.lean","def_pos":[98,2],"def_end_pos":[98,13]},{"full_name":"sub_nonpos","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[486,14],"def_end_pos":[486,24]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n","state_after":"case zero\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N 0 = upperCrossingTime a b f N 0 ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N 0 = lowerCrossingTime a b f N 0\n\ncase succ\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\nk : ℕ\nih :\n  upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) = upperCrossingTime a b f N (k + 1) ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) = lowerCrossingTime a b f N (k + 1)","tactic":"induction' n with k ih","premises":[]}]}
{"url":"Mathlib/CategoryTheory/SingleObj.lean","commit":"","full_name":"MulEquiv.toSingleObjEquiv_inverse_map","start":[200,0],"end":[212,14],"file_path":"Mathlib/CategoryTheory/SingleObj.lean","tactics":[{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ 𝟭 (SingleObj M) = e.toMonoidHom.toFunctor ⋙ e.symm.toMonoidHom.toFunctor","state_after":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (MonoidHom.id M).toFunctor = (e.symm.toMonoidHom.comp e.toMonoidHom).toFunctor","tactic":"rw [← MonoidHom.comp_toFunctor, ← MonoidHom.id_toFunctor]","premises":[{"full_name":"MonoidHom.comp_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[184,8],"def_end_pos":[184,22]},{"full_name":"MonoidHom.id_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[191,8],"def_end_pos":[191,20]}]},{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (MonoidHom.id M).toFunctor = (e.symm.toMonoidHom.comp e.toMonoidHom).toFunctor","state_after":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ MonoidHom.id M = e.symm.toMonoidHom.comp e.toMonoidHom","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ MonoidHom.id M = e.symm.toMonoidHom.comp e.toMonoidHom","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.symm.toMonoidHom.toFunctor ⋙ e.toMonoidHom.toFunctor = 𝟭 (SingleObj N)","state_after":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (e.toMonoidHom.comp e.symm.toMonoidHom).toFunctor = (MonoidHom.id N).toFunctor","tactic":"rw [← MonoidHom.comp_toFunctor, ← MonoidHom.id_toFunctor]","premises":[{"full_name":"MonoidHom.comp_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[184,8],"def_end_pos":[184,22]},{"full_name":"MonoidHom.id_toFunctor","def_path":"Mathlib/CategoryTheory/SingleObj.lean","def_pos":[191,8],"def_end_pos":[191,20]}]},{"state_before":"M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ (e.toMonoidHom.comp e.symm.toMonoidHom).toFunctor = (MonoidHom.id N).toFunctor","state_after":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.toMonoidHom.comp e.symm.toMonoidHom = MonoidHom.id N","tactic":"congr 1","premises":[]},{"state_before":"case e_f\nM : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.toMonoidHom.comp e.symm.toMonoidHom = MonoidHom.id N","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Data/Multiset/Basic.lean","commit":"","full_name":"Multiset.card_pair","start":[690,0],"end":[691,48],"file_path":"Mathlib/Data/Multiset/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type v\nγ : Type u_2\na b : α\n⊢ card {a, b} = 2","state_after":"no goals","tactic":"rw [insert_eq_cons, card_cons, card_singleton]","premises":[{"full_name":"Multiset.card_cons","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[683,8],"def_end_pos":[683,17]},{"full_name":"Multiset.card_singleton","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[687,8],"def_end_pos":[687,22]},{"full_name":"Multiset.insert_eq_cons","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[128,8],"def_end_pos":[128,22]}]}]}
{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.sqrt_eq_zero","start":[1408,0],"end":[1411,24],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\nh : n.sqrt = 0\n⊢ n.sqrt < 1","state_after":"a b c d m n k : ℕ\np q : ℕ → Prop\nh : n.sqrt = 0\n⊢ 0 < 1","tactic":"rw [h]","premises":[]},{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\nh : n.sqrt = 0\n⊢ 0 < 1","state_after":"no goals","tactic":"decide","premises":[]},{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\n⊢ n = 0 → n.sqrt = 0","state_after":"a b c d m k : ℕ\np q : ℕ → Prop\n⊢ sqrt 0 = 0","tactic":"rintro rfl","premises":[]},{"state_before":"a b c d m k : ℕ\np q : ℕ → Prop\n⊢ sqrt 0 = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Computability/TuringMachine.lean","commit":"","full_name":"Turing.TM1to1.exists_enc_dec","start":[1453,0],"end":[1462,91],"file_path":"Mathlib/Computability/TuringMachine.lean","tactics":[{"state_before":"Γ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","state_after":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","tactic":"rcases Finite.exists_equiv_fin Γ with ⟨n, ⟨e⟩⟩","premises":[{"full_name":"Finite.exists_equiv_fin","def_path":"Mathlib/Data/Finite/Defs.lean","def_pos":[85,8],"def_end_pos":[85,31]}]},{"state_before":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","state_after":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","tactic":"letI : DecidableEq Γ := e.decidableEq","premises":[{"full_name":"DecidableEq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[868,7],"def_end_pos":[868,18]},{"full_name":"Equiv.decidableEq","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[203,14],"def_end_pos":[203,25]}]},{"state_before":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","state_after":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","tactic":"let G : Fin n ↪ Fin n → Bool :=\n    ⟨fun a b ↦ a = b, fun a b h ↦\n      Bool.of_decide_true <| (congr_fun h b).trans <| Bool.decide_true rfl⟩","premises":[{"full_name":"Bool","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[567,10],"def_end_pos":[567,14]},{"full_name":"Bool.decide_true","def_path":"Mathlib/Data/Bool/Basic.lean","def_pos":[73,8],"def_end_pos":[73,19]},{"full_name":"Bool.of_decide_true","def_path":"Mathlib/Data/Bool/Basic.lean","def_pos":[76,8],"def_end_pos":[76,22]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Function.Embedding","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[21,10],"def_end_pos":[21,19]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","state_after":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\nH : Γ ↪ Vector Bool n := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin Bool n).symm.toEmbedding\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","tactic":"let H := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin _ _).symm.toEmbedding","premises":[{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"Equiv.toEmbedding","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[69,14],"def_end_pos":[69,31]},{"full_name":"Equiv.vectorEquivFin","def_path":"Mathlib/Data/Vector/Basic.lean","def_pos":[137,4],"def_end_pos":[137,31]},{"full_name":"Function.Embedding.trans","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[137,14],"def_end_pos":[137,19]}]},{"state_before":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\nH : Γ ↪ Vector Bool n := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin Bool n).symm.toEmbedding\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","state_after":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\nH : Γ ↪ Vector Bool n := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin Bool n).symm.toEmbedding\nenc : Γ ↪ Vector Bool n := H.setValue default (Vector.replicate n false)\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","tactic":"let enc := H.setValue default (Vector.replicate n false)","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.Embedding.setValue","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[174,4],"def_end_pos":[174,12]},{"full_name":"Inhabited.default","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[697,2],"def_end_pos":[697,9]},{"full_name":"Mathlib.Vector.replicate","def_path":"Mathlib/Data/Vector/Defs.lean","def_pos":[110,4],"def_end_pos":[110,13]}]},{"state_before":"case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\nH : Γ ↪ Vector Bool n := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin Bool n).symm.toEmbedding\nenc : Γ ↪ Vector Bool n := H.setValue default (Vector.replicate n false)\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a","state_after":"no goals","tactic":"exact ⟨_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Function.Embedding.inj'","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[25,2],"def_end_pos":[25,6]},{"full_name":"Function.Embedding.setValue_eq","def_path":"Mathlib/Logic/Embedding/Basic.lean","def_pos":[181,8],"def_end_pos":[181,19]},{"full_name":"Function.invFun","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[367,18],"def_end_pos":[367,24]},{"full_name":"Function.leftInverse_invFun","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[391,8],"def_end_pos":[391,26]}]}]}
{"url":"Mathlib/Data/Nat/Factorization/Basic.lean","commit":"","full_name":"Nat.ord_proj_dvd_ord_proj_of_dvd","start":[260,0],"end":[265,50],"file_path":"Mathlib/Data/Nat/Factorization/Basic.lean","tactics":[{"state_before":"a✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p","state_after":"case inl\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : ¬Prime p\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p\n\ncase inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p","tactic":"rcases em' p.Prime with (pp | pp)","premises":[{"full_name":"Nat.Prime","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[36,4],"def_end_pos":[36,9]},{"full_name":"em'","def_path":"Mathlib/Logic/Basic.lean","def_pos":[157,8],"def_end_pos":[157,11]}]},{"state_before":"case inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p","state_after":"case inr.inl\na b✝ m n p✝ b : ℕ\nhb0 : b ≠ 0\np : ℕ\npp : Prime p\nhab : 0 ∣ b\n⊢ p ^ (factorization 0) p ∣ p ^ b.factorization p\n\ncase inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\nha0 : a ≠ 0\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p","tactic":"rcases eq_or_ne a 0 with (rfl | ha0)","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\nha0 : a ≠ 0\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p","state_after":"case inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\nha0 : a ≠ 0\n⊢ a.factorization p ≤ b.factorization p","tactic":"rw [pow_dvd_pow_iff_le_right pp.one_lt]","premises":[{"full_name":"Nat.Prime.one_lt","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Nat.pow_dvd_pow_iff_le_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[756,8],"def_end_pos":[756,32]}]},{"state_before":"case inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\nha0 : a ≠ 0\n⊢ a.factorization p ≤ b.factorization p","state_after":"no goals","tactic":"exact (factorization_le_iff_dvd ha0 hb0).2 hab p","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Nat.factorization_le_iff_dvd","def_path":"Mathlib/Data/Nat/Factorization/Defs.lean","def_pos":[150,8],"def_end_pos":[150,32]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Bochner.lean","commit":"","full_name":"MeasureTheory.integral_exp_pos","start":[1154,0],"end":[1160,96],"file_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\n⊢ 0 < ∫ (x : α), Real.exp (f x) ∂μ","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\n⊢ 0 < μ (Function.support fun x => Real.exp (f x))","tactic":"rw [integral_pos_iff_support_of_nonneg (fun x ↦ (Real.exp_pos _).le) hf]","premises":[{"full_name":"MeasureTheory.integral_pos_iff_support_of_nonneg","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[1150,8],"def_end_pos":[1150,42]},{"full_name":"Real.exp_pos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[982,8],"def_end_pos":[982,15]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\n⊢ 0 < μ (Function.support fun x => Real.exp (f x))","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\n⊢ (Function.support fun x => Real.exp (f x)) = univ","tactic":"suffices (Function.support fun x ↦ Real.exp (f x)) = Set.univ by simp [this, hμ.out]","premises":[{"full_name":"Function.support","def_path":"Mathlib/Algebra/Group/Support.lean","def_pos":[28,2],"def_end_pos":[28,13]},{"full_name":"NeZero.out","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[25,2],"def_end_pos":[25,5]},{"full_name":"Real.exp","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[102,11],"def_end_pos":[102,14]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\n⊢ (Function.support fun x => Real.exp (f x)) = univ","state_after":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\nx : α\n⊢ (x ∈ Function.support fun x => Real.exp (f x)) ↔ x ∈ univ","tactic":"ext1 x","premises":[]},{"state_before":"case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\nhE : CompleteSpace E\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : SMulCommClass ℝ 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf✝ g : α → E\nm : MeasurableSpace α\nμ✝ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nμ : Measure α\nf : α → ℝ\nhμ : NeZero μ\nhf : Integrable (fun x => Real.exp (f x)) μ\nx : α\n⊢ (x ∈ Function.support fun x => Real.exp (f x)) ↔ x ∈ univ","state_after":"no goals","tactic":"simp only [Function.mem_support, ne_eq, (Real.exp_pos _).ne', not_false_eq_true, Set.mem_univ]","premises":[{"full_name":"Function.mem_support","def_path":"Mathlib/Algebra/Group/Support.lean","def_pos":[43,2],"def_end_pos":[43,13]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Real.exp_pos","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[982,8],"def_end_pos":[982,15]},{"full_name":"Set.mem_univ","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[80,28],"def_end_pos":[80,36]},{"full_name":"ne_eq","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[89,16],"def_end_pos":[89,21]},{"full_name":"not_false_eq_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[134,16],"def_end_pos":[134,33]}]}]}
{"url":"Mathlib/Data/DFinsupp/Basic.lean","commit":"","full_name":"DFinsupp.subtypeSupportEqEquiv_symm_apply_coe","start":[980,0],"end":[998,27],"file_path":"Mathlib/Data/DFinsupp/Basic.lean","tactics":[{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\nf : (i : { x // x ∈ s }) → { x // x ≠ 0 }\ni : ι\n⊢ i ∈ (mk s fun i => ↑(f i)).support ↔ i ∈ s","state_after":"no goals","tactic":"calc\n      i ∈ support (mk s fun i ↦ (f i).1) ↔ ∃ h : i ∈ s, (f ⟨i, h⟩).1 ≠ 0 := by simp\n      _ ↔ ∃ _ : i ∈ s, True := exists_congr fun h ↦ (iff_true _).mpr (f _).2\n      _ ↔ i ∈ s := by simp","premises":[{"full_name":"DFinsupp.mk","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[494,4],"def_end_pos":[494,6]},{"full_name":"DFinsupp.support","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[949,4],"def_end_pos":[949,11]},{"full_name":"Eq.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[675,20],"def_end_pos":[675,26]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]},{"full_name":"True","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[189,10],"def_end_pos":[189,14]},{"full_name":"exists_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[210,8],"def_end_pos":[210,20]},{"full_name":"iff_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[127,16],"def_end_pos":[127,24]}]},{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\n⊢ Function.LeftInverse (fun f => ⟨mk s fun i => ↑(f i), ⋯⟩) fun x =>\n    match (motive := { f // f.support = s } → (i : { x // x ∈ s }) → { x // x ≠ 0 }) x with\n    | ⟨f, hf⟩ => fun x =>\n      match x with\n      | ⟨i, hi⟩ => ⟨f i, ⋯⟩","state_after":"case mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\n⊢ (fun f_1 => ⟨mk f.support fun i => ↑(f_1 i), ⋯⟩)\n      ((fun x =>\n          match (motive := { f_1 // f_1.support = f.support } → (i : { x // x ∈ f.support }) → { x // x ≠ 0 }) x with\n          | ⟨f_1, hf⟩ => fun x =>\n            match x with\n            | ⟨i, hi⟩ => ⟨f_1 i, ⋯⟩)\n        ⟨f, ⋯⟩) =\n    ⟨f, ⋯⟩","tactic":"rintro ⟨f, rfl⟩","premises":[]},{"state_before":"case mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\n⊢ (fun f_1 => ⟨mk f.support fun i => ↑(f_1 i), ⋯⟩)\n      ((fun x =>\n          match (motive := { f_1 // f_1.support = f.support } → (i : { x // x ∈ f.support }) → { x // x ≠ 0 }) x with\n          | ⟨f_1, hf⟩ => fun x =>\n            match x with\n            | ⟨i, hi⟩ => ⟨f_1 i, ⋯⟩)\n        ⟨f, ⋯⟩) =\n    ⟨f, ⋯⟩","state_after":"case mk.a.h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\n⊢ ↑((fun f_1 => ⟨mk f.support fun i => ↑(f_1 i), ⋯⟩)\n          ((fun x =>\n              match (motive := { f_1 // f_1.support = f.support } → (i : { x // x ∈ f.support }) → { x // x ≠ 0 })\n                x with\n              | ⟨f_1, hf⟩ => fun x =>\n                match x with\n                | ⟨i, hi⟩ => ⟨f_1 i, ⋯⟩)\n            ⟨f, ⋯⟩))\n      i =\n    ↑⟨f, ⋯⟩ i","tactic":"ext i","premises":[]},{"state_before":"case mk.a.h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\n⊢ ↑((fun f_1 => ⟨mk f.support fun i => ↑(f_1 i), ⋯⟩)\n          ((fun x =>\n              match (motive := { f_1 // f_1.support = f.support } → (i : { x // x ∈ f.support }) → { x // x ≠ 0 })\n                x with\n              | ⟨f_1, hf⟩ => fun x =>\n                match x with\n                | ⟨i, hi⟩ => ⟨f_1 i, ⋯⟩)\n            ⟨f, ⋯⟩))\n      i =\n    ↑⟨f, ⋯⟩ i","state_after":"no goals","tactic":"simpa using Eq.symm","premises":[{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]}]},{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\nf : (i : { x // x ∈ s }) → { x // x ≠ 0 }\n⊢ (fun x =>\n        match (motive := { f // f.support = s } → (i : { x // x ∈ s }) → { x // x ≠ 0 }) x with\n        | ⟨f, hf⟩ => fun x =>\n          match x with\n          | ⟨i, hi⟩ => ⟨f i, ⋯⟩)\n      ((fun f => ⟨mk s fun i => ↑(f i), ⋯⟩) f) =\n    f","state_after":"case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\nf : (i : { x // x ∈ s }) → { x // x ≠ 0 }\nx✝ : { x // x ∈ s }\n⊢ (fun x =>\n        match (motive := { f // f.support = s } → (i : { x // x ∈ s }) → { x // x ≠ 0 }) x with\n        | ⟨f, hf⟩ => fun x =>\n          match x with\n          | ⟨i, hi⟩ => ⟨f i, ⋯⟩)\n      ((fun f => ⟨mk s fun i => ↑(f i), ⋯⟩) f) x✝ =\n    f x✝","tactic":"ext1","premises":[]},{"state_before":"case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\nf : (i : { x // x ∈ s }) → { x // x ≠ 0 }\nx✝ : { x // x ∈ s }\n⊢ (fun x =>\n        match (motive := { f // f.support = s } → (i : { x // x ∈ s }) → { x // x ≠ 0 }) x with\n        | ⟨f, hf⟩ => fun x =>\n          match x with\n          | ⟨i, hi⟩ => ⟨f i, ⋯⟩)\n      ((fun f => ⟨mk s fun i => ↑(f i), ⋯⟩) f) x✝ =\n    f x✝","state_after":"case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\nf : (i : { x // x ∈ s }) → { x // x ≠ 0 }\nx✝ : { x // x ∈ s }\n⊢ f ⟨↑x✝, ⋯⟩ = f x✝","tactic":"simp [Subtype.eta]","premises":[{"full_name":"Subtype.eta","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1113,8],"def_end_pos":[1113,11]}]},{"state_before":"case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Finset ι\nf : (i : { x // x ∈ s }) → { x // x ≠ 0 }\nx✝ : { x // x ∈ s }\n⊢ f ⟨↑x✝, ⋯⟩ = f x✝","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/MetricSpace/Infsep.lean","commit":"","full_name":"Set.Nontrivial.infsep_exists_of_finite","start":[433,0],"end":[441,54],"file_path":"Mathlib/Topology/MetricSpace/Infsep.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : s.Nontrivial\n⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y","state_after":"no goals","tactic":"classical\n    cases nonempty_fintype s\n    simp_rw [hs.infsep_of_fintype]\n    rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with\n      ⟨w, hxy, hed⟩\n    simp_rw [mem_toFinset] at hxy\n    exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Dist.dist","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[80,2],"def_end_pos":[80,6]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Finset.exists_mem_eq_inf'","def_path":"Mathlib/Data/Finset/Lattice.lean","def_pos":[1158,8],"def_end_pos":[1158,26]},{"full_name":"Function.uncurry","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[189,4],"def_end_pos":[189,11]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"Set.Nontrivial.infsep_of_fintype","def_path":"Mathlib/Topology/MetricSpace/Infsep.lean","def_pos":[398,8],"def_end_pos":[398,36]},{"full_name":"Set.mem_toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[540,8],"def_end_pos":[540,20]},{"full_name":"Set.offDiag","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[203,4],"def_end_pos":[203,11]},{"full_name":"Set.toFinset","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[532,4],"def_end_pos":[532,12]},{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]}]}
{"url":"Mathlib/Topology/Algebra/Group/Basic.lean","commit":"","full_name":"TopologicalAddGroup.t1Space","start":[1305,0],"end":[1307,55],"file_path":"Mathlib/Topology/Algebra/Group/Basic.lean","tactics":[{"state_before":"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : ContinuousMul G\nh : IsClosed {1}\nx : G\n⊢ IsClosed {x}","state_after":"no goals","tactic":"simpa using isClosedMap_mul_right x _ h","premises":[{"full_name":"isClosedMap_mul_right","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[108,8],"def_end_pos":[108,29]}]}]}
{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"norm_div_le_norm_div_add_norm_div","start":[397,0],"end":[399,57],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b c : E\n⊢ ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖","state_after":"no goals","tactic":"simpa only [dist_eq_norm_div] using dist_triangle a b c","premises":[{"full_name":"dist_eq_norm_div","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[343,8],"def_end_pos":[343,24]},{"full_name":"dist_triangle","def_path":"Mathlib/Topology/MetricSpace/Pseudo/Defs.lean","def_pos":[172,8],"def_end_pos":[172,21]}]}]}
{"url":"Mathlib/RingTheory/Valuation/Quotient.lean","commit":"","full_name":"Valuation.supp_quot","start":[58,0],"end":[67,12],"file_path":"Mathlib/RingTheory/Valuation/Quotient.lean","tactics":[{"state_before":"R : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nJ : Ideal R\nhJ : J ≤ v.supp\n⊢ (v.onQuot hJ).supp = Ideal.map (Ideal.Quotient.mk J) v.supp","state_after":"case a\nR : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nJ : Ideal R\nhJ : J ≤ v.supp\n⊢ (v.onQuot hJ).supp ≤ Ideal.map (Ideal.Quotient.mk J) v.supp\n\ncase a\nR : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nJ : Ideal R\nhJ : J ≤ v.supp\n⊢ Ideal.map (Ideal.Quotient.mk J) v.supp ≤ (v.onQuot hJ).supp","tactic":"apply le_antisymm","premises":[{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
{"url":"Mathlib/Data/Set/Function.lean","commit":"","full_name":"Set.InjOn.image_diff","start":[671,0],"end":[674,75],"file_path":"Mathlib/Data/Set/Function.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt✝ t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nt : Set α\nh : InjOn f s\n⊢ f '' (s \\ t) = f '' s \\ f '' (s ∩ t)","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt✝ t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nt : Set α\nh : InjOn f s\n⊢ Disjoint (f '' (s \\ t)) (f '' (s ∩ t))","tactic":"refine subset_antisymm (subset_diff.2 ⟨image_subset f diff_subset, ?_⟩)\n    (diff_subset_iff.2 (by rw [← image_union, inter_union_diff]))","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]},{"full_name":"Set.diff_subset_iff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1517,8],"def_end_pos":[1517,23]},{"full_name":"Set.image_subset","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[250,8],"def_end_pos":[250,20]},{"full_name":"Set.image_union","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[257,8],"def_end_pos":[257,19]},{"full_name":"Set.inter_union_diff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1468,8],"def_end_pos":[1468,24]},{"full_name":"Set.subset_diff","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1277,6],"def_end_pos":[1277,17]},{"full_name":"subset_antisymm","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[548,6],"def_end_pos":[548,21]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁ s₂ : Set α\nt✝ t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nt : Set α\nh : InjOn f s\n⊢ Disjoint (f '' (s \\ t)) (f '' (s ∩ t))","state_after":"no goals","tactic":"exact Disjoint.image disjoint_sdiff_inter h diff_subset inter_subset_left","premises":[{"full_name":"Disjoint.image","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[666,8],"def_end_pos":[666,29]},{"full_name":"Set.diff_subset","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1432,8],"def_end_pos":[1432,19]},{"full_name":"Set.disjoint_sdiff_inter","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1259,8],"def_end_pos":[1259,28]},{"full_name":"Set.inter_subset_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[761,8],"def_end_pos":[761,25]}]}]}
{"url":"Mathlib/Data/Part.lean","commit":"","full_name":"Part.inter_mem_inter","start":[683,0],"end":[684,49],"file_path":"Mathlib/Data/Part.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Inter α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ma ∩ mb ∈ a ∩ b","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Inter α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1 ∈ a, ∃ a ∈ b, a_1 ∩ a = ma ∩ mb","tactic":"simp [inter_def]","premises":[{"full_name":"Part.inter_def","def_path":"Mathlib/Data/Part.lean","def_pos":[601,8],"def_end_pos":[601,17]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Inter α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1 ∈ a, ∃ a ∈ b, a_1 ∩ a = ma ∩ mb","state_after":"no goals","tactic":"aesop","premises":[]}]}
{"url":"Mathlib/GroupTheory/MonoidLocalization/Order.lean","commit":"","full_name":"Localization.mkOrderEmbedding_apply","start":[102,0],"end":[108,63],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Order.lean","tactics":[{"state_before":"α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns : Submonoid α\na₁ b₁ : α\na₂ b₂ b✝ : ↥s\na b : α\n⊢ { toFun := fun a => mk a b✝, inj' := ⋯ } a ≤ { toFun := fun a => mk a b✝, inj' := ⋯ } b ↔ a ≤ b","state_after":"no goals","tactic":"simp [-mk_eq_monoidOf_mk', mk_le_mk]","premises":[{"full_name":"Localization.mk_eq_monoidOf_mk'","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[1334,8],"def_end_pos":[1334,26]},{"full_name":"Localization.mk_le_mk","def_path":"Mathlib/GroupTheory/MonoidLocalization/Order.lean","def_pos":[50,8],"def_end_pos":[50,16]}]}]}
{"url":"Mathlib/Data/Matroid/Closure.lean","commit":"","full_name":"Matroid.Indep.insert_diff_indep_iff","start":[371,0],"end":[377,26],"file_path":"Mathlib/Data/Matroid/Closure.lean","tactics":[{"state_before":"ι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X Y : Set α\ne : α\nι : Sort u_3\nI J B : Set α\nf x y : α\nhI : M.Indep (I \\ {e})\nheI : e ∈ I\n⊢ M.Indep (insert f I \\ {e}) ↔ f ∈ M.E \\ M.closure (I \\ {e}) ∨ f ∈ I","state_after":"case inl\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X Y : Set α\ne : α\nι : Sort u_3\nI J B : Set α\nx y : α\nhI : M.Indep (I \\ {e})\nheI : e ∈ I\n⊢ M.Indep (insert e I \\ {e}) ↔ e ∈ M.E \\ M.closure (I \\ {e}) ∨ e ∈ I\n\ncase inr\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X Y : Set α\ne : α\nι : Sort u_3\nI J B : Set α\nf x y : α\nhI : M.Indep (I \\ {e})\nheI : e ∈ I\nhne : e ≠ f\n⊢ M.Indep (insert f I \\ {e}) ↔ f ∈ M.E \\ M.closure (I \\ {e}) ∨ f ∈ I","tactic":"obtain rfl | hne := eq_or_ne e f","premises":[{"full_name":"eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[167,8],"def_end_pos":[167,16]}]},{"state_before":"case inr\nι✝ : Type u_1\nα : Type u_2\nM : Matroid α\nF X Y : Set α\ne : α\nι : Sort u_3\nI J B : Set α\nf x y : α\nhI : M.Indep (I \\ {e})\nheI : e ∈ I\nhne : e ≠ f\n⊢ M.Indep (insert f I \\ {e}) ↔ f ∈ M.E \\ M.closure (I \\ {e}) ∨ f ∈ I","state_after":"no goals","tactic":"rw [← insert_diff_singleton_comm hne.symm, hI.insert_indep_iff, mem_diff_singleton,\n    and_iff_left hne.symm]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Matroid.Indep.insert_indep_iff","def_path":"Mathlib/Data/Matroid/Closure.lean","def_pos":[360,6],"def_end_pos":[360,28]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"Set.insert_diff_singleton_comm","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1624,8],"def_end_pos":[1624,34]},{"full_name":"Set.mem_diff_singleton","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1639,8],"def_end_pos":[1639,26]},{"full_name":"and_iff_left","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[69,8],"def_end_pos":[69,20]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Smeval.lean","commit":"","full_name":"Polynomial.smeval_smul","start":[117,0],"end":[123,68],"file_path":"Mathlib/Algebra/Polynomial/Smeval.lean","tactics":[{"state_before":"R : Type u_1\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nr : R\n⊢ (r • p).smeval x = r • p.smeval x","state_after":"no goals","tactic":"induction p using Polynomial.induction_on' with\n  | h_add p q ph qh =>\n    rw [smul_add, smeval_add, ph, qh, ← smul_add, smeval_add]\n  | h_monomial n a =>\n    rw [smul_monomial, smeval_monomial, smeval_monomial, smul_assoc]","premises":[{"full_name":"Polynomial.induction_on'","def_path":"Mathlib/Algebra/Polynomial/Induction.lean","def_pos":[60,18],"def_end_pos":[60,31]},{"full_name":"Polynomial.smeval_add","def_path":"Mathlib/Algebra/Polynomial/Smeval.lean","def_pos":[103,8],"def_end_pos":[103,18]},{"full_name":"Polynomial.smeval_monomial","def_path":"Mathlib/Algebra/Polynomial/Smeval.lean","def_pos":[59,8],"def_end_pos":[59,23]},{"full_name":"Polynomial.smul_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[409,8],"def_end_pos":[409,21]},{"full_name":"smul_add","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[130,8],"def_end_pos":[130,16]},{"full_name":"smul_assoc","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[219,6],"def_end_pos":[219,16]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/CoverLifting.lean","commit":"","full_name":"CategoryTheory.ran_isSheaf_of_isCocontinuous","start":[213,0],"end":[225,57],"file_path":"Mathlib/CategoryTheory/Sites/CoverLifting.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\n⊢ Presheaf.IsSheaf K (G.op.ran.obj ℱ.val)","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\n⊢ ∀ (X : D) (S : K.Cover X), Nonempty (IsLimit (S.multifork (G.op.ran.obj ℱ.val)))","tactic":"rw [Presheaf.isSheaf_iff_multifork]","premises":[{"full_name":"CategoryTheory.Presheaf.isSheaf_iff_multifork","def_path":"Mathlib/CategoryTheory/Sites/Sheaf.lean","def_pos":[534,8],"def_end_pos":[534,29]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\n⊢ ∀ (X : D) (S : K.Cover X), Nonempty (IsLimit (S.multifork (G.op.ran.obj ℱ.val)))","state_after":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\nX : D\nS : K.Cover X\n⊢ Nonempty (IsLimit (S.multifork (G.op.ran.obj ℱ.val)))","tactic":"intros X S","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\nX : D\nS : K.Cover X\n⊢ Nonempty (IsLimit (S.multifork (G.op.ran.obj ℱ.val)))","state_after":"no goals","tactic":"exact ⟨RanIsSheafOfIsCocontinuous.isLimitMultifork ℱ.2\n    (G.op.isPointwiseRightKanExtensionRanCounit ℱ.val) S⟩","premises":[{"full_name":"CategoryTheory.Functor.isPointwiseRightKanExtensionRanCounit","def_path":"Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.lean","def_pos":[160,18],"def_end_pos":[160,55]},{"full_name":"CategoryTheory.Functor.op","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[159,14],"def_end_pos":[159,16]},{"full_name":"CategoryTheory.RanIsSheafOfIsCocontinuous.isLimitMultifork","def_path":"Mathlib/CategoryTheory/Sites/CoverLifting.lean","def_pos":[205,4],"def_end_pos":[205,20]},{"full_name":"CategoryTheory.Sheaf.cond","def_path":"Mathlib/CategoryTheory/Sites/Sheaf.lean","def_pos":[309,2],"def_end_pos":[309,6]},{"full_name":"CategoryTheory.Sheaf.val","def_path":"Mathlib/CategoryTheory/Sites/Sheaf.lean","def_pos":[307,2],"def_end_pos":[307,5]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","commit":"","full_name":"Submodule.IsOrtho.map","start":[325,0],"end":[329,9],"file_path":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : U ⟂ V\n⊢ Submodule.map f U ⟂ Submodule.map f V","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫_𝕜 = 0\n⊢ ∀ u ∈ Submodule.map f U, ∀ v ∈ Submodule.map f V, ⟪u, v⟫_𝕜 = 0","tactic":"rw [isOrtho_iff_inner_eq] at *","premises":[{"full_name":"Submodule.isOrtho_iff_inner_eq","def_path":"Mathlib/Analysis/InnerProductSpace/Orthogonal.lean","def_pos":[235,8],"def_end_pos":[235,28]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫_𝕜 = 0\n⊢ ∀ u ∈ Submodule.map f U, ∀ v ∈ Submodule.map f V, ⟪u, v⟫_𝕜 = 0","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫_𝕜 = 0\n⊢ ∀ a ∈ U, ∀ a_2 ∈ V, ⟪a, a_2⟫_𝕜 = 0","tactic":"simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,\n    LinearIsometry.inner_map_map]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LinearIsometry.inner_map_map","def_path":"Mathlib/Analysis/InnerProductSpace/Basic.lean","def_pos":[1088,8],"def_end_pos":[1088,36]},{"full_name":"Submodule.mem_map","def_path":"Mathlib/Algebra/Module/Submodule/Map.lean","def_pos":[85,8],"def_end_pos":[85,15]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]},{"full_name":"forall_apply_eq_imp_iff₂","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[327,16],"def_end_pos":[327,40]},{"full_name":"forall_exists_index","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[189,16],"def_end_pos":[189,35]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫_𝕜 = 0\n⊢ ∀ a ∈ U, ∀ a_2 ∈ V, ⟪a, a_2⟫_𝕜 = 0","state_after":"no goals","tactic":"exact h","premises":[]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.length_lookmap","start":[2100,0],"end":[2101,67],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → Option α\nl : List α\n⊢ (lookmap f l).length = l.length","state_after":"case h\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → Option α\nl : List α\n⊢ ∀ (a b : α), b ∈ f a → () = ()","tactic":"rw [← length_map, lookmap_map_eq _ fun _ => (), length_map]","premises":[{"full_name":"List.length_map","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[725,16],"def_end_pos":[725,26]},{"full_name":"List.lookmap_map_eq","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[2089,8],"def_end_pos":[2089,22]},{"full_name":"Unit.unit","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[148,24],"def_end_pos":[148,33]}]},{"state_before":"case h\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → Option α\nl : List α\n⊢ ∀ (a b : α), b ∈ f a → () = ()","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Matrix/Basis.lean","commit":"","full_name":"Matrix.induction_on'","start":[90,0],"end":[98,21],"file_path":"Mathlib/Data/Matrix/Basis.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\n⊢ P M","state_after":"case intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝ : Fintype m\n⊢ P M","tactic":"cases nonempty_fintype m","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝ : Fintype m\n⊢ P M","state_after":"case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ P M","tactic":"cases nonempty_fintype n","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ P M","state_after":"case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ P (∑ x ∈ Finset.univ ×ˢ Finset.univ, stdBasisMatrix x.1 x.2 (M x.1 x.2))","tactic":"rw [matrix_eq_sum_std_basis M, ← Finset.sum_product']","premises":[{"full_name":"Finset.sum_product'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[783,2],"def_end_pos":[783,13]},{"full_name":"Matrix.matrix_eq_sum_std_basis","def_path":"Mathlib/Data/Matrix/Basis.lean","def_pos":[59,8],"def_end_pos":[59,31]}]},{"state_before":"case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ P (∑ x ∈ Finset.univ ×ˢ Finset.univ, stdBasisMatrix x.1 x.2 (M x.1 x.2))","state_after":"case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ ∀ x ∈ Finset.univ ×ˢ Finset.univ, P (stdBasisMatrix x.1 x.2 (M x.1 x.2))","tactic":"apply Finset.sum_induction _ _ h_add h_zero","premises":[{"full_name":"Finset.sum_induction","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1401,2],"def_end_pos":[1401,13]}]}]}
{"url":"Mathlib/NumberTheory/Primorial.lean","commit":"","full_name":"primorial_add","start":[47,0],"end":[51,65],"file_path":"Mathlib/NumberTheory/Primorial.lean","tactics":[{"state_before":"m n : ℕ\n⊢ (m + n)# = m# * ∏ p ∈ filter Nat.Prime (Ico (m + 1) (m + n + 1)), p","state_after":"case hab\nm n : ℕ\n⊢ 0 ≤ m + 1\n\ncase hbc\nm n : ℕ\n⊢ m + 1 ≤ m + n + 1\n\nm n : ℕ\n⊢ Disjoint (filter Nat.Prime (Ico 0 (m + 1))) (filter Nat.Prime (Ico (m + 1) (m + n + 1)))","tactic":"rw [primorial, primorial, ← Ico_zero_eq_range, ← prod_union, ← filter_union, Ico_union_Ico_eq_Ico]","premises":[{"full_name":"Finset.Ico_union_Ico_eq_Ico","def_path":"Mathlib/Order/Interval/Finset/Basic.lean","def_pos":[659,8],"def_end_pos":[659,28]},{"full_name":"Finset.filter_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2297,8],"def_end_pos":[2297,20]},{"full_name":"Finset.prod_union","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[411,8],"def_end_pos":[411,18]},{"full_name":"Nat.Ico_zero_eq_range","def_path":"Mathlib/Order/Interval/Finset/Nat.lean","def_pos":[59,8],"def_end_pos":[59,25]},{"full_name":"primorial","def_path":"Mathlib/NumberTheory/Primorial.lean","def_pos":[34,4],"def_end_pos":[34,13]}]},{"state_before":"case hab\nm n : ℕ\n⊢ 0 ≤ m + 1\n\ncase hbc\nm n : ℕ\n⊢ m + 1 ≤ m + n + 1\n\nm n : ℕ\n⊢ Disjoint (filter Nat.Prime (Ico 0 (m + 1))) (filter Nat.Prime (Ico (m + 1) (m + n + 1)))","state_after":"no goals","tactic":"exacts [Nat.zero_le _, add_le_add_right (Nat.le_add_right _ _) _,\n    disjoint_filter_filter <| Ico_disjoint_Ico_consecutive _ _ _]","premises":[{"full_name":"Finset.Ico_disjoint_Ico_consecutive","def_path":"Mathlib/Order/Interval/Finset/Basic.lean","def_pos":[453,8],"def_end_pos":[453,36]},{"full_name":"Finset.disjoint_filter_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2254,8],"def_end_pos":[2254,30]},{"full_name":"Nat.le_add_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[395,8],"def_end_pos":[395,20]},{"full_name":"Nat.zero_le","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1663,8],"def_end_pos":[1663,19]},{"full_name":"add_le_add_right","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[74,31],"def_end_pos":[74,47]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean","commit":"","full_name":"Matrix.SpecialLinearGroup.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero","start":[368,0],"end":[375,46],"file_path":"Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean","tactics":[{"state_before":"n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\ng : SL(2, R)\nhg : ↑g 1 0 = 0\na b : R\nh : a ≠ 0\n⊢ !![a, b; 0, a⁻¹].det = 1","state_after":"no goals","tactic":"simp [h]","premises":[]},{"state_before":"n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\ng : SL(2, R)\nhg : ↑g 1 0 = 0\n⊢ ∃ a b, ∃ (h : a ≠ 0), g = ⟨!![a, b; 0, a⁻¹], ⋯⟩","state_after":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑⟨!![a, b; c, d], ⋯⟩ 1 0 = 0\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩","tactic":"induction' g using Matrix.SpecialLinearGroup.fin_two_induction with a b c d h_det","premises":[{"full_name":"Matrix.SpecialLinearGroup.fin_two_induction","def_path":"Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean","def_pos":[361,8],"def_end_pos":[361,25]}]},{"state_before":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑⟨!![a, b; c, d], ⋯⟩ 1 0 = 0\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩","state_after":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩","tactic":"replace hg : c = 0 := by simpa using hg","premises":[]},{"state_before":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩","state_after":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩","tactic":"have had : a * d = 1 := by rwa [hg, mul_zero, sub_zero] at h_det","premises":[{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"sub_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[353,2],"def_end_pos":[353,13]}]},{"state_before":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩","state_after":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a, b; 0, a⁻¹], ⋯⟩","tactic":"refine ⟨a, b, left_ne_zero_of_mul_eq_one had, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"left_ne_zero_of_mul_eq_one","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[131,8],"def_end_pos":[131,34]}]},{"state_before":"case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a, b; 0, a⁻¹], ⋯⟩","state_after":"no goals","tactic":"simp_rw [eq_inv_of_mul_eq_one_right had, hg]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"eq_inv_of_mul_eq_one_right","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[369,8],"def_end_pos":[369,34]}]}]}
{"url":"Mathlib/Data/Set/Finite.lean","commit":"","full_name":"Set.Finite.toFinset_diff","start":[227,0],"end":[230,6],"file_path":"Mathlib/Data/Set/Finite.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs✝ : s.Finite\nht✝ : t.Finite\ninst✝ : DecidableEq α\nhs : s.Finite\nht : t.Finite\nh : (s \\ t).Finite\n⊢ h.toFinset = hs.toFinset \\ ht.toFinset","state_after":"case a\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs✝ : s.Finite\nht✝ : t.Finite\ninst✝ : DecidableEq α\nhs : s.Finite\nht : t.Finite\nh : (s \\ t).Finite\na✝ : α\n⊢ a✝ ∈ h.toFinset ↔ a✝ ∈ hs.toFinset \\ ht.toFinset","tactic":"ext","premises":[]},{"state_before":"case a\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs✝ : s.Finite\nht✝ : t.Finite\ninst✝ : DecidableEq α\nhs : s.Finite\nht : t.Finite\nh : (s \\ t).Finite\na✝ : α\n⊢ a✝ ∈ h.toFinset ↔ a✝ ∈ hs.toFinset \\ ht.toFinset","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/NumberTheory/Dioph.lean","commit":"","full_name":"Dioph.pow_dioph","start":[668,0],"end":[694,73],"file_path":"Mathlib/NumberTheory/Dioph.lean","tactics":[{"state_before":"α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\n⊢ DiophFn fun v => f v ^ g v","state_after":"α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\nproof :\n  Dioph\n    (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪\n      {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩\n        (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪\n          {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩\n            {v |\n              ∃ x,\n                (x :: v) ∈\n                  {v |\n                    ∃ x,\n                      (x :: v) ∈\n                        {v |\n                          ∃ x,\n                            (x :: v) ∈\n                              {v |\n                                ∃ x,\n                                  (x :: v) ∈\n                                    {v |\n                                      ∃ x,\n                                        (x :: v) ∈\n                                          {v |\n                                            ∃ x,\n                                              (x :: v) ∈\n                                                {v |\n                                                    (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v =>\n                                                      ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩\n                                                  ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩\n                                                    ((fun v =>\n                                                        const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 =\n                                                          v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩\n                                                      ({v | v &6 < v &3} ∩\n                                                        ({v | v &7 ≤ v &5} ∩\n                                                          ({v | v &8 ≤ v &5} ∩ fun v =>\n                                                            v &4 * v &4 -\n                                                                ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                        (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) -\n                                                                      const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                    (v &5 * v &2) *\n                                                                  (v &5 * v &2) =\n                                                              const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))\n⊢ DiophFn fun v => f v ^ g v","tactic":"have proof :=\n    let D_pell := pell_dioph.reindex_dioph (Fin2 9) [&4, &8, &1, &0]\n    (D&2 D= D.0 D∧ D&0 D= D.1) D∨ (D.0 D< D&2 D∧\n    ((D&1 D= D.0 D∧ D&0 D= D.0) D∨ (D.0 D< D&1 D∧\n    ((D∃) 3 <| (D∃) 4 <| (D∃) 5 <| (D∃) 6 <| (D∃) 7 <| (D∃) 8 <| D_pell D∧\n    (D≡ (D&1) (D&0 D* (D&4 D- D&7) D+ D&6) (D&3)) D∧\n    D.2 D* D&4 D* D&7 D= D&3 D+ (D&7 D* D&7 D+ D.1) D∧\n    D&6 D< D&3 D∧ D&7 D≤ D&5 D∧ D&8 D≤ D&5 D∧\n    D&4 D* D&4 D- ((D&5 D+ D.1) D* (D&5 D+ D.1) D- D.1) D* (D&5 D* D&2) D* (D&5 D* D&2) D= D.1))))","premises":[{"full_name":"Dioph.add_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[539,8],"def_end_pos":[539,17]},{"full_name":"Dioph.const_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[519,8],"def_end_pos":[519,19]},{"full_name":"Dioph.eq_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[532,8],"def_end_pos":[532,16]},{"full_name":"Dioph.inter","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[337,8],"def_end_pos":[337,13]},{"full_name":"Dioph.le_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[549,8],"def_end_pos":[549,16]},{"full_name":"Dioph.lt_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[555,8],"def_end_pos":[555,16]},{"full_name":"Dioph.modEq_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[604,8],"def_end_pos":[604,19]},{"full_name":"Dioph.mul_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[544,8],"def_end_pos":[544,17]},{"full_name":"Dioph.pell_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[634,8],"def_end_pos":[634,18]},{"full_name":"Dioph.proj_dioph_of_nat","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[514,8],"def_end_pos":[514,25]},{"full_name":"Dioph.reindex_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[292,8],"def_end_pos":[292,21]},{"full_name":"Dioph.sub_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[564,8],"def_end_pos":[564,17]},{"full_name":"Dioph.union","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[339,8],"def_end_pos":[339,13]},{"full_name":"Dioph.vec_ex1_dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[452,8],"def_end_pos":[452,21]},{"full_name":"Fin2","def_path":"Mathlib/Data/Fin/Fin2.lean","def_pos":[36,10],"def_end_pos":[36,14]},{"full_name":"Fin2.ofNat'","def_path":"Mathlib/Data/Fin/Fin2.lean","def_pos":[111,4],"def_end_pos":[111,10]},{"full_name":"Vector3.cons","def_path":"Mathlib/Data/Vector3.lean","def_pos":[42,4],"def_end_pos":[42,8]},{"full_name":"Vector3.nil","def_path":"Mathlib/Data/Vector3.lean","def_pos":[37,4],"def_end_pos":[37,7]}]},{"state_before":"α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\nproof :\n  Dioph\n    (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪\n      {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩\n        (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪\n          {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩\n            {v |\n              ∃ x,\n                (x :: v) ∈\n                  {v |\n                    ∃ x,\n                      (x :: v) ∈\n                        {v |\n                          ∃ x,\n                            (x :: v) ∈\n                              {v |\n                                ∃ x,\n                                  (x :: v) ∈\n                                    {v |\n                                      ∃ x,\n                                        (x :: v) ∈\n                                          {v |\n                                            ∃ x,\n                                              (x :: v) ∈\n                                                {v |\n                                                    (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v =>\n                                                      ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩\n                                                  ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩\n                                                    ((fun v =>\n                                                        const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 =\n                                                          v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩\n                                                      ({v | v &6 < v &3} ∩\n                                                        ({v | v &7 ≤ v &5} ∩\n                                                          ({v | v &8 ≤ v &5} ∩ fun v =>\n                                                            v &4 * v &4 -\n                                                                ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                        (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) -\n                                                                      const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                    (v &5 * v &2) *\n                                                                  (v &5 * v &2) =\n                                                              const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))\n⊢ DiophFn fun v => f v ^ g v","state_after":"α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\nproof :\n  Dioph\n    (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪\n      {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩\n        (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪\n          {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩\n            {v |\n              ∃ x,\n                (x :: v) ∈\n                  {v |\n                    ∃ x,\n                      (x :: v) ∈\n                        {v |\n                          ∃ x,\n                            (x :: v) ∈\n                              {v |\n                                ∃ x,\n                                  (x :: v) ∈\n                                    {v |\n                                      ∃ x,\n                                        (x :: v) ∈\n                                          {v |\n                                            ∃ x,\n                                              (x :: v) ∈\n                                                {v |\n                                                    (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v =>\n                                                      ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩\n                                                  ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩\n                                                    ((fun v =>\n                                                        const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 =\n                                                          v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩\n                                                      ({v | v &6 < v &3} ∩\n                                                        ({v | v &7 ≤ v &5} ∩\n                                                          ({v | v &8 ≤ v &5} ∩ fun v =>\n                                                            v &4 * v &4 -\n                                                                ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                        (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) -\n                                                                      const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                    (v &5 * v &2) *\n                                                                  (v &5 * v &2) =\n                                                              const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))\nthis :\n  Dioph\n    {v |\n      v &2 = 0 ∧ v &0 = 1 ∨\n        0 < v &2 ∧\n          (v &1 = 0 ∧ v &0 = 0 ∨\n            0 < v &1 ∧\n              ∃ w a t z x y,\n                (∃ (a1 : 1 < a), xn a1 (v &2) = x ∧ yn a1 (v &2) = y) ∧\n                  x ≡ y * (a - v &1) + v &0 [MOD t] ∧\n                    2 * a * v &1 = t + (v &1 * v &1 + 1) ∧\n                      v &0 < t ∧ v &1 ≤ w ∧ v &2 ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)}\n⊢ DiophFn fun v => f v ^ g v","tactic":"have : Dioph {v : Vector3 ℕ 3 |\n    v &2 = 0 ∧ v &0 = 1 ∨ 0 < v &2 ∧\n    (v &1 = 0 ∧ v &0 = 0 ∨ 0 < v &1 ∧\n    ∃ w a t z x y : ℕ,\n      (∃ a1 : 1 < a, xn a1 (v &2) = x ∧ yn a1 (v &2) = y) ∧\n      x ≡ y * (a - v &1) + v &0 [MOD t] ∧\n      2 * a * v &1 = t + (v &1 * v &1 + 1) ∧\n      v &0 < t ∧ v &1 ≤ w ∧ v &2 ≤ w ∧\n      a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)} := by\n    exact proof","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Dioph","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[261,4],"def_end_pos":[261,9]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Fin2.ofNat'","def_path":"Mathlib/Data/Fin/Fin2.lean","def_pos":[111,4],"def_end_pos":[111,10]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Nat.ModEq","def_path":"Mathlib/Data/Nat/ModEq.lean","def_pos":[32,4],"def_end_pos":[32,9]},{"full_name":"Or","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[532,10],"def_end_pos":[532,12]},{"full_name":"Pell.xn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[103,4],"def_end_pos":[103,6]},{"full_name":"Pell.yn","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[107,4],"def_end_pos":[107,6]},{"full_name":"Vector3","def_path":"Mathlib/Data/Vector3.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\nproof :\n  Dioph\n    (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪\n      {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩\n        (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪\n          {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩\n            {v |\n              ∃ x,\n                (x :: v) ∈\n                  {v |\n                    ∃ x,\n                      (x :: v) ∈\n                        {v |\n                          ∃ x,\n                            (x :: v) ∈\n                              {v |\n                                ∃ x,\n                                  (x :: v) ∈\n                                    {v |\n                                      ∃ x,\n                                        (x :: v) ∈\n                                          {v |\n                                            ∃ x,\n                                              (x :: v) ∈\n                                                {v |\n                                                    (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v =>\n                                                      ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩\n                                                  ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩\n                                                    ((fun v =>\n                                                        const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 =\n                                                          v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩\n                                                      ({v | v &6 < v &3} ∩\n                                                        ({v | v &7 ≤ v &5} ∩\n                                                          ({v | v &8 ≤ v &5} ∩ fun v =>\n                                                            v &4 * v &4 -\n                                                                ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                        (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) -\n                                                                      const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                    (v &5 * v &2) *\n                                                                  (v &5 * v &2) =\n                                                              const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))\nthis :\n  Dioph\n    {v |\n      v &2 = 0 ∧ v &0 = 1 ∨\n        0 < v &2 ∧\n          (v &1 = 0 ∧ v &0 = 0 ∨\n            0 < v &1 ∧\n              ∃ w a t z x y,\n                (∃ (a1 : 1 < a), xn a1 (v &2) = x ∧ yn a1 (v &2) = y) ∧\n                  x ≡ y * (a - v &1) + v &0 [MOD t] ∧\n                    2 * a * v &1 = t + (v &1 * v &1 + 1) ∧\n                      v &0 < t ∧ v &1 ≤ w ∧ v &2 ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)}\n⊢ DiophFn fun v => f v ^ g v","state_after":"no goals","tactic":"exact diophFn_comp2 df dg <| (diophFn_vec _).2 <| Dioph.ext this fun v => Iff.symm <|\n    eq_pow_of_pell.trans <| or_congr Iff.rfl <| and_congr Iff.rfl <| or_congr Iff.rfl <|\n       and_congr Iff.rfl <|\n        ⟨fun ⟨w, a, t, z, a1, h⟩ => ⟨w, a, t, z, _, _, ⟨a1, rfl, rfl⟩, h⟩,\n        fun ⟨w, a, t, z, _, _, ⟨a1, rfl, rfl⟩, h⟩ => ⟨w, a, t, z, a1, h⟩⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Dioph.diophFn_comp2","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[529,8],"def_end_pos":[529,21]},{"full_name":"Dioph.diophFn_vec","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[460,8],"def_end_pos":[460,19]},{"full_name":"Dioph.ext","def_path":"Mathlib/NumberTheory/Dioph.lean","def_pos":[270,8],"def_end_pos":[270,11]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"Iff.trans","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[803,8],"def_end_pos":[803,17]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Pell.eq_pow_of_pell","def_path":"Mathlib/NumberTheory/PellMatiyasevic.lean","def_pos":[859,8],"def_end_pos":[859,22]},{"full_name":"and_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[43,8],"def_end_pos":[43,17]},{"full_name":"or_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[77,8],"def_end_pos":[77,16]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Topology/Instances/NNReal.lean","commit":"","full_name":"Real.tendsto_toNNReal_atTop_iff","start":[141,0],"end":[144,72],"file_path":"Mathlib/Topology/Instances/NNReal.lean","tactics":[{"state_before":"α : Type u_1\nl : Filter α\nf : α → ℝ\n⊢ Tendsto (fun x => (f x).toNNReal) l atTop ↔ Tendsto f l atTop","state_after":"no goals","tactic":"rw [← Real.comap_toNNReal_atTop, tendsto_comap_iff, Function.comp_def]","premises":[{"full_name":"Filter.tendsto_comap_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2686,8],"def_end_pos":[2686,25]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Real.comap_toNNReal_atTop","def_path":"Mathlib/Topology/Instances/NNReal.lean","def_pos":[135,8],"def_end_pos":[135,40]}]}]}
{"url":"Mathlib/Analysis/Complex/Circle.lean","commit":"","full_name":"normSq_eq_of_mem_circle","start":[58,0],"end":[59,86],"file_path":"Mathlib/Analysis/Complex/Circle.lean","tactics":[{"state_before":"z : ↥circle\n⊢ normSq ↑z = 1","state_after":"no goals","tactic":"simp [normSq_eq_abs]","premises":[{"full_name":"Complex.normSq_eq_abs","def_path":"Mathlib/Data/Complex/Abs.lean","def_pos":[215,8],"def_end_pos":[215,21]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Cover/Open.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.OpenCover.pullbackCoverAffineRefinementObjIso_inv_pullbackHom","start":[341,0],"end":[352,49],"file_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","tactics":[{"state_before":"X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\n⊢ (pullbackCoverAffineRefinementObjIso f 𝒰 i).inv ≫ 𝒰.affineRefinement.openCover.pullbackHom f i =\n    (𝒰.obj i.fst).affineCover.pullbackHom (𝒰.pullbackHom f i.fst) i.snd","state_after":"X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\n⊢ (pullbackSymmetry ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)).inv ≫\n      (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫\n        pullback.fst (𝒰.affineRefinement.map i) f =\n    pullback.snd (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd)","tactic":"simp only [pullbackCover_obj, pullbackHom, AffineOpenCover.openCover_obj,\n    AffineOpenCover.openCover_map, pullbackCoverAffineRefinementObjIso, Iso.trans_inv, asIso_inv,\n    Iso.symm_inv, Category.assoc, pullbackSymmetry_inv_comp_snd, IsIso.inv_comp_eq, limit.lift_π,\n    id_eq, PullbackCone.mk_pt, PullbackCone.mk_π_app, Category.comp_id]","premises":[{"full_name":"AlgebraicGeometry.Scheme.AffineOpenCover.openCover_map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[289,2],"def_end_pos":[289,7]},{"full_name":"AlgebraicGeometry.Scheme.AffineOpenCover.openCover_obj","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[289,2],"def_end_pos":[289,7]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.pullbackCoverAffineRefinementObjIso","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[320,4],"def_end_pos":[320,49]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.pullbackCover_obj","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[166,2],"def_end_pos":[166,7]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.pullbackHom","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[185,4],"def_end_pos":[185,25]},{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.IsIso.inv_comp_eq","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[373,8],"def_end_pos":[373,19]},{"full_name":"CategoryTheory.Iso.symm_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[93,8],"def_end_pos":[93,16]},{"full_name":"CategoryTheory.Iso.trans_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[128,9],"def_end_pos":[128,14]},{"full_name":"CategoryTheory.Limits.PullbackCone.mk_pt","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.lean","def_pos":[108,2],"def_end_pos":[108,7]},{"full_name":"CategoryTheory.Limits.PullbackCone.mk_π_app","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.lean","def_pos":[108,2],"def_end_pos":[108,7]},{"full_name":"CategoryTheory.Limits.limit.lift_π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[169,8],"def_end_pos":[169,20]},{"full_name":"CategoryTheory.Limits.pullbackSymmetry_inv_comp_snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[465,8],"def_end_pos":[465,37]},{"full_name":"CategoryTheory.asIso_inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[292,8],"def_end_pos":[292,17]},{"full_name":"id_eq","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[297,16],"def_end_pos":[297,21]}]},{"state_before":"X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\n⊢ (pullbackSymmetry ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)).inv ≫\n      (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫\n        pullback.fst (𝒰.affineRefinement.map i) f =\n    pullback.snd (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd)","state_after":"case h.e'_2.h.h.e'_7.h\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\ne_1✝ :\n  (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ Spec (𝒰.affineRefinement.obj i)) =\n    (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶\n      (𝒰.obj i.fst).affineCover.obj i.snd)\ne_5✝ : Spec (𝒰.affineRefinement.obj i) = (𝒰.obj i.fst).affineCover.obj i.snd\n⊢ (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫\n      pullback.fst (𝒰.affineRefinement.map i) f =\n    pullback.fst ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)","tactic":"convert pullbackSymmetry_inv_comp_fst ((𝒰.obj i.1).affineCover.map i.2) (pullback.fst _ _)","premises":[{"full_name":"AlgebraicGeometry.Scheme.OpenCover.map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.obj","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[46,2],"def_end_pos":[46,5]},{"full_name":"AlgebraicGeometry.Scheme.affineCover","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[64,4],"def_end_pos":[64,15]},{"full_name":"CategoryTheory.Limits.pullback.fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[108,7],"def_end_pos":[108,19]},{"full_name":"CategoryTheory.Limits.pullbackSymmetry_inv_comp_fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[461,8],"def_end_pos":[461,37]},{"full_name":"Sigma.fst","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[175,2],"def_end_pos":[175,5]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]}]},{"state_before":"case h.e'_2.h.h.e'_7.h\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\ne_1✝ :\n  (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ Spec (𝒰.affineRefinement.obj i)) =\n    (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶\n      (𝒰.obj i.fst).affineCover.obj i.snd)\ne_5✝ : Spec (𝒰.affineRefinement.obj i) = (𝒰.obj i.fst).affineCover.obj i.snd\n⊢ (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫\n      pullback.fst (𝒰.affineRefinement.map i) f =\n    pullback.fst ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)","state_after":"no goals","tactic":"exact pullbackRightPullbackFstIso_hom_fst _ _ _","premises":[{"full_name":"CategoryTheory.Limits.pullbackRightPullbackFstIso_hom_fst","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.lean","def_pos":[216,8],"def_end_pos":[216,43]}]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean","commit":"","full_name":"SimpleGraph.Walk.IsHamiltonian.mem_support","start":[37,0],"end":[39,61],"file_path":"Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nG : SimpleGraph α\na b : α\np : G.Walk a b\nhp : p.IsHamiltonian\nc : α\n⊢ c ∈ p.support","state_after":"no goals","tactic":"simp only [← List.count_pos_iff_mem, hp _, Nat.zero_lt_one]","premises":[{"full_name":"List.count_pos_iff_mem","def_path":".lake/packages/batteries/Batteries/Data/List/Count.lean","def_pos":[167,8],"def_end_pos":[167,25]},{"full_name":"Nat.zero_lt_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[513,18],"def_end_pos":[513,29]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","commit":"","full_name":"CochainComplex.HomComplex.Cochain.leftShift_comp","start":[366,0],"end":[379,46],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\n⊢ n' + m = t'","state_after":"no goals","tactic":"rw [← ht', ← h, ← hn', add_assoc, add_comm a, add_assoc]","premises":[{"full_name":"add_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[258,2],"def_end_pos":[258,13]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]},{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\n⊢ (γ.comp γ' h).leftShift a t' ht' = (a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = ((a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯).v p q hpq","tactic":"ext p q hpq","premises":[]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = ((a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯).v p q hpq","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = ((a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯).v p q hpq","tactic":"have h' : n' + m = t' := by linarith","premises":[]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = ((a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯).v p q hpq","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = (a * m).negOnePow • ((γ.leftShift a n' hn').comp γ' ⋯).v p q hpq","tactic":"dsimp","premises":[]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = (a * m).negOnePow • ((γ.leftShift a n' hn').comp γ' ⋯).v p q hpq","state_after":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((a * (n' + m)).negOnePow * (a * (a - 1) / 2).negOnePow) •\n      (K.shiftFunctorObjXIso a p (p + a) ⋯).hom ≫ γ.v (p + a) (p + n') ⋯ ≫ γ'.v (p + n') q ⋯ =\n    ((a * m).negOnePow * (a * n').negOnePow * (a * (a - 1) / 2).negOnePow) •\n      (K.shiftFunctorObjXIso a p (p + a) ⋯).hom ≫ γ.v (p + a) (p + n') ⋯ ≫ γ'.v (p + n') q ⋯","tactic":"simp only [Cochain.comp_v _ _ h' p (p + n') q rfl (by omega),\n    γ.leftShift_v a n' hn' p (p + n') rfl (p + a) (by omega),\n    (γ.comp γ' h).leftShift_v a t' (by omega) p q hpq (p + a) (by omega),\n    smul_smul, Linear.units_smul_comp, assoc, Int.negOnePow_add, ← mul_assoc, ← h',\n    comp_v _ _ h (p + a) (p + n') q (by omega) (by omega)]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Linear.units_smul_comp","def_path":"Mathlib/CategoryTheory/Linear/Basic.lean","def_pos":[164,6],"def_end_pos":[164,21]},{"full_name":"CochainComplex.HomComplex.Cochain.comp","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","def_pos":[219,4],"def_end_pos":[219,8]},{"full_name":"CochainComplex.HomComplex.Cochain.comp_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","def_pos":[236,6],"def_end_pos":[236,12]},{"full_name":"CochainComplex.HomComplex.Cochain.leftShift_v","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean","def_pos":[61,6],"def_end_pos":[61,17]},{"full_name":"Int.negOnePow_add","def_path":"Mathlib/Algebra/Ring/NegOnePow.lean","def_pos":[26,6],"def_end_pos":[26,19]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"smul_smul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[374,6],"def_end_pos":[374,15]}]},{"state_before":"case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((a * (n' + m)).negOnePow * (a * (a - 1) / 2).negOnePow) •\n      (K.shiftFunctorObjXIso a p (p + a) ⋯).hom ≫ γ.v (p + a) (p + n') ⋯ ≫ γ'.v (p + n') q ⋯ =\n    ((a * m).negOnePow * (a * n').negOnePow * (a * (a - 1) / 2).negOnePow) •\n      (K.shiftFunctorObjXIso a p (p + a) ⋯).hom ≫ γ.v (p + a) (p + n') ⋯ ≫ γ'.v (p + n') q ⋯","state_after":"case h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ (a * (n' + m)).negOnePow = (a * m).negOnePow * (a * n').negOnePow","tactic":"congr 2","premises":[]},{"state_before":"case h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ (a * (n' + m)).negOnePow = (a * m).negOnePow * (a * n').negOnePow","state_after":"no goals","tactic":"rw [add_comm n', mul_add, Int.negOnePow_add]","premises":[{"full_name":"Int.negOnePow_add","def_path":"Mathlib/Algebra/Ring/NegOnePow.lean","def_pos":[26,6],"def_end_pos":[26,19]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]}]}]}
{"url":"Mathlib/LinearAlgebra/Determinant.lean","commit":"","full_name":"LinearEquiv.det_coe_symm","start":[391,0],"end":[394,43],"file_path":"Mathlib/LinearAlgebra/Determinant.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type u_3\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne : Basis ι R M\n𝕜 : Type u_5\ninst✝¹ : Field 𝕜\ninst✝ : Module 𝕜 M\nf : M ≃ₗ[𝕜] M\n⊢ LinearMap.det ↑f.symm = (LinearMap.det ↑f)⁻¹","state_after":"no goals","tactic":"field_simp [IsUnit.ne_zero f.isUnit_det']","premises":[{"full_name":"IsUnit.ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","def_pos":[50,8],"def_end_pos":[50,15]},{"full_name":"LinearEquiv.isUnit_det'","def_path":"Mathlib/LinearAlgebra/Determinant.lean","def_pos":[387,8],"def_end_pos":[387,31]}]}]}
{"url":"Mathlib/Algebra/Group/Defs.lean","commit":"","full_name":"nsmul_add_comm'","start":[577,0],"end":[578,92],"file_path":"Mathlib/Algebra/Group/Defs.lean","tactics":[{"state_before":"G : Type u_1\nM : Type u_2\ninst✝ : Monoid M\na✝ b c : M\nm n✝ : ℕ\na : M\nn : ℕ\n⊢ a ^ n * a = a * a ^ n","state_after":"no goals","tactic":"rw [← pow_succ, pow_succ']","premises":[{"full_name":"pow_succ","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[567,8],"def_end_pos":[567,16]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Derivation.lean","commit":"","full_name":"Derivation.compAEval_apply","start":[104,0],"end":[122,29],"file_path":"Mathlib/Algebra/Polynomial/Derivation.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nd : Derivation R A M\na : A\n⊢ ∀ (x y : R[X]),\n    (fun f => (AEval.of R M a) (d ((aeval a) f))) (x + y) =\n      (fun f => (AEval.of R M a) (d ((aeval a) f))) x + (fun f => (AEval.of R M a) (d ((aeval a) f))) y","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nd : Derivation R A M\na : A\n⊢ ∀ (m : R) (x : R[X]),\n    { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯ }.toFun (m • x) =\n      (RingHom.id R) m • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯ }.toFun x","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nd : Derivation R A M\na : A\n⊢ ∀ (a_1 b : R[X]),\n    { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } (a_1 * b) =\n      a_1 • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } b +\n        b • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } a_1","state_after":"no goals","tactic":"simp [AEval.of_aeval_smul, -Derivation.map_aeval]","premises":[{"full_name":"Derivation.map_aeval","def_path":"Mathlib/RingTheory/Derivation/Basic.lean","def_pos":[147,8],"def_end_pos":[147,17]},{"full_name":"Module.AEval.of_aeval_smul","def_path":"Mathlib/Algebra/Polynomial/Module/AEval.lean","def_pos":[62,6],"def_end_pos":[62,19]}]},{"state_before":"R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nd : Derivation R A M\na : A\n⊢ { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } 1 = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/RingTheory/EisensteinCriterion.lean","commit":"","full_name":"Polynomial.EisensteinCriterionAux.eval_zero_mem_ideal_of_eq_mul_X_pow","start":[59,0],"end":[62,80],"file_path":"Mathlib/RingTheory/EisensteinCriterion.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nP : Ideal R\nq : R[X]\nc : (R ⧸ P)[X]\nhq : map (mk P) q = c * X ^ n\nhn0 : n ≠ 0\n⊢ eval 0 q ∈ P","state_after":"no goals","tactic":"rw [← coeff_zero_eq_eval_zero, ← eq_zero_iff_mem, ← coeff_map, hq,\n    coeff_zero_eq_eval_zero, eval_mul, eval_pow, eval_X, zero_pow hn0, mul_zero]","premises":[{"full_name":"Ideal.Quotient.eq_zero_iff_mem","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[112,8],"def_end_pos":[112,23]},{"full_name":"MulZeroClass.mul_zero","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[37,2],"def_end_pos":[37,10]},{"full_name":"Polynomial.coeff_map","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[671,8],"def_end_pos":[671,17]},{"full_name":"Polynomial.coeff_zero_eq_eval_zero","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[430,8],"def_end_pos":[430,31]},{"full_name":"Polynomial.eval_X","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[324,8],"def_end_pos":[324,14]},{"full_name":"Polynomial.eval_mul","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[938,8],"def_end_pos":[938,16]},{"full_name":"Polynomial.eval_pow","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[953,8],"def_end_pos":[953,16]},{"full_name":"zero_pow","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[142,14],"def_end_pos":[142,22]}]}]}
{"url":"Mathlib/MeasureTheory/Group/Action.lean","commit":"","full_name":"MeasureTheory.eventuallyConst_vadd_set_ae","start":[136,0],"end":[139,78],"file_path":"Mathlib/MeasureTheory/Group/Action.lean","tactics":[{"state_before":"G : Type u\nM : Type v\nα : Type w\ns✝ : Set α\nm : MeasurableSpace α\ninst✝³ : MeasurableSpace G\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ EventuallyConst (c • s) (ae μ) ↔ EventuallyConst s (ae μ)","state_after":"no goals","tactic":"rw [← preimage_smul_inv, eventuallyConst_preimage, Filter.map_smul, smul_ae]","premises":[{"full_name":"Filter.eventuallyConst_preimage","def_path":"Mathlib/Order/Filter/EventuallyConst.lean","def_pos":[73,8],"def_end_pos":[73,32]},{"full_name":"Filter.map_smul","def_path":"Mathlib/Order/Filter/Pointwise.lean","def_pos":[942,18],"def_end_pos":[942,26]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"MeasureTheory.smul_ae","def_path":"Mathlib/MeasureTheory/Group/Action.lean","def_pos":[132,8],"def_end_pos":[132,15]},{"full_name":"Set.preimage_smul_inv","def_path":"Mathlib/Data/Set/Pointwise/SMul.lean","def_pos":[717,8],"def_end_pos":[717,25]}]}]}
{"url":"Mathlib/CategoryTheory/Opposites.lean","commit":"","full_name":"CategoryTheory.op_inv","start":[140,0],"end":[143,41],"file_path":"Mathlib/CategoryTheory/Opposites.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ (inv f).op = inv f.op","state_after":"case hom_inv_id\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ f.op ≫ (inv f).op = 𝟙 (op Y)","tactic":"apply IsIso.eq_inv_of_hom_inv_id","premises":[{"full_name":"CategoryTheory.IsIso.eq_inv_of_hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[322,8],"def_end_pos":[322,28]}]},{"state_before":"case hom_inv_id\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ f.op ≫ (inv f).op = 𝟙 (op Y)","state_after":"no goals","tactic":"rw [← op_comp, IsIso.inv_hom_id, op_id]","premises":[{"full_name":"CategoryTheory.IsIso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[245,8],"def_end_pos":[245,18]},{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]},{"full_name":"CategoryTheory.op_id","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[76,8],"def_end_pos":[76,13]}]}]}
{"url":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","commit":"","full_name":"Submodule.subtype_comp_inclusion","start":[293,0],"end":[296,5],"file_path":"Mathlib/Algebra/Module/Submodule/LinearMap.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np✝ p' p q : Submodule R M\nh : p ≤ q\n⊢ q.subtype ∘ₗ inclusion h = p.subtype","state_after":"case h.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np✝ p' p q : Submodule R M\nh : p ≤ q\nb : M\nhb : b ∈ p\n⊢ (q.subtype ∘ₗ inclusion h) ⟨b, hb⟩ = p.subtype ⟨b, hb⟩","tactic":"ext ⟨b, hb⟩","premises":[]},{"state_before":"case h.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np✝ p' p q : Submodule R M\nh : p ≤ q\nb : M\nhb : b ∈ p\n⊢ (q.subtype ∘ₗ inclusion h) ⟨b, hb⟩ = p.subtype ⟨b, hb⟩","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Topology/MetricSpace/Bounded.lean","commit":"","full_name":"Metric.tendsto_dist_right_atTop_iff","start":[115,0],"end":[118,71],"file_path":"Mathlib/Topology/MetricSpace/Bounded.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr : ℝ\nc : α\nf : β → α\nl : Filter β\n⊢ Tendsto (fun x => dist (f x) c) l atTop ↔ Tendsto f l (cobounded α)","state_after":"no goals","tactic":"rw [← comap_dist_right_atTop c, tendsto_comap_iff, Function.comp_def]","premises":[{"full_name":"Filter.tendsto_comap_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2686,8],"def_end_pos":[2686,25]},{"full_name":"Function.comp_def","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[37,8],"def_end_pos":[37,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Metric.comap_dist_right_atTop","def_path":"Mathlib/Topology/MetricSpace/Bounded.lean","def_pos":[107,8],"def_end_pos":[107,30]}]}]}
{"url":"Mathlib/Logic/Equiv/Functor.lean","commit":"","full_name":"Functor.mapEquiv_refl","start":[51,0],"end":[55,30],"file_path":"Mathlib/Logic/Equiv/Functor.lean","tactics":[{"state_before":"α β : Type u\nf : Type u → Type v\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\n⊢ mapEquiv f (Equiv.refl α) = Equiv.refl (f α)","state_after":"case H\nα β : Type u\nf : Type u → Type v\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\nx : f α\n⊢ (mapEquiv f (Equiv.refl α)) x = (Equiv.refl (f α)) x","tactic":"ext x","premises":[]},{"state_before":"case H\nα β : Type u\nf : Type u → Type v\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\nx : f α\n⊢ (mapEquiv f (Equiv.refl α)) x = (Equiv.refl (f α)) x","state_after":"case H\nα β : Type u\nf : Type u → Type v\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\nx : f α\n⊢ ⇑(Equiv.refl α) <$> x = x","tactic":"simp only [mapEquiv_apply, refl_apply]","premises":[{"full_name":"Equiv.refl_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[231,16],"def_end_pos":[231,26]},{"full_name":"Functor.mapEquiv_apply","def_path":"Mathlib/Logic/Equiv/Functor.lean","def_pos":[43,8],"def_end_pos":[43,22]}]},{"state_before":"case H\nα β : Type u\nf : Type u → Type v\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\nx : f α\n⊢ ⇑(Equiv.refl α) <$> x = x","state_after":"no goals","tactic":"exact LawfulFunctor.id_map x","premises":[{"full_name":"LawfulFunctor.id_map","def_path":".lake/packages/lean4/src/lean/Init/Control/Lawful/Basic.lean","def_pos":[26,2],"def_end_pos":[26,8]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Indicator.lean","commit":"","full_name":"MeasureTheory.tendsto_measure_of_tendsto_indicator","start":[39,0],"end":[49,58],"file_path":"Mathlib/MeasureTheory/Integral/Indicator.lean","tactics":[{"state_before":"α : Type u_1\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nμ : Measure α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ⊤\nAs_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\n⊢ Tendsto (fun i => μ (As i)) L (𝓝 (μ A))","state_after":"α : Type u_1\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nμ : Measure α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ⊤\nAs_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\n⊢ MeasurableSet A","tactic":"apply tendsto_measure_of_ae_tendsto_indicator L ?_ As_mble B_mble B_finmeas As_le_B\n        (ae_of_all μ h_lim)","premises":[{"full_name":"MeasureTheory.ae_of_all","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[87,8],"def_end_pos":[87,17]},{"full_name":"MeasureTheory.tendsto_measure_of_ae_tendsto_indicator","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[2025,6],"def_end_pos":[2025,45]}]},{"state_before":"α : Type u_1\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nμ : Measure α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ⊤\nAs_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\n⊢ MeasurableSet A","state_after":"no goals","tactic":"exact measurableSet_of_tendsto_indicator L As_mble h_lim","premises":[{"full_name":"measurableSet_of_tendsto_indicator","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean","def_pos":[139,6],"def_end_pos":[139,40]}]}]}
{"url":"Mathlib/Control/Traversable/Lemmas.lean","commit":"","full_name":"Traversable.traverse_id","start":[91,0],"end":[94,21],"file_path":"Mathlib/Control/Traversable/Lemmas.lean","tactics":[{"state_before":"t : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : LawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf : β → γ\nx : t β\n⊢ traverse pure = pure","state_after":"case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : LawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf : β → γ\nx : t β\nx✝ : t α\n⊢ traverse pure x✝ = pure x✝","tactic":"ext","premises":[]},{"state_before":"case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : LawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf : β → γ\nx : t β\nx✝ : t α\n⊢ traverse pure x✝ = pure x✝","state_after":"no goals","tactic":"exact id_traverse _","premises":[{"full_name":"LawfulTraversable.id_traverse","def_path":"Mathlib/Control/Traversable/Basic.lean","def_pos":[227,2],"def_end_pos":[227,13]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Scheme.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.Hom.app_eq_appLE","start":[133,0],"end":[135,18],"file_path":"Mathlib/AlgebraicGeometry/Scheme.lean","tactics":[{"state_before":"X Y : Scheme\nf : X.Hom Y\nU✝ U' : Y.Opens\nV V' : X.Opens\nU : Y.Opens\n⊢ f.app U = f.appLE U (f ⁻¹ᵁ U) ⋯","state_after":"no goals","tactic":"simp [Hom.appLE]","premises":[{"full_name":"AlgebraicGeometry.Scheme.Hom.appLE","def_path":"Mathlib/AlgebraicGeometry/Scheme.lean","def_pos":[107,4],"def_end_pos":[107,9]}]}]}
{"url":"Mathlib/Algebra/Homology/TotalComplex.lean","commit":"","full_name":"HomologicalComplex₂.total.mapIso_inv","start":[413,0],"end":[420,56],"file_path":"Mathlib/Algebra/Homology/TotalComplex.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁶ : Category.{?u.130934, u_1} C\ninst✝⁵ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝⁴ : DecidableEq I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : K.HasTotal c₁₂\ninst✝¹ : L.HasTotal c₁₂\ninst✝ : M.HasTotal c₁₂\n⊢ map e.hom c₁₂ ≫ map e.inv c₁₂ = 𝟙 (K.total c₁₂)","state_after":"no goals","tactic":"rw [← map_comp, e.hom_inv_id, map_id]","premises":[{"full_name":"CategoryTheory.Iso.hom_inv_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[55,2],"def_end_pos":[55,12]},{"full_name":"HomologicalComplex₂.total.map_comp","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[409,6],"def_end_pos":[409,14]},{"full_name":"HomologicalComplex₂.total.map_id","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[402,6],"def_end_pos":[402,12]}]},{"state_before":"C : Type u_1\ninst✝⁶ : Category.{?u.130934, u_1} C\ninst✝⁵ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝⁴ : DecidableEq I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : K.HasTotal c₁₂\ninst✝¹ : L.HasTotal c₁₂\ninst✝ : M.HasTotal c₁₂\n⊢ map e.inv c₁₂ ≫ map e.hom c₁₂ = 𝟙 (L.total c₁₂)","state_after":"no goals","tactic":"rw [← map_comp, e.inv_hom_id, map_id]","premises":[{"full_name":"CategoryTheory.Iso.inv_hom_id","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[58,2],"def_end_pos":[58,12]},{"full_name":"HomologicalComplex₂.total.map_comp","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[409,6],"def_end_pos":[409,14]},{"full_name":"HomologicalComplex₂.total.map_id","def_path":"Mathlib/Algebra/Homology/TotalComplex.lean","def_pos":[402,6],"def_end_pos":[402,12]}]}]}
{"url":"Mathlib/Data/Int/Log.lean","commit":"","full_name":"Int.log_one_right","start":[121,0],"end":[123,85],"file_path":"Mathlib/Data/Int/Log.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\n⊢ log b 1 = 0","state_after":"no goals","tactic":"rw [log_of_one_le_right _ le_rfl, Nat.floor_one, Nat.log_one_right, Int.ofNat_zero]","premises":[{"full_name":"Int.log_of_one_le_right","def_path":"Mathlib/Data/Int/Log.lean","def_pos":[59,8],"def_end_pos":[59,27]},{"full_name":"Int.ofNat_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[73,16],"def_end_pos":[73,26]},{"full_name":"Nat.floor_one","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[162,8],"def_end_pos":[162,17]},{"full_name":"Nat.log_one_right","def_path":"Mathlib/Data/Nat/Log.lean","def_pos":[72,8],"def_end_pos":[72,21]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Category.lean","commit":"","full_name":"CategoryTheory.MonoidalCategory.id_tensor_comp_tensor_id","start":[752,0],"end":[755,6],"file_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","tactics":[{"state_before":"C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X Y Z : C\nf : W ⟶ X\ng : Y ⟶ Z\n⊢ (𝟙 Y ⊗ f) ≫ (g ⊗ 𝟙 X) = g ⊗ f","state_after":"C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X Y Z : C\nf : W ⟶ X\ng : Y ⟶ Z\n⊢ 𝟙 Y ≫ g ⊗ f ≫ 𝟙 X = g ⊗ f","tactic":"rw [← tensor_comp]","premises":[{"full_name":"CategoryTheory.MonoidalCategory.tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[166,2],"def_end_pos":[166,13]}]},{"state_before":"C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X Y Z : C\nf : W ⟶ X\ng : Y ⟶ Z\n⊢ 𝟙 Y ≫ g ⊗ f ≫ 𝟙 X = g ⊗ f","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/DirectSum/Internal.lean","commit":"","full_name":"SetLike.natCast_mem_graded","start":[67,0],"end":[73,49],"file_path":"Mathlib/Algebra/DirectSum/Internal.lean","tactics":[{"state_before":"ι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddMonoidWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : GradedOne A\nn : ℕ\n⊢ ↑n ∈ A 0","state_after":"case zero\nι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddMonoidWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : GradedOne A\n⊢ ↑0 ∈ A 0\n\ncase succ\nι : Type u_1\nσ : Type u_2\nS : Type u_3\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddMonoidWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : GradedOne A\nn✝ : ℕ\nn_ih : ↑n✝ ∈ A 0\n⊢ ↑(n✝ + 1) ∈ A 0","tactic":"induction' n with _ n_ih","premises":[]}]}
{"url":"Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean","commit":"","full_name":"MeasureTheory.OuterMeasure.comap_boundedBy","start":[276,0],"end":[285,23],"file_path":"Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean","tactics":[{"state_before":"α : Type u_1\nm : Set α → ℝ≥0∞\nβ : Type u_2\nf : β → α\nh : (Monotone fun s => m ↑s) ∨ Surjective f\n⊢ (comap f) (boundedBy m) = boundedBy fun s => m (f '' s)","state_after":"case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nβ : Type u_2\nf : β → α\nh : (Monotone fun s => m ↑s) ∨ Surjective f\n⊢ (Monotone fun s => ⨆ (_ : s.Nonempty), m s) ∨ Surjective f\n\ncase refine_2\nα : Type u_1\nm : Set α → ℝ≥0∞\nβ : Type u_2\nf : β → α\nh : (Monotone fun s => m ↑s) ∨ Surjective f\n⊢ OuterMeasure.ofFunction (fun s => ⨆ (_ : (f '' s).Nonempty), m (f '' s)) ⋯ = boundedBy fun s => m (f '' s)","tactic":"refine (comap_ofFunction _ ?_).trans ?_","premises":[{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"MeasureTheory.OuterMeasure.comap_ofFunction","def_path":"Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean","def_pos":[162,8],"def_end_pos":[162,24]}]}]}
{"url":"Mathlib/Deprecated/Submonoid.lean","commit":"","full_name":"IsAddSubmonoid.list_sum_mem","start":[194,0],"end":[202,47],"file_path":"Mathlib/Deprecated/Submonoid.lean","tactics":[{"state_before":"M : Type u_1\ninst✝¹ : Monoid M\ns : Set M\nA : Type u_2\ninst✝ : AddMonoid A\nt : Set A\nhs : IsSubmonoid s\na : M\nl : List M\nh : ∀ x ∈ a :: l, x ∈ s\nthis : a * l.prod ∈ s\n⊢ (a :: l).prod ∈ s","state_after":"no goals","tactic":"simpa","premises":[]},{"state_before":"M : Type u_1\ninst✝¹ : Monoid M\ns : Set M\nA : Type u_2\ninst✝ : AddMonoid A\nt : Set A\nhs : IsSubmonoid s\na : M\nl : List M\nh : ∀ x ∈ a :: l, x ∈ s\n⊢ a ∈ s ∧ ∀ x ∈ l, x ∈ s","state_after":"no goals","tactic":"simpa using h","premises":[]}]}
{"url":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","commit":"","full_name":"MeasureTheory.measure_symmDiff_eq_zero_iff","start":[159,0],"end":[161,32],"file_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns✝ t✝ s t : Set α\n⊢ μ (s ∆ t) = 0 ↔ s =ᶠ[ae μ] t","state_after":"no goals","tactic":"simp [ae_eq_set, symmDiff_def]","premises":[{"full_name":"MeasureTheory.ae_eq_set","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[155,8],"def_end_pos":[155,17]},{"full_name":"symmDiff_def","def_path":"Mathlib/Order/SymmDiff.lean","def_pos":[74,8],"def_end_pos":[74,20]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Subsheaf.lean","commit":"","full_name":"CategoryTheory.GrothendieckTopology.Subpresheaf.to_sheafify_lift_unique","start":[289,0],"end":[298,53],"file_path":"Mathlib/CategoryTheory/Sites/Subsheaf.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\n⊢ l₁ = l₂","state_after":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\n⊢ l₁.app U ⟨s, hs⟩ = l₂.app U ⟨s, hs⟩","tactic":"ext U ⟨s, hs⟩","premises":[]},{"state_before":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\n⊢ l₁.app U ⟨s, hs⟩ = l₂.app U ⟨s, hs⟩","state_after":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop U⦄,\n    (G.sieveOfSection s).arrows f → F'.map f.op (l₁.app U ⟨s, hs⟩) = F'.map f.op (l₂.app U ⟨s, hs⟩)","tactic":"apply (h _ hs).isSeparatedFor.ext","premises":[{"full_name":"CategoryTheory.Presieve.IsSeparatedFor.ext","def_path":"Mathlib/CategoryTheory/Sites/IsSheafFor.lean","def_pos":[378,8],"def_end_pos":[378,26]},{"full_name":"CategoryTheory.Presieve.IsSheafFor.isSeparatedFor","def_path":"Mathlib/CategoryTheory/Sites/IsSheafFor.lean","def_pos":[551,8],"def_end_pos":[551,33]}]},{"state_before":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop U⦄,\n    (G.sieveOfSection s).arrows f → F'.map f.op (l₁.app U ⟨s, hs⟩) = F'.map f.op (l₂.app U ⟨s, hs⟩)","state_after":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : (G.sieveOfSection s).arrows i\n⊢ F'.map i.op (l₁.app U ⟨s, hs⟩) = F'.map i.op (l₂.app U ⟨s, hs⟩)","tactic":"rintro V i hi","premises":[]},{"state_before":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : (G.sieveOfSection s).arrows i\n⊢ F'.map i.op (l₁.app U ⟨s, hs⟩) = F'.map i.op (l₂.app U ⟨s, hs⟩)","state_after":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : F.map i.op s ∈ G.obj (op V)\n⊢ F'.map i.op (l₁.app U ⟨s, hs⟩) = F'.map i.op (l₂.app U ⟨s, hs⟩)","tactic":"dsimp at hi","premises":[]},{"state_before":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : F.map i.op s ∈ G.obj (op V)\n⊢ F'.map i.op (l₁.app U ⟨s, hs⟩) = F'.map i.op (l₂.app U ⟨s, hs⟩)","state_after":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : F.map i.op s ∈ G.obj (op V)\n⊢ l₁.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩) =\n    l₂.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩)","tactic":"erw [← FunctorToTypes.naturality, ← FunctorToTypes.naturality]","premises":[{"full_name":"CategoryTheory.FunctorToTypes.naturality","def_path":"Mathlib/CategoryTheory/Types.lean","def_pos":[145,8],"def_end_pos":[145,18]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]}]},{"state_before":"case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : F.map i.op s ∈ G.obj (op V)\n⊢ l₁.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩) =\n    l₂.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩)","state_after":"no goals","tactic":"exact (congr_fun (congr_app e <| op V) ⟨_, hi⟩ : _)","premises":[{"full_name":"CategoryTheory.congr_app","def_path":"Mathlib/CategoryTheory/NatTrans.lean","def_pos":[56,8],"def_end_pos":[56,17]},{"full_name":"Opposite.op","def_path":"Mathlib/Data/Opposite.lean","def_pos":[35,2],"def_end_pos":[35,4]}]}]}
{"url":"Mathlib/Computability/Language.lean","commit":"","full_name":"Language.one_add_self_mul_kstar_eq_kstar","start":[239,0],"end":[242,27],"file_path":"Mathlib/Computability/Language.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ 1 + l * l∗ = l∗","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ l ^ 0 + ⨆ i, l ^ (i + 1) = ⨆ i, l ^ i","tactic":"simp only [kstar_eq_iSup_pow, mul_iSup, ← pow_succ', ← pow_zero l]","premises":[{"full_name":"Language.kstar_eq_iSup_pow","def_path":"Mathlib/Computability/Language.lean","def_pos":[221,8],"def_end_pos":[221,25]},{"full_name":"Language.mul_iSup","def_path":"Mathlib/Computability/Language.lean","def_pos":[192,8],"def_end_pos":[192,16]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]},{"full_name":"pow_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[563,8],"def_end_pos":[563,16]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ l ^ 0 + ⨆ i, l ^ (i + 1) = ⨆ i, l ^ i","state_after":"no goals","tactic":"exact sup_iSup_nat_succ _","premises":[{"full_name":"sup_iSup_nat_succ","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[1387,8],"def_end_pos":[1387,25]}]}]}
{"url":"Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean","commit":"","full_name":"groupCohomology.resolution.forget₂ToModuleCatHomotopyEquiv_f_0_eq","start":[575,0],"end":[602,93],"file_path":"Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean","tactics":[{"state_before":"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (forget₂ToModuleCatHomotopyEquiv k G).hom.f 0 = (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)","state_after":"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ ((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫\n          (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫\n            (HomotopyEquiv.ofIso\n                ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom).f\n      0 =\n    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)","tactic":"show (HomotopyEquiv.hom _ ≫ HomotopyEquiv.hom _ ≫ HomotopyEquiv.hom _).f 0 = _","premises":[{"full_name":"CategoryTheory.CategoryStruct.comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[89,2],"def_end_pos":[89,6]},{"full_name":"HomologicalComplex.Hom.f","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[216,2],"def_end_pos":[216,3]},{"full_name":"HomotopyEquiv.hom","def_path":"Mathlib/Algebra/Homology/Homotopy.lean","def_pos":[689,2],"def_end_pos":[689,5]}]},{"state_before":"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ ((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫\n          (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫\n            (HomotopyEquiv.ofIso\n                ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom).f\n      0 =\n    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)","state_after":"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫\n      (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫\n        (HomotopyEquiv.ofIso\n                ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f\n          0 =\n    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)","tactic":"simp only [HomologicalComplex.comp_f]","premises":[{"full_name":"HomologicalComplex.comp_f","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[260,8],"def_end_pos":[260,14]}]},{"state_before":"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫\n      (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫\n        (HomotopyEquiv.ofIso\n                ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f\n          0 =\n    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)","state_after":"case h.e'_2.h.e'_6.h\nk G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\ne_4✝ : (compForgetAugmented.toModule k G).left.obj (Opposite.op (SimplexCategory.mk 0)) = (forget₂ToModuleCat k G).X 0\n⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 = 𝟙 ((forget₂ToModuleCat k G).X 0)\n\ncase h.e'_3\nk G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) =\n    (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫\n      (HomotopyEquiv.ofIso\n              ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f\n        0","tactic":"convert Category.id_comp (X := (forget₂ToModuleCat k G).X 0) _","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"HomologicalComplex.X","def_path":"Mathlib/Algebra/Homology/HomologicalComplex.lean","def_pos":[56,2],"def_end_pos":[56,3]},{"full_name":"groupCohomology.resolution.forget₂ToModuleCat","def_path":"Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean","def_pos":[537,4],"def_end_pos":[537,22]}]},{"state_before":"case h.e'_3\nk G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) =\n    (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫\n      (HomotopyEquiv.ofIso\n              ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f\n        0","state_after":"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) =\n    (Finsupp.total (⊤_ Type u) k k fun x => 1) ∘ₗ (ModuleCat.free k).map (terminal.from (Fin 1 → G))\n\nk G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (Finsupp.total (⊤_ Type u) k k fun x => 1) ∘ₗ (ModuleCat.free k).map (terminal.from (Fin 1 → G)) =\n    (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫\n      (HomotopyEquiv.ofIso\n              ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f\n        0","tactic":"trans (Finsupp.total _ _ _ fun _ => (1 : k)).comp ((ModuleCat.free k).map (terminal.from _))","premises":[{"full_name":"CategoryTheory.Limits.terminal.from","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean","def_pos":[326,7],"def_end_pos":[326,20]},{"full_name":"Finsupp.total","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[597,14],"def_end_pos":[597,19]},{"full_name":"LinearMap.comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[489,4],"def_end_pos":[489,8]},{"full_name":"ModuleCat.free","def_path":"Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean","def_pos":[39,4],"def_end_pos":[39,8]},{"full_name":"Prefunctor.map","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[57,2],"def_end_pos":[57,5]}]}]}
{"url":"Mathlib/NumberTheory/Pell.lean","commit":"","full_name":"Pell.Solution₁.x_mul","start":[158,0],"end":[161,5],"file_path":"Mathlib/NumberTheory/Pell.lean","tactics":[{"state_before":"d : ℤ\na b : Solution₁ d\n⊢ (a * b).x = a.x * b.x + d * (a.y * b.y)","state_after":"d : ℤ\na b : Solution₁ d\n⊢ (a * b).x = a.x * b.x + d * a.y * b.y","tactic":"rw [← mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]},{"state_before":"d : ℤ\na b : Solution₁ d\n⊢ (a * b).x = a.x * b.x + d * a.y * b.y","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Logic/Equiv/Basic.lean","commit":"","full_name":"Equiv.swap_apply_ne_self_iff","start":[1470,0],"end":[1480,46],"file_path":"Mathlib/Logic/Equiv/Basic.lean","tactics":[{"state_before":"α : Sort u_1\ninst✝ : DecidableEq α\na b x : α\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","state_after":"case pos\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : a = b\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)\n\ncase neg\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","tactic":"by_cases hab : a = b","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","state_after":"case pos\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\nhax : x = a\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)\n\ncase neg\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\nhax : ¬x = a\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","tactic":"by_cases hax : x = a","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\nhax : ¬x = a\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","state_after":"case pos\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\nhax : ¬x = a\nhbx : x = b\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)\n\ncase neg\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\nhax : ¬x = a\nhbx : ¬x = b\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","tactic":"by_cases hbx : x = b","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nα : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nhab : ¬a = b\nhax : ¬x = a\nhbx : ¬x = b\n⊢ (swap a b) x ≠ x ↔ a ≠ b ∧ (x = a ∨ x = b)","state_after":"no goals","tactic":"simp [hab, hax, hbx, swap_apply_of_ne_of_ne]","premises":[{"full_name":"Equiv.swap_apply_of_ne_of_ne","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[1405,8],"def_end_pos":[1405,30]}]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieIdeal.inclusion_injective","start":[1142,0],"end":[1145,77],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\nh : I₁ ≤ I₂\nx y : ↥↑I₁\n⊢ (inclusion h) x = (inclusion h) y → x = y","state_after":"no goals","tactic":"simp only [inclusion_apply, imp_self, Subtype.mk_eq_mk, SetLike.coe_eq_coe]","premises":[{"full_name":"LieIdeal.inclusion_apply","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1138,8],"def_end_pos":[1138,23]},{"full_name":"SetLike.coe_eq_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[172,8],"def_end_pos":[172,18]},{"full_name":"Subtype.mk_eq_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[93,8],"def_end_pos":[93,16]},{"full_name":"imp_self","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1418,21],"def_end_pos":[1418,29]}]}]}
{"url":"Mathlib/Data/Nat/Digits.lean","commit":"","full_name":"Nat.digits_len_le_digits_len_succ","start":[430,0],"end":[436,61],"file_path":"Mathlib/Data/Nat/Digits.lean","tactics":[{"state_before":"n✝ b n : ℕ\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length","state_after":"case inl\nn b : ℕ\n⊢ (b.digits 0).length ≤ (b.digits (0 + 1)).length\n\ncase inr\nn✝ b n : ℕ\nhn : n ≠ 0\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length","tactic":"rcases Decidable.eq_or_ne n 0 with (rfl | hn)","premises":[{"full_name":"Decidable.eq_or_ne","def_path":"Mathlib/Logic/Basic.lean","def_pos":[161,8],"def_end_pos":[161,26]}]},{"state_before":"case inr\nn✝ b n : ℕ\nhn : n ≠ 0\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length","state_after":"case inr.inl\nn✝ b n : ℕ\nhn : n ≠ 0\nhb : b ≤ 1\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length\n\ncase inr.inr\nn✝ b n : ℕ\nhn : n ≠ 0\nhb : 1 < b\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length","tactic":"rcases le_or_lt b 1 with hb | hb","premises":[{"full_name":"le_or_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[290,8],"def_end_pos":[290,16]}]},{"state_before":"case inr.inr\nn✝ b n : ℕ\nhn : n ≠ 0\nhb : 1 < b\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length","state_after":"no goals","tactic":"simpa [digits_len, hb, hn] using log_mono_right (le_succ _)","premises":[{"full_name":"Nat.digits_len","def_path":"Mathlib/Data/Nat/Digits.lean","def_pos":[307,8],"def_end_pos":[307,18]},{"full_name":"Nat.le_succ","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1684,8],"def_end_pos":[1684,19]},{"full_name":"Nat.log_mono_right","def_path":"Mathlib/Data/Nat/Log.lean","def_pos":[164,8],"def_end_pos":[164,22]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Content.lean","commit":"","full_name":"Polynomial.Monic.isPrimitive","start":[50,0],"end":[52,82],"file_path":"Mathlib/RingTheory/Polynomial/Content.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nhp : p.Monic\n⊢ p.IsPrimitive","state_after":"case intro\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nhp : p.Monic\nr : R\nq : R[X]\nh : p = C r * q\n⊢ IsUnit r","tactic":"rintro r ⟨q, h⟩","premises":[]},{"state_before":"case intro\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nhp : p.Monic\nr : R\nq : R[X]\nh : p = C r * q\n⊢ IsUnit r","state_after":"no goals","tactic":"exact isUnit_of_mul_eq_one r (q.coeff p.natDegree) (by rwa [← coeff_C_mul, ← h])","premises":[{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.coeff_C_mul","def_path":"Mathlib/Algebra/Polynomial/Coeff.lean","def_pos":[149,8],"def_end_pos":[149,19]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"isUnit_of_mul_eq_one","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[563,8],"def_end_pos":[563,28]}]}]}
{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"mapClusterPt_iff_ultrafilter","start":[945,0],"end":[948,51],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι✝ : Sort w\nα : Type u_1\nβ : Type u_2\nx✝ : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\nι : Type u_3\nx : X\nF : Filter ι\nu : ι → X\n⊢ MapClusterPt x F u ↔ ∃ U, ↑U ≤ F ∧ Tendsto u (↑U) (𝓝 x)","state_after":"no goals","tactic":"simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,\n    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ClusterPt","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[239,4],"def_end_pos":[239,13]},{"full_name":"Filter.exists_ultrafilter_iff","def_path":"Mathlib/Order/Filter/Ultrafilter.lean","def_pos":[396,8],"def_end_pos":[396,30]},{"full_name":"Filter.map_neBot_iff","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2284,8],"def_end_pos":[2284,21]},{"full_name":"Filter.push_pull'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2394,18],"def_end_pos":[2394,28]},{"full_name":"Filter.tendsto_iff_comap","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2621,8],"def_end_pos":[2621,25]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MapClusterPt","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[244,4],"def_end_pos":[244,16]},{"full_name":"inf_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[385,8],"def_end_pos":[385,16]},{"full_name":"le_inf_iff","def_path":"Mathlib/Order/Lattice.lean","def_pos":[333,8],"def_end_pos":[333,18]}]}]}
{"url":"Mathlib/Order/ModularLattice.lean","commit":"","full_name":"infIooOrderIsoIooSup_apply_coe","start":[278,0],"end":[301,36],"file_path":"Mathlib/Order/ModularLattice.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo (a ⊓ b) a)\n⊢ ↑((fun c => ⟨a ⊓ ↑c, ⋯⟩) ((fun c => ⟨↑c ⊔ b, ⋯⟩) c)) = ↑c","state_after":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo (a ⊓ b) a)\n⊢ a ⊓ (↑c ⊔ b) = ↑c","tactic":"dsimp","premises":[]},{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo (a ⊓ b) a)\n⊢ a ⊓ (↑c ⊔ b) = ↑c","state_after":"no goals","tactic":"rw [sup_comm, ← inf_sup_assoc_of_le _ c.prop.2.le, sup_eq_right.2 c.prop.1.le]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Subtype.prop","def_path":"Mathlib/Data/Subtype.lean","def_pos":[37,8],"def_end_pos":[37,12]},{"full_name":"inf_sup_assoc_of_le","def_path":"Mathlib/Order/ModularLattice.lean","def_pos":[189,8],"def_end_pos":[189,27]},{"full_name":"sup_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[193,8],"def_end_pos":[193,16]},{"full_name":"sup_eq_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[142,8],"def_end_pos":[142,20]}]},{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo b (a ⊔ b))\n⊢ ↑((fun c => ⟨↑c ⊔ b, ⋯⟩) ((fun c => ⟨a ⊓ ↑c, ⋯⟩) c)) = ↑c","state_after":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo b (a ⊔ b))\n⊢ a ⊓ ↑c ⊔ b = ↑c","tactic":"dsimp","premises":[]},{"state_before":"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo b (a ⊔ b))\n⊢ a ⊓ ↑c ⊔ b = ↑c","state_after":"no goals","tactic":"rw [inf_comm, inf_sup_assoc_of_le _ c.prop.1.le, inf_eq_left.2 c.prop.2.le]","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Subtype.prop","def_path":"Mathlib/Data/Subtype.lean","def_pos":[37,8],"def_end_pos":[37,12]},{"full_name":"inf_comm","def_path":"Mathlib/Order/Lattice.lean","def_pos":[385,8],"def_end_pos":[385,16]},{"full_name":"inf_eq_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[337,8],"def_end_pos":[337,19]},{"full_name":"inf_sup_assoc_of_le","def_path":"Mathlib/Order/ModularLattice.lean","def_pos":[189,8],"def_end_pos":[189,27]}]}]}
{"url":"Mathlib/Topology/Category/Profinite/Nobeling.lean","commit":"","full_name":"Profinite.NobelingProof.GoodProducts.span","start":[664,0],"end":[673,44],"file_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","tactics":[{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\n⊢ ⊤ ≤ Submodule.span ℤ (Set.range (eval C))","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\n⊢ ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))","tactic":"rw [span_iff_products]","premises":[{"full_name":"Profinite.NobelingProof.GoodProducts.span_iff_products","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[430,8],"def_end_pos":[430,38]}]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\n⊢ ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))","state_after":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nf : LocallyConstant ↑C ℤ\na✝ : f ∈ ⊤\n⊢ f ∈ Submodule.span ℤ (Set.range (Products.eval C))","tactic":"intro f _","premises":[]},{"state_before":"I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nf : LocallyConstant ↑C ℤ\na✝ : f ∈ ⊤\n⊢ f ∈ Submodule.span ℤ (Set.range (Products.eval C))","state_after":"case intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nK : Finset I\nf' : LocallyConstant ↑(π C fun x => x ∈ K) ℤ\na✝ : (πJ C K) f' ∈ ⊤\n⊢ (πJ C K) f' ∈ Submodule.span ℤ (Set.range (Products.eval C))","tactic":"obtain ⟨K, f', rfl⟩ : ∃ K f', f = πJ C K f' := fin_comap_jointlySurjective C hC f","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Profinite.NobelingProof.fin_comap_jointlySurjective","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[654,8],"def_end_pos":[654,35]},{"full_name":"Profinite.NobelingProof.πJ","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[474,4],"def_end_pos":[474,6]}]},{"state_before":"case intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nK : Finset I\nf' : LocallyConstant ↑(π C fun x => x ∈ K) ℤ\na✝ : (πJ C K) f' ∈ ⊤\n⊢ (πJ C K) f' ∈ Submodule.span ℤ (Set.range (Products.eval C))","state_after":"case intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nK : Finset I\nf' : LocallyConstant ↑(π C fun x => x ∈ K) ℤ\na✝ : (πJ C K) f' ∈ ⊤\n⊢ ⇑(πJ C K) '' Set.range (eval (π C fun x => x ∈ K)) ⊆ Set.range (Products.eval C)","tactic":"refine Submodule.span_mono ?_ <| Submodule.apply_mem_span_image_of_mem_span (πJ C K) <|\n    spanFin C K (Submodule.mem_top : f' ∈ ⊤)","premises":[{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Profinite.NobelingProof.GoodProducts.spanFin","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[600,8],"def_end_pos":[600,28]},{"full_name":"Profinite.NobelingProof.πJ","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[474,4],"def_end_pos":[474,6]},{"full_name":"Submodule.apply_mem_span_image_of_mem_span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[572,8],"def_end_pos":[572,40]},{"full_name":"Submodule.mem_top","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[144,8],"def_end_pos":[144,15]},{"full_name":"Submodule.span_mono","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[77,8],"def_end_pos":[77,17]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]}]},{"state_before":"case intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nK : Finset I\nf' : LocallyConstant ↑(π C fun x => x ∈ K) ℤ\na✝ : (πJ C K) f' ∈ ⊤\n⊢ ⇑(πJ C K) '' Set.range (eval (π C fun x => x ∈ K)) ⊆ Set.range (Products.eval C)","state_after":"case intro.intro.intro.intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nK : Finset I\nf' : LocallyConstant ↑(π C fun x => x ∈ K) ℤ\na✝ : (πJ C K) f' ∈ ⊤\nm : { l // Products.isGood (π C fun x => x ∈ K) l }\n⊢ (πJ C K) (eval (π C fun x => x ∈ K) m) ∈ Set.range (Products.eval C)","tactic":"rintro l ⟨y, ⟨m, rfl⟩, rfl⟩","premises":[]},{"state_before":"case intro.intro.intro.intro.intro\nI : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nhC : IsClosed C\nK : Finset I\nf' : LocallyConstant ↑(π C fun x => x ∈ K) ℤ\na✝ : (πJ C K) f' ∈ ⊤\nm : { l // Products.isGood (π C fun x => x ∈ K) l }\n⊢ (πJ C K) (eval (π C fun x => x ∈ K) m) ∈ Set.range (Products.eval C)","state_after":"no goals","tactic":"exact ⟨m.val, eval_eq_πJ C K m.val m.prop⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Profinite.NobelingProof.eval_eq_πJ","def_path":"Mathlib/Topology/Category/Profinite/Nobeling.lean","def_pos":[477,8],"def_end_pos":[477,18]},{"full_name":"Subtype.prop","def_path":"Mathlib/Data/Subtype.lean","def_pos":[37,8],"def_end_pos":[37,12]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]}]}
{"url":"Mathlib/RingTheory/PowerSeries/Basic.lean","commit":"","full_name":"PowerSeries.map_X","start":[458,0],"end":[461,29],"file_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝² : Semiring R\nS : Type u_2\nT : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\n⊢ (map f) X = X","state_after":"case h\nR : Type u_1\ninst✝² : Semiring R\nS : Type u_2\nT : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn✝ : ℕ\n⊢ (coeff S n✝) ((map f) X) = (coeff S n✝) X","tactic":"ext","premises":[]},{"state_before":"case h\nR : Type u_1\ninst✝² : Semiring R\nS : Type u_2\nT : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn✝ : ℕ\n⊢ (coeff S n✝) ((map f) X) = (coeff S n✝) X","state_after":"no goals","tactic":"simp [coeff_X, apply_ite f]","premises":[{"full_name":"PowerSeries.coeff_X","def_path":"Mathlib/RingTheory/PowerSeries/Basic.lean","def_pos":[250,8],"def_end_pos":[250,15]},{"full_name":"apply_ite","def_path":".lake/packages/lean4/src/lean/Init/ByCases.lean","def_pos":[36,8],"def_end_pos":[36,17]}]}]}
{"url":"Mathlib/LinearAlgebra/Basis.lean","commit":"","full_name":"Basis.prod_apply_inl_fst","start":[672,0],"end":[678,53],"file_path":"Mathlib/LinearAlgebra/Basis.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni : ι\n⊢ b.repr ((b.prod b') (Sum.inl i)).1 = b.repr (b i)","state_after":"case h\nι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni j : ι\n⊢ (b.repr ((b.prod b') (Sum.inl i)).1) j = (b.repr (b i)) j","tactic":"ext j","premises":[]},{"state_before":"case h\nι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni j : ι\n⊢ (b.repr ((b.prod b') (Sum.inl i)).1) j = (b.repr (b i)) j","state_after":"case h\nι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni j : ι\n⊢ (Finsupp.single (Sum.inl i) 1) (Sum.inl j) = (Finsupp.single i 1) j","tactic":"simp only [Basis.prod, Basis.coe_ofRepr, LinearEquiv.symm_trans_apply, LinearEquiv.prod_symm,\n      LinearEquiv.prod_apply, b.repr.apply_symm_apply, LinearEquiv.symm_symm, repr_self,\n      Equiv.toFun_as_coe, Finsupp.fst_sumFinsuppLEquivProdFinsupp]","premises":[{"full_name":"Basis.coe_ofRepr","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[117,8],"def_end_pos":[117,18]},{"full_name":"Basis.prod","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[661,14],"def_end_pos":[661,18]},{"full_name":"Basis.repr","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[91,4],"def_end_pos":[91,8]},{"full_name":"Basis.repr_self","def_path":"Mathlib/LinearAlgebra/Basis.lean","def_pos":[133,8],"def_end_pos":[133,17]},{"full_name":"Equiv.toFun_as_coe","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[171,28],"def_end_pos":[171,40]},{"full_name":"Finsupp.fst_sumFinsuppLEquivProdFinsupp","def_path":"Mathlib/LinearAlgebra/Finsupp.lean","def_pos":[947,8],"def_end_pos":[947,39]},{"full_name":"LinearEquiv.apply_symm_apply","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[342,8],"def_end_pos":[342,24]},{"full_name":"LinearEquiv.prod_apply","def_path":"Mathlib/LinearAlgebra/Prod.lean","def_pos":[706,8],"def_end_pos":[706,18]},{"full_name":"LinearEquiv.prod_symm","def_path":"Mathlib/LinearAlgebra/Prod.lean","def_pos":[702,8],"def_end_pos":[702,17]},{"full_name":"LinearEquiv.symm_symm","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[455,8],"def_end_pos":[455,17]},{"full_name":"LinearEquiv.symm_trans_apply","def_path":"Mathlib/Algebra/Module/Equiv/Defs.lean","def_pos":[353,8],"def_end_pos":[353,24]}]},{"state_before":"case h\nι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni j : ι\n⊢ (Finsupp.single (Sum.inl i) 1) (Sum.inl j) = (Finsupp.single i 1) j","state_after":"no goals","tactic":"apply Finsupp.single_apply_left Sum.inl_injective","premises":[{"full_name":"Finsupp.single_apply_left","def_path":"Mathlib/Data/Finsupp/Defs.lean","def_pos":[249,8],"def_end_pos":[249,25]},{"full_name":"Sum.inl_injective","def_path":"Mathlib/Data/Sum/Basic.lean","def_pos":[28,8],"def_end_pos":[28,21]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.eq_prod_range_div'","start":[1448,0],"end":[1452,42],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nM : Type u_6\ninst✝ : CommGroup M\nf : ℕ → M\nn : ℕ\n⊢ f n = ∏ i ∈ range (n + 1), if i = 0 then f 0 else f i / f (i - 1)","state_after":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nM : Type u_6\ninst✝ : CommGroup M\nf : ℕ → M\nn : ℕ\n⊢ f 0 * ∏ i ∈ range n, f (i + 1) / f i = ∏ i ∈ range (n + 1), if i = 0 then f 0 else f i / f (i - 1)","tactic":"conv_lhs => rw [Finset.eq_prod_range_div f]","premises":[{"full_name":"Finset.eq_prod_range_div","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1445,8],"def_end_pos":[1445,25]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nM : Type u_6\ninst✝ : CommGroup M\nf : ℕ → M\nn : ℕ\n⊢ f 0 * ∏ i ∈ range n, f (i + 1) / f i = ∏ i ∈ range (n + 1), if i = 0 then f 0 else f i / f (i - 1)","state_after":"no goals","tactic":"simp [Finset.prod_range_succ', mul_comm]","premises":[{"full_name":"Finset.prod_range_succ'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1302,8],"def_end_pos":[1302,24]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","commit":"","full_name":"lipschitzOnWith_univ","start":[83,0],"end":[85,39],"file_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\ns t : Set α\nf : α → β\n⊢ LipschitzOnWith K f univ ↔ LipschitzWith K f","state_after":"no goals","tactic":"simp [LipschitzOnWith, LipschitzWith]","premises":[{"full_name":"LipschitzOnWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[58,4],"def_end_pos":[58,19]},{"full_name":"LipschitzWith","def_path":"Mathlib/Topology/EMetricSpace/Lipschitz.lean","def_pos":[54,4],"def_end_pos":[54,17]}]}]}
{"url":"Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean","commit":"","full_name":"ContinuousAffineMap.norm_eq","start":[153,0],"end":[158,54],"file_path":"Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean","tactics":[{"state_before":"𝕜 : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW₂ : Type u_5\nP : Type u_6\nQ : Type u_7\nQ₂ : Type u_8\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →ᴬ[𝕜] W\nh : f 0 = 0\n⊢ ‖f‖ = max ‖f 0‖ ‖f.contLinear‖","state_after":"no goals","tactic":"rw [norm_def]","premises":[{"full_name":"ContinuousAffineMap.norm_def","def_path":"Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean","def_pos":[144,8],"def_end_pos":[144,16]}]},{"state_before":"𝕜 : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW₂ : Type u_5\nP : Type u_6\nQ : Type u_7\nQ₂ : Type u_8\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →ᴬ[𝕜] W\nh : f 0 = 0\n⊢ max ‖f 0‖ ‖f.contLinear‖ = max 0 ‖f.contLinear‖","state_after":"no goals","tactic":"rw [h, norm_zero]","premises":[{"full_name":"norm_zero","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[436,29],"def_end_pos":[436,38]}]}]}
{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"nhds_basis_closeds","start":[696,0],"end":[698,84],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx✝ : X\ns s₁ s₂ t✝ : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\nx : X\nt : Set X\n⊢ (∃ x_1, (x ∈ x_1ᶜ ∧ IsOpen x_1ᶜ) ∧ x_1ᶜ ⊆ t) ↔ ∃ i, (x ∉ i ∧ IsClosed i) ∧ iᶜ ⊆ t","state_after":"no goals","tactic":"simp only [isOpen_compl_iff, mem_compl_iff]","premises":[{"full_name":"Set.mem_compl_iff","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[84,16],"def_end_pos":[84,29]},{"full_name":"isOpen_compl_iff","def_path":"Mathlib/Topology/Basic.lean","def_pos":[143,16],"def_end_pos":[143,32]}]}]}
{"url":"Mathlib/Data/Stream/Init.lean","commit":"","full_name":"Stream'.eta","start":[28,0],"end":[29,36],"file_path":"Mathlib/Data/Stream/Init.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\ni : ℕ\n⊢ (s.head :: s.tail) i = s i","state_after":"no goals","tactic":"cases i <;> rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieModuleHom.comp_ker_incl","start":[1253,0],"end":[1254,88],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : M →ₗ⁅R,L⁆ N\n⊢ f.comp f.ker.incl = 0","state_after":"case h.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : M →ₗ⁅R,L⁆ N\nm : M\nhm : m ∈ ↑f.ker\n⊢ (f.comp f.ker.incl) ⟨m, hm⟩ = 0 ⟨m, hm⟩","tactic":"ext ⟨m, hm⟩","premises":[]},{"state_before":"case h.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nf : M →ₗ⁅R,L⁆ N\nm : M\nhm : m ∈ ↑f.ker\n⊢ (f.comp f.ker.incl) ⟨m, hm⟩ = 0 ⟨m, hm⟩","state_after":"no goals","tactic":"exact (mem_ker m).mp hm","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"LieModuleHom.mem_ker","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1246,8],"def_end_pos":[1246,15]}]}]}
{"url":"Mathlib/Topology/Homeomorph.lean","commit":"","full_name":"Homeomorph.image_frontier","start":[361,0],"end":[362,41],"file_path":"Mathlib/Topology/Homeomorph.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝¹ : TopologicalSpace X'\ninst✝ : TopologicalSpace Y'\nh : X ≃ₜ Y\ns : Set X\n⊢ ⇑h '' frontier s = frontier (⇑h '' s)","state_after":"no goals","tactic":"rw [← preimage_symm, preimage_frontier]","premises":[{"full_name":"Homeomorph.preimage_frontier","def_path":"Mathlib/Topology/Homeomorph.lean","def_pos":[358,8],"def_end_pos":[358,25]},{"full_name":"Homeomorph.preimage_symm","def_path":"Mathlib/Topology/Homeomorph.lean","def_pos":[194,8],"def_end_pos":[194,21]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.support_C_mul_X_pow'","start":[746,0],"end":[747,66],"file_path":"Mathlib/Algebra/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nc : R\n⊢ (C c * X ^ n).support ⊆ {n}","state_after":"no goals","tactic":"simpa only [C_mul_X_pow_eq_monomial] using support_monomial' n c","premises":[{"full_name":"Polynomial.C_mul_X_pow_eq_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[646,8],"def_end_pos":[646,31]},{"full_name":"Polynomial.support_monomial'","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[732,8],"def_end_pos":[732,25]}]}]}
{"url":"Mathlib/Data/Real/Sqrt.lean","commit":"","full_name":"Real.sqrt_eq_zero","start":[250,0],"end":[251,85],"file_path":"Mathlib/Data/Real/Sqrt.lean","tactics":[{"state_before":"x y : ℝ\nh : 0 ≤ x\n⊢ √x = 0 ↔ x = 0","state_after":"no goals","tactic":"simpa using sqrt_inj h le_rfl","premises":[{"full_name":"Real.sqrt_inj","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[247,8],"def_end_pos":[247,16]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean","commit":"","full_name":"CategoryTheory.coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily","start":[78,0],"end":[99,64],"file_path":"Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Precoherent C\nX : C\nS : Sieve X\n⊢ S ∈ (coherentTopology C).sieves X ↔ ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)","state_after":"case mp\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Precoherent C\nX : C\nS : Sieve X\n⊢ S ∈ (coherentTopology C).sieves X → ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)\n\ncase mpr\nC : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Precoherent C\nX : C\nS : Sieve X\n⊢ (∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)) → S ∈ (coherentTopology C).sieves X","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean","commit":"","full_name":"HasDerivAt.rpow_const","start":[529,0],"end":[532,24],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean","tactics":[{"state_before":"f g : ℝ → ℝ\nf' g' x y p : ℝ\ns : Set ℝ\nhf : HasDerivAt f f' x\nhx : f x ≠ 0 ∨ 1 ≤ p\n⊢ HasDerivAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) x","state_after":"f g : ℝ → ℝ\nf' g' x y p : ℝ\ns : Set ℝ\nhf : HasDerivWithinAt f f' Set.univ x\nhx : f x ≠ 0 ∨ 1 ≤ p\n⊢ HasDerivWithinAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) Set.univ x","tactic":"rw [← hasDerivWithinAt_univ] at *","premises":[{"full_name":"hasDerivWithinAt_univ","def_path":"Mathlib/Analysis/Calculus/Deriv/Basic.lean","def_pos":[340,8],"def_end_pos":[340,29]}]},{"state_before":"f g : ℝ → ℝ\nf' g' x y p : ℝ\ns : Set ℝ\nhf : HasDerivWithinAt f f' Set.univ x\nhx : f x ≠ 0 ∨ 1 ≤ p\n⊢ HasDerivWithinAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) Set.univ x","state_after":"no goals","tactic":"exact hf.rpow_const hx","premises":[{"full_name":"HasDerivWithinAt.rpow_const","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean","def_pos":[524,8],"def_end_pos":[524,35]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Homology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.HomologyData.op_right","start":[181,0],"end":[187,39],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Homology.lean","tactics":[{"state_before":"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ S₄ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nh : S.HomologyData\n⊢ (h.right.op.π ≫ h.iso.op.hom ≫ h.left.op.ι).unop = (h.right.op.i ≫ h.left.op.p).unop","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","commit":"","full_name":"AlgebraicClosure.AdjoinMonic.isIntegral","start":[118,0],"end":[128,54],"file_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","tactics":[{"state_before":"k : Type u\ninst✝ : Field k\nz : AdjoinMonic k\n⊢ IsIntegral k z","state_after":"k : Type u\ninst✝ : Field k\nz : AdjoinMonic k\np : MvPolynomial (MonicIrreducible k) k\nhp : (Ideal.Quotient.mk (maxIdeal k)) p = z\n⊢ IsIntegral k z","tactic":"let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z","premises":[{"full_name":"Ideal.Quotient.mk_surjective","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[141,8],"def_end_pos":[141,21]}]},{"state_before":"k : Type u\ninst✝ : Field k\nz : AdjoinMonic k\np : MvPolynomial (MonicIrreducible k) k\nhp : (Ideal.Quotient.mk (maxIdeal k)) p = z\n⊢ IsIntegral k z","state_after":"k : Type u\ninst✝ : Field k\nz : AdjoinMonic k\np : MvPolynomial (MonicIrreducible k) k\nhp : (Ideal.Quotient.mk (maxIdeal k)) p = z\n⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)","tactic":"rw [← hp]","premises":[]},{"state_before":"k : Type u\ninst✝ : Field k\nz : AdjoinMonic k\np : MvPolynomial (MonicIrreducible k) k\nhp : (Ideal.Quotient.mk (maxIdeal k)) p = z\n⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)","state_after":"no goals","tactic":"induction p using MvPolynomial.induction_on generalizing z with\n    | h_C => exact isIntegral_algebraMap\n    | h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)\n    | h_X p f ih =>\n      refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _) (ih _ rfl) ?_\n      refine ⟨f, f.2.1, ?_⟩\n      erw [AdjoinMonic.algebraMap, ← hom_eval₂, Ideal.Quotient.eq_zero_iff_mem]\n      exact le_maxIdeal k (Ideal.subset_span ⟨f, rfl⟩)","premises":[{"full_name":"AlgebraicClosure.AdjoinMonic.algebraMap","def_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","def_pos":[115,8],"def_end_pos":[115,30]},{"full_name":"AlgebraicClosure.le_maxIdeal","def_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","def_pos":[94,8],"def_end_pos":[94,19]},{"full_name":"AlgebraicClosure.maxIdeal","def_path":"Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","def_pos":[88,4],"def_end_pos":[88,12]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Ideal.Quotient.eq_zero_iff_mem","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[112,8],"def_end_pos":[112,23]},{"full_name":"Ideal.Quotient.mk","def_path":"Mathlib/RingTheory/Ideal/Quotient.lean","def_pos":[84,4],"def_end_pos":[84,6]},{"full_name":"Ideal.subset_span","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[125,8],"def_end_pos":[125,19]},{"full_name":"IsIntegral.add","def_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","def_pos":[81,15],"def_end_pos":[81,29]},{"full_name":"IsIntegral.mul","def_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","def_pos":[107,15],"def_end_pos":[107,29]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MvPolynomial.induction_on","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[399,8],"def_end_pos":[399,20]},{"full_name":"Polynomial.hom_eval₂","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[872,8],"def_end_pos":[872,17]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"isIntegral_algebraMap","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean","def_pos":[29,8],"def_end_pos":[29,29]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","commit":"","full_name":"WeierstrassCurve.baseChange_preΨ","start":[566,0],"end":[567,33],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","tactics":[{"state_before":"R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsScalarTower R S B\nf : A →ₐ[S] B\nn : ℤ\n⊢ (W.baseChange B).preΨ n = Polynomial.map (↑f) ((W.baseChange A).preΨ n)","state_after":"no goals","tactic":"rw [← map_preΨ, map_baseChange]","premises":[{"full_name":"WeierstrassCurve.map_baseChange","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean","def_pos":[381,6],"def_end_pos":[381,20]},{"full_name":"WeierstrassCurve.map_preΨ","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean","def_pos":[518,6],"def_end_pos":[518,14]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean","commit":"","full_name":"jacobiTheta_eq_jacobiTheta₂","start":[27,0],"end":[28,42],"file_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean","tactics":[{"state_before":"τ : ℂ\n⊢ ∀ (b : ℤ), cexp (↑π * I * ↑b ^ 2 * τ) = jacobiTheta₂_term b 0 τ","state_after":"no goals","tactic":"simp [jacobiTheta₂_term]","premises":[{"full_name":"jacobiTheta₂_term","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[40,4],"def_end_pos":[40,21]}]}]}
{"url":"Mathlib/CategoryTheory/Galois/Prorepresentability.lean","commit":"","full_name":"CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_symm_app","start":[374,0],"end":[379,5],"file_path":"Mathlib/CategoryTheory/Galois/Prorepresentability.lean","tactics":[{"state_before":"C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nx : AutGalois F\nA : C\ninst✝ : IsGalois A\na : ↑(F.obj A)\n⊢ ((autMulEquivAutGalois F).symm { unop' := x }).hom.app A a =\n    F.map ((AutGalois.π F { obj := A, pt := a, isGalois := ⋯ }) x).hom a","state_after":"C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nx : AutGalois F\nA : C\ninst✝ : IsGalois A\na : ↑(F.obj A)\n⊢ F.map ((AutGalois.π F { obj := A, pt := a, isGalois := ⋯ }) (MulOpposite.unop { unop' := x })).hom a =\n    F.map ((AutGalois.π F { obj := A, pt := a, isGalois := ⋯ }) x).hom a","tactic":"rw [← autMulEquivAutGalois_π, MulEquiv.apply_symm_apply]","premises":[{"full_name":"CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_π","def_path":"Mathlib/CategoryTheory/Galois/Prorepresentability.lean","def_pos":[367,6],"def_end_pos":[367,28]},{"full_name":"MulEquiv.apply_symm_apply","def_path":"Mathlib/Algebra/Group/Equiv/Basic.lean","def_pos":[308,8],"def_end_pos":[308,24]}]},{"state_before":"C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nx : AutGalois F\nA : C\ninst✝ : IsGalois A\na : ↑(F.obj A)\n⊢ F.map ((AutGalois.π F { obj := A, pt := a, isGalois := ⋯ }) (MulOpposite.unop { unop' := x })).hom a =\n    F.map ((AutGalois.π F { obj := A, pt := a, isGalois := ⋯ }) x).hom a","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","commit":"","full_name":"cfcₙHom_nonneg_iff","start":[510,0],"end":[516,53],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : OrderedCommSemiring R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : StarOrderedRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : TopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ∀ (α : Type ?u.1065866) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : StarModule R A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus R p\ninst✝ : NonnegSpectrumClass R A\na : A\nha : p a\nf : C(↑(σₙ R a), R)₀\n⊢ 0 ≤ (cfcₙHom ha) f ↔ 0 ≤ f","state_after":"case mp\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : OrderedCommSemiring R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : StarOrderedRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : TopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ∀ (α : Type ?u.1065866) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : StarModule R A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus R p\ninst✝ : NonnegSpectrumClass R A\na : A\nha : p a\nf : C(↑(σₙ R a), R)₀\n⊢ 0 ≤ (cfcₙHom ha) f → 0 ≤ f\n\ncase mpr\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : OrderedCommSemiring R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : StarOrderedRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : TopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ∀ (α : Type ?u.1065866) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : StarModule R A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus R p\ninst✝ : NonnegSpectrumClass R A\na : A\nha : p a\nf : C(↑(σₙ R a), R)₀\n⊢ 0 ≤ f → 0 ≤ (cfcₙHom ha) f","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Data/Complex/Abs.lean","commit":"","full_name":"_private.Mathlib.Data.Complex.Abs.0.Complex.AbsTheory.abs_mul","start":[41,0],"end":[42,50],"file_path":"Mathlib/Data/Complex/Abs.lean","tactics":[{"state_before":"z w : ℂ\n⊢ (abs z * w) = (abs z) * abs w","state_after":"no goals","tactic":"rw [normSq_mul, Real.sqrt_mul (normSq_nonneg _)]","premises":[{"full_name":"Complex.normSq_mul","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[601,8],"def_end_pos":[601,18]},{"full_name":"Complex.normSq_nonneg","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[582,8],"def_end_pos":[582,21]},{"full_name":"Real.sqrt_mul","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[317,8],"def_end_pos":[317,16]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean","commit":"","full_name":"LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure","start":[27,0],"end":[71,66],"file_path":"Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\nx : E\n⊢ Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 ↑(g x))","state_after":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\n⊢ Tendsto (fun x => birkhoffAverage 𝕜 (⇑f) _root_.id x (y + z)) atTop (𝓝 ↑(g (y + z)))","tactic":"obtain ⟨y, hy, z, hz, rfl⟩ : ∃ y, g y = 0 ∧ ∃ z, IsFixedPt f z ∧ x = y + z :=\n    ⟨x - g x, by simp [hg_proj], g x, (g x).2, by simp⟩","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Function.IsFixedPt","def_path":"Mathlib/Dynamics/FixedPoints/Basic.lean","def_pos":[37,4],"def_end_pos":[37,13]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]}]},{"state_before":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\n⊢ Tendsto (fun x => birkhoffAverage 𝕜 (⇑f) _root_.id x y) atTop (𝓝 0)","state_after":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\nx : E\n⊢ (f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}","tactic":"refine closure_minimal (Set.forall_mem_range.2 fun x ↦ ?_) this (hg_ker hy)","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.forall_mem_range","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[564,8],"def_end_pos":[564,24]},{"full_name":"closure_minimal","def_path":"Mathlib/Topology/Basic.lean","def_pos":[353,8],"def_end_pos":[353,23]}]},{"state_before":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis✝ : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\nx : E\nthis : Bornology.IsBounded (Set.range fun x_1 => _root_.id ((⇑f)^[x_1] x))\n⊢ (f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}","state_after":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis✝ : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\nx : E\nthis : Bornology.IsBounded (Set.range fun x_1 => _root_.id ((⇑f)^[x_1] x))\nH : ∀ (n : ℕ) (x y : E), (⇑f)^[n] (x - y) = (⇑f)^[n] x - (⇑f)^[n] y\n⊢ (f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}","tactic":"have H : ∀ n x y, f^[n] (x - y) = f^[n] x - f^[n] y := iterate_map_sub (f : E →+ E)","premises":[{"full_name":"AddMonoidHom","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[132,10],"def_end_pos":[132,22]},{"full_name":"Nat.iterate","def_path":"Mathlib/Logic/Function/Iterate.lean","def_pos":[36,4],"def_end_pos":[36,15]},{"full_name":"iterate_map_sub","def_path":"Mathlib/Algebra/GroupPower/IterateHom.lean","def_pos":[62,2],"def_end_pos":[62,13]}]},{"state_before":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis✝ : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\nx : E\nthis : Bornology.IsBounded (Set.range fun x_1 => _root_.id ((⇑f)^[x_1] x))\nH : ∀ (n : ℕ) (x y : E), (⇑f)^[n] (x - y) = (⇑f)^[n] x - (⇑f)^[n] y\n⊢ (f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}","state_after":"no goals","tactic":"simpa [birkhoffAverage, birkhoffSum, Finset.sum_sub_distrib, smul_sub, H]\n    using tendsto_birkhoffAverage_apply_sub_birkhoffAverage 𝕜 this","premises":[{"full_name":"Finset.sum_sub_distrib","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1807,2],"def_end_pos":[1807,13]},{"full_name":"birkhoffAverage","def_path":"Mathlib/Dynamics/BirkhoffSum/Average.lean","def_pos":[42,4],"def_end_pos":[42,19]},{"full_name":"birkhoffSum","def_path":"Mathlib/Dynamics/BirkhoffSum/Basic.lean","def_pos":[27,4],"def_end_pos":[27,15]},{"full_name":"smul_sub","def_path":"Mathlib/Algebra/GroupWithZero/Action/Defs.lean","def_pos":[279,8],"def_end_pos":[279,16]},{"full_name":"tendsto_birkhoffAverage_apply_sub_birkhoffAverage","def_path":"Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean","def_pos":[75,8],"def_end_pos":[75,57]}]}]}
{"url":"Mathlib/Logic/Equiv/Fintype.lean","commit":"","full_name":"Equiv.extendSubtype_apply_of_mem","start":[112,0],"end":[116,73],"file_path":"Mathlib/Logic/Equiv/Fintype.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ e.extendSubtype x = ↑(e ⟨x, hx⟩)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ (e.subtypeCongr e.toCompl) x = ↑(e ⟨x, hx⟩)","tactic":"dsimp only [extendSubtype]","premises":[{"full_name":"Equiv.extendSubtype","def_path":"Mathlib/Logic/Equiv/Fintype.lean","def_pos":[109,21],"def_end_pos":[109,34]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ (e.subtypeCongr e.toCompl) x = ↑(e ⟨x, hx⟩)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ (sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e ⟨x, hx⟩)","tactic":"simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply]","premises":[{"full_name":"Equiv.subtypeCongr","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[511,4],"def_end_pos":[511,16]},{"full_name":"Equiv.sumCongr_apply","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[249,8],"def_end_pos":[249,13]},{"full_name":"Equiv.trans_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[237,16],"def_end_pos":[237,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ (sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e ⟨x, hx⟩)","state_after":"no goals","tactic":"rw [sumCompl_apply_symm_of_pos _ _ hx, Sum.map_inl, sumCompl_apply_inl]","premises":[{"full_name":"Equiv.sumCompl_apply_inl","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[490,8],"def_end_pos":[490,26]},{"full_name":"Equiv.sumCompl_apply_symm_of_pos","def_path":"Mathlib/Logic/Equiv/Basic.lean","def_pos":[500,8],"def_end_pos":[500,34]},{"full_name":"Sum.map_inl","def_path":".lake/packages/batteries/Batteries/Data/Sum/Basic.lean","def_pos":[95,16],"def_end_pos":[95,23]}]}]}
{"url":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","commit":"","full_name":"AddSubmonoid.LocalizationMap.lift_of_comp","start":[777,0],"end":[782,30],"file_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","tactics":[{"state_before":"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\n⊢ f.lift ⋯ = j","state_after":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ (f.lift ⋯) x✝ = j x✝","tactic":"ext","premises":[]},{"state_before":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ (f.lift ⋯) x✝ = j x✝","state_after":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ (j.comp f.toMap) (f.sec x✝).1 = (j.comp f.toMap) ↑(f.sec x✝).2 * j x✝","tactic":"rw [lift_spec]","premises":[{"full_name":"Submonoid.LocalizationMap.lift_spec","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[726,8],"def_end_pos":[726,17]}]},{"state_before":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ (j.comp f.toMap) (f.sec x✝).1 = (j.comp f.toMap) ↑(f.sec x✝).2 * j x✝","state_after":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ j (f.toMap (f.sec x✝).1) = j (f.toMap ↑(f.sec x✝).2) * j x✝","tactic":"show j _ = j _ * _","premises":[]},{"state_before":"case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ j (f.toMap (f.sec x✝).1) = j (f.toMap ↑(f.sec x✝).2) * j x✝","state_after":"no goals","tactic":"erw [← j.map_mul, sec_spec']","premises":[{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"MonoidHom.map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[630,18],"def_end_pos":[630,35]},{"full_name":"Submonoid.LocalizationMap.sec_spec'","def_path":"Mathlib/GroupTheory/MonoidLocalization/Basic.lean","def_pos":[440,8],"def_end_pos":[440,17]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Fin.lean","commit":"","full_name":"Fin.sum_ofFn","start":[41,0],"end":[43,30],"file_path":"Mathlib/Algebra/BigOperators/Fin.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin n → β\n⊢ (List.ofFn f).prod = ∏ i : Fin n, f i","state_after":"no goals","tactic":"simp [prod_eq_multiset_prod]","premises":[{"full_name":"Finset.prod_eq_multiset_prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[267,8],"def_end_pos":[267,29]}]}]}
{"url":"Mathlib/Analysis/Convolution.lean","commit":"","full_name":"MeasureTheory.convolutionExistsAt_flip","start":[367,0],"end":[370,31],"file_path":"Mathlib/Analysis/Convolution.lean","tactics":[{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedAddCommGroup E'\ninst✝¹² : NormedAddCommGroup E''\ninst✝¹¹ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace 𝕜 E''\ninst✝⁶ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁵ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁴ : AddCommGroup G\ninst✝³ : MeasurableNeg G\ninst✝² : μ.IsAddLeftInvariant\ninst✝¹ : MeasurableAdd G\ninst✝ : μ.IsNegInvariant\n⊢ ConvolutionExistsAt g f x L.flip μ ↔ ConvolutionExistsAt f g x L μ","state_after":"no goals","tactic":"simp_rw [ConvolutionExistsAt, ← integrable_comp_sub_left (fun t => L (f t) (g (x - t))) x,\n    sub_sub_cancel, flip_apply]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ContinuousLinearMap.flip_apply","def_path":"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean","def_pos":[163,8],"def_end_pos":[163,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasureTheory.ConvolutionExistsAt","def_path":"Mathlib/Analysis/Convolution.lean","def_pos":[158,4],"def_end_pos":[158,23]},{"full_name":"MeasureTheory.integrable_comp_sub_left","def_path":"Mathlib/MeasureTheory/Group/Integral.lean","def_pos":[116,2],"def_end_pos":[116,13]},{"full_name":"sub_sub_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[910,2],"def_end_pos":[910,13]}]}]}
{"url":"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean","commit":"","full_name":"mem_zmultiples_of_prime_card","start":[140,0],"end":[143,63],"file_path":"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean","tactics":[{"state_before":"α : Type u\na : α\ninst✝¹ : Group α\nG : Type u_1\ninst✝ : Group G\nx✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card G = p\ng g' : G\nhg : g ≠ 1\n⊢ g' ∈ zpowers g","state_after":"no goals","tactic":"simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Subgroup.mem_top","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[660,8],"def_end_pos":[660,15]},{"full_name":"zpowers_eq_top_of_prime_card","def_path":"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean","def_pos":[134,8],"def_end_pos":[134,36]}]}]}
{"url":"Mathlib/Analysis/InnerProductSpace/Calculus.lean","commit":"","full_name":"contDiffAt_norm","start":[143,0],"end":[145,81],"file_path":"Mathlib/Analysis/InnerProductSpace/Calculus.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : _root_.RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : E\nhx : x ≠ 0\n⊢ ContDiffAt ℝ n Norm.norm x","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : _root_.RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : E\nhx : x ≠ 0\nthis : ‖id x‖ ^ 2 ≠ 0\n⊢ ContDiffAt ℝ n Norm.norm x","tactic":"have : ‖id x‖ ^ 2 ≠ 0 := pow_ne_zero 2 (norm_pos_iff.2 hx).ne'","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"LT.lt.ne'","def_path":"Mathlib/Order/Basic.lean","def_pos":[267,8],"def_end_pos":[267,11]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"norm_pos_iff","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[1216,29],"def_end_pos":[1216,41]},{"full_name":"pow_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[172,6],"def_end_pos":[172,17]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : _root_.RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : E\nhx : x ≠ 0\nthis : ‖id x‖ ^ 2 ≠ 0\n⊢ ContDiffAt ℝ n Norm.norm x","state_after":"no goals","tactic":"simpa only [id, sqrt_sq, norm_nonneg] using (contDiffAt_id.norm_sq 𝕜).sqrt this","premises":[{"full_name":"ContDiffAt.norm_sq","def_path":"Mathlib/Analysis/InnerProductSpace/Calculus.lean","def_pos":[140,15],"def_end_pos":[140,33]},{"full_name":"ContDiffAt.sqrt","def_path":"Mathlib/Analysis/SpecialFunctions/Sqrt.lean","def_pos":[140,8],"def_end_pos":[140,23]},{"full_name":"Real.sqrt_sq","def_path":"Mathlib/Data/Real/Sqrt.lean","def_pos":[173,8],"def_end_pos":[173,15]},{"full_name":"contDiffAt_id","def_path":"Mathlib/Analysis/Calculus/ContDiff/Basic.lean","def_pos":[170,8],"def_end_pos":[170,21]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]}]}]}
{"url":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","commit":"","full_name":"_private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.optimalGHDist_mem_candidatesB","start":[416,0],"end":[418,12],"file_path":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","tactics":[{"state_before":"X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\n⊢ GromovHausdorff.optimalGHDist X Y ∈ GromovHausdorff.candidatesB X Y","state_after":"case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nleft✝ : Classical.choose ⋯ ∈ GromovHausdorff.candidatesB X Y\nright✝ : ∀ g ∈ GromovHausdorff.candidatesB X Y, HD (Classical.choose ⋯) ≤ HD g\n⊢ GromovHausdorff.optimalGHDist X Y ∈ GromovHausdorff.candidatesB X Y","tactic":"cases Classical.choose_spec (exists_minimizer X Y)","premises":[{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"_private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.exists_minimizer","def_path":"Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean","def_pos":[410,16],"def_end_pos":[410,32]}]},{"state_before":"case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nleft✝ : Classical.choose ⋯ ∈ GromovHausdorff.candidatesB X Y\nright✝ : ∀ g ∈ GromovHausdorff.candidatesB X Y, HD (Classical.choose ⋯) ≤ HD g\n⊢ GromovHausdorff.optimalGHDist X Y ∈ GromovHausdorff.candidatesB X Y","state_after":"no goals","tactic":"assumption","premises":[]}]}
{"url":"Mathlib/Combinatorics/SimpleGraph/Turan.lean","commit":"","full_name":"SimpleGraph.IsTuranMaximal.equivalence_not_adj","start":[172,0],"end":[176,26],"file_path":"Mathlib/Combinatorics/SimpleGraph/Turan.lean","tactics":[{"state_before":"V : Type u_1\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn r : ℕ\ns t u : V\nh : G.IsTuranMaximal r\n⊢ ∀ (x : V), ¬G.Adj x x","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"V : Type u_1\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn r : ℕ\ns t u : V\nh : G.IsTuranMaximal r\n⊢ ∀ {x y : V}, ¬G.Adj x y → ¬G.Adj y x","state_after":"no goals","tactic":"simp [adj_comm]","premises":[{"full_name":"SimpleGraph.adj_comm","def_path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","def_pos":[166,8],"def_end_pos":[166,16]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","commit":"","full_name":"Polynomial.abs_isBoundedUnder_iff","start":[82,0],"end":[88,67],"file_path":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ (IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => |eval x P|) ↔ P.degree ≤ 0","state_after":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => |eval x P|\n⊢ P.degree ≤ 0","tactic":"refine ⟨fun h => ?_, fun h => ⟨|P.coeff 0|, eventually_map.mpr (eventually_of_forall\n    (forall_imp (fun _ => le_of_eq) fun x => congr_arg abs <| _root_.trans (congr_arg (eval x)\n    (eq_C_of_degree_le_zero h)) eval_C))⟩⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.eventually_map","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1650,8],"def_end_pos":[1650,22]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Polynomial.coeff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[557,4],"def_end_pos":[557,9]},{"full_name":"Polynomial.eq_C_of_degree_le_zero","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[537,8],"def_end_pos":[537,30]},{"full_name":"Polynomial.eval","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[266,4],"def_end_pos":[266,8]},{"full_name":"Polynomial.eval_C","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[308,8],"def_end_pos":[308,14]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]},{"full_name":"forall_imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[183,8],"def_end_pos":[183,18]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]},{"full_name":"trans","def_path":"Mathlib/Init/Algebra/Classes.lean","def_pos":[261,8],"def_end_pos":[261,13]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => |eval x P|\n⊢ P.degree ≤ 0","state_after":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : 0 < P.degree\n⊢ ¬IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => |eval x P|","tactic":"contrapose! h","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : 0 < P.degree\n⊢ ¬IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => |eval x P|","state_after":"no goals","tactic":"exact not_isBoundedUnder_of_tendsto_atTop (abs_tendsto_atTop P h)","premises":[{"full_name":"Filter.not_isBoundedUnder_of_tendsto_atTop","def_path":"Mathlib/Order/LiminfLimsup.lean","def_pos":[138,8],"def_end_pos":[138,43]},{"full_name":"Polynomial.abs_tendsto_atTop","def_path":"Mathlib/Analysis/SpecialFunctions/Polynomials.lean","def_pos":[76,8],"def_end_pos":[76,25]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one","start":[1715,0],"end":[1730,62],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\n⊢ Pairwise (AEDisjoint μ on fun g => g • s)","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\nhg : g₁ ≠ g₂\n⊢ (AEDisjoint μ on fun g => g • s) g₁ g₂","tactic":"intro g₁ g₂ hg","premises":[]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\nhg : g₁ ≠ g₂\n⊢ (AEDisjoint μ on fun g => g • s) g₁ g₂","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\nhg : g₁ ≠ g₂\ng : G := g₂⁻¹ * g₁\n⊢ (AEDisjoint μ on fun g => g • s) g₁ g₂","tactic":"let g := g₂⁻¹ * g₁","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\nhg : g₁ ≠ g₂\ng : G := g₂⁻¹ * g₁\n⊢ (AEDisjoint μ on fun g => g • s) g₁ g₂","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\ng : G := g₂⁻¹ * g₁\nhg : g ≠ 1\n⊢ (AEDisjoint μ on fun g => g • s) g₁ g₂","tactic":"replace hg : g ≠ 1 := by\n    rw [Ne, inv_mul_eq_one]\n    exact hg.symm","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"inv_mul_eq_one","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[673,8],"def_end_pos":[673,22]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\ng : G := g₂⁻¹ * g₁\nhg : g ≠ 1\nthis : (fun x => g₂⁻¹ • x) ⁻¹' (g • s ∩ s) = g₁ • s ∩ g₂ • s\n⊢ (AEDisjoint μ on fun g => g • s) g₁ g₂","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\ng : G := g₂⁻¹ * g₁\nhg : g ≠ 1\nthis : (fun x => g₂⁻¹ • x) ⁻¹' (g • s ∩ s) = g₁ • s ∩ g₂ • s\n⊢ μ (g₁ • s ∩ g₂ • s) = 0","tactic":"change μ (g₁ • s ∩ g₂ • s) = 0","premises":[{"full_name":"Inter.inter","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[407,2],"def_end_pos":[407,7]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\nf : α✝ → β\nG : Type u_8\nα : Type u_9\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ\ng₁ g₂ : G\ng : G := g₂⁻¹ * g₁\nhg : g ≠ 1\nthis : (fun x => g₂⁻¹ • x) ⁻¹' (g • s ∩ s) = g₁ • s ∩ g₂ • s\n⊢ μ (g₁ • s ∩ g₂ • s) = 0","state_after":"no goals","tactic":"exact this ▸ (h_qmp g₂⁻¹).preimage_null (h_ae_disjoint g hg)","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"MeasureTheory.Measure.QuasiMeasurePreserving.preimage_null","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1634,8],"def_end_pos":[1634,21]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/WithDensity.lean","commit":"","full_name":"MeasureTheory.withDensity_apply'","start":[66,0],"end":[73,79],"file_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","tactics":[{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s ≤ ∫⁻ (a : α) in s, f a ∂μ","tactic":"apply le_antisymm ?_ (withDensity_apply_le f s)","premises":[{"full_name":"MeasureTheory.withDensity_apply_le","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[42,8],"def_end_pos":[42,28]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s ≤ ∫⁻ (a : α) in s, f a ∂μ","state_after":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ (μ.withDensity f) s ≤ ∫⁻ (a : α) in s, f a ∂μ","tactic":"let t := toMeasurable μ s","premises":[{"full_name":"MeasureTheory.toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[294,16],"def_end_pos":[294,28]}]},{"state_before":"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ (μ.withDensity f) s ≤ ∫⁻ (a : α) in s, f a ∂μ","state_after":"no goals","tactic":"calc\n  μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)\n  _ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)\n  _ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFinite s","premises":[{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.Measure.restrict_toMeasurable_of_sFinite","def_path":"Mathlib/MeasureTheory/Measure/Typeclasses.lean","def_pos":[880,8],"def_end_pos":[880,40]},{"full_name":"MeasureTheory.Measure.withDensity","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[33,4],"def_end_pos":[33,23]},{"full_name":"MeasureTheory.lintegral","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[59,16],"def_end_pos":[59,25]},{"full_name":"MeasureTheory.measurableSet_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[310,8],"def_end_pos":[310,34]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"MeasureTheory.subset_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[300,8],"def_end_pos":[300,27]},{"full_name":"MeasureTheory.withDensity_apply","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[38,8],"def_end_pos":[38,25]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","commit":"","full_name":"MeasureTheory.lintegral_sub'","start":[953,0],"end":[957,75],"file_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\nhg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤\nh_le : g ≤ᶠ[ae μ] f\n⊢ ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\nhg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤\nh_le : g ≤ᶠ[ae μ] f\n⊢ ∫⁻ (a : α), f a - g a ∂μ + ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ","tactic":"refine ENNReal.eq_sub_of_add_eq hg_fin ?_","premises":[{"full_name":"ENNReal.eq_sub_of_add_eq","def_path":"Mathlib/Data/ENNReal/Operations.lean","def_pos":[320,18],"def_end_pos":[320,34]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\nhg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤\nh_le : g ≤ᶠ[ae μ] f\n⊢ ∫⁻ (a : α), f a - g a ∂μ + ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\nhg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤\nh_le : g ≤ᶠ[ae μ] f\n⊢ ∫⁻ (a : α), f a - g a + g a ∂μ = ∫⁻ (a : α), f a ∂μ","tactic":"rw [← lintegral_add_right' _ hg]","premises":[{"full_name":"MeasureTheory.lintegral_add_right'","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[568,8],"def_end_pos":[568,28]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\nhg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤\nh_le : g ≤ᶠ[ae μ] f\n⊢ ∫⁻ (a : α), f a - g a + g a ∂μ = ∫⁻ (a : α), f a ∂μ","state_after":"no goals","tactic":"exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)","premises":[{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"MeasureTheory.lintegral_congr_ae","def_path":"Mathlib/MeasureTheory/Integral/Lebesgue.lean","def_pos":[309,8],"def_end_pos":[309,26]},{"full_name":"tsub_add_cancel_of_le","def_path":"Mathlib/Algebra/Order/Sub/Canonical.lean","def_pos":[28,8],"def_end_pos":[28,29]}]}]}
{"url":"Mathlib/Data/Nat/Defs.lean","commit":"","full_name":"Nat.not_exists_sq'","start":[1474,0],"end":[1475,46],"file_path":"Mathlib/Data/Nat/Defs.lean","tactics":[{"state_before":"a b c d m n k : ℕ\np q : ℕ → Prop\n⊢ m ^ 2 < n → n < (m + 1) ^ 2 → ¬∃ t, t ^ 2 = n","state_after":"no goals","tactic":"simpa only [Nat.pow_two] using not_exists_sq","premises":[{"full_name":"Nat.not_exists_sq","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[1468,6],"def_end_pos":[1468,19]},{"full_name":"Nat.pow_two","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean","def_pos":[602,18],"def_end_pos":[602,25]}]}]}
{"url":"Mathlib/Analysis/BoxIntegral/Basic.lean","commit":"","full_name":"BoxIntegral.integralSum_biUnionTagged","start":[81,0],"end":[85,33],"file_path":"Mathlib/Analysis/BoxIntegral/Basic.lean","tactics":[{"state_before":"ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\n⊢ integralSum f vol (π.biUnionTagged πi) = ∑ J ∈ π.boxes, integralSum f vol (πi J)","state_after":"ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J✝ : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nJ : Box ι\nhJ : J ∈ π.boxes\nJ' : Box ι\nhJ' : J' ∈ (πi J).boxes\n⊢ (vol J') (f ((π.biUnionTagged πi).tag J')) = (vol J') (f ((πi J).tag J'))","tactic":"refine (π.sum_biUnion_boxes _ _).trans <| sum_congr rfl fun J hJ => sum_congr rfl fun J' hJ' => ?_","premises":[{"full_name":"BoxIntegral.Prepartition.sum_biUnion_boxes","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Basic.lean","def_pos":[307,8],"def_end_pos":[307,25]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]},{"full_name":"Finset.sum_congr","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[380,2],"def_end_pos":[380,13]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J✝ : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nJ : Box ι\nhJ : J ∈ π.boxes\nJ' : Box ι\nhJ' : J' ∈ (πi J).boxes\n⊢ (vol J') (f ((π.biUnionTagged πi).tag J')) = (vol J') (f ((πi J).tag J'))","state_after":"no goals","tactic":"rw [π.tag_biUnionTagged hJ hJ']","premises":[{"full_name":"BoxIntegral.Prepartition.tag_biUnionTagged","def_path":"Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean","def_pos":[125,8],"def_end_pos":[125,25]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Finprod.lean","commit":"","full_name":"finsum_mem_finset_product","start":[1040,0],"end":[1046,8],"file_path":"Mathlib/Algebra/BigOperators/Finprod.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset (α × β)\nf : α × β → M\n⊢ ∏ᶠ (ab : α × β) (_ : ab ∈ s), f ab = ∏ᶠ (a : α) (b : β) (_ : (a, b) ∈ s), f (a, b)","state_after":"no goals","tactic":"classical\n    rw [finprod_mem_finset_product']\n    simp","premises":[{"full_name":"finprod_mem_finset_product'","def_path":"Mathlib/Algebra/BigOperators/Finprod.lean","def_pos":[1023,8],"def_end_pos":[1023,35]}]}]}
{"url":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","commit":"","full_name":"Ordinal.add_le_of_forall_add_lt","start":[2194,0],"end":[2204,21],"file_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","tactics":[{"state_before":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\n⊢ a + b ≤ c","state_after":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\n⊢ a + b ≤ c","tactic":"have H : a + (c - a) = c :=\n    Ordinal.add_sub_cancel_of_le\n      (by\n        rw [← add_zero a]\n        exact (h _ hb).le)","premises":[{"full_name":"Ordinal.add_sub_cancel_of_le","def_path":"Mathlib/SetTheory/Ordinal/Arithmetic.lean","def_pos":[482,18],"def_end_pos":[482,38]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]}]},{"state_before":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\n⊢ a + b ≤ c","state_after":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\n⊢ a + b ≤ a + (c - a)","tactic":"rw [← H]","premises":[]},{"state_before":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\n⊢ a + b ≤ a + (c - a)","state_after":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\n⊢ b ≤ c - a","tactic":"apply add_le_add_left _ a","premises":[{"full_name":"add_le_add_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[61,31],"def_end_pos":[61,46]}]},{"state_before":"a b c : Ordinal.{u_1}\nhb : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\n⊢ b ≤ c - a","state_after":"a b c : Ordinal.{u_1}\nhb✝ : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\nhb : c - a < b\n⊢ False","tactic":"by_contra! hb","premises":[{"full_name":"Decidable.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[880,8],"def_end_pos":[880,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"a b c : Ordinal.{u_1}\nhb✝ : 0 < b\nh : ∀ d < b, a + d < c\nH : a + (c - a) = c\nhb : c - a < b\n⊢ False","state_after":"no goals","tactic":"exact (h _ hb).ne H","premises":[]}]}
{"url":"Mathlib/Logic/Encodable/Lattice.lean","commit":"","full_name":"Encodable.iUnion_decode₂_disjoint_on","start":[48,0],"end":[54,55],"file_path":"Mathlib/Logic/Encodable/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\n⊢ Pairwise (Disjoint on fun i => ⋃ b ∈ decode₂ β i, f b)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\ni j : ℕ\nij : i ≠ j\n⊢ (Disjoint on fun i => ⋃ b ∈ decode₂ β i, f b) i j","tactic":"rintro i j ij","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\ni j : ℕ\nij : i ≠ j\n⊢ (Disjoint on fun i => ⋃ b ∈ decode₂ β i, f b) i j","state_after":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\ni j : ℕ\nij : i ≠ j\nx : α\n⊢ x ∈ (fun i => ⋃ b ∈ decode₂ β i, f b) i → x ∉ (fun i => ⋃ b ∈ decode₂ β i, f b) j","tactic":"refine disjoint_left.mpr fun x => ?_","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Set.disjoint_left","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1216,8],"def_end_pos":[1216,21]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\ni j : ℕ\nij : i ≠ j\nx : α\n⊢ x ∈ (fun i => ⋃ b ∈ decode₂ β i, f b) i → x ∉ (fun i => ⋃ b ∈ decode₂ β i, f b) j","state_after":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\ni j : ℕ\nij : i ≠ j\nx : α\n⊢ ∀ (a : β), encode a = i → x ∈ f a → ∀ (b : β), encode b = j → x ∉ f b","tactic":"suffices ∀ a, encode a = i → x ∈ f a → ∀ b, encode b = j → x ∉ f b by simpa [decode₂_eq_some]","premises":[{"full_name":"Encodable.decode₂_eq_some","def_path":"Mathlib/Logic/Encodable/Basic.lean","def_pos":[174,8],"def_end_pos":[174,23]},{"full_name":"Encodable.encode","def_path":"Mathlib/Logic/Encodable/Basic.lean","def_pos":[45,2],"def_end_pos":[45,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\ni j : ℕ\nij : i ≠ j\nx : α\n⊢ ∀ (a : β), encode a = i → x ∈ f a → ∀ (b : β), encode b = j → x ∉ f b","state_after":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\nx : α\na : β\nha : x ∈ f a\nb : β\nij : encode a ≠ encode b\nhb : x ∈ f b\n⊢ False","tactic":"rintro a rfl ha b rfl hb","premises":[]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : Encodable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\nx : α\na : β\nha : x ∈ f a\nb : β\nij : encode a ≠ encode b\nhb : x ∈ f b\n⊢ False","state_after":"no goals","tactic":"exact (hd (mt (congr_arg encode) ij)).le_bot ⟨ha, hb⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Disjoint.le_bot","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[119,8],"def_end_pos":[119,23]},{"full_name":"Encodable.encode","def_path":"Mathlib/Logic/Encodable/Basic.lean","def_pos":[45,2],"def_end_pos":[45,8]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
{"url":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","commit":"","full_name":"MeasureTheory.SimpleFunc.simpleFunc_bot","start":[141,0],"end":[144,83],"file_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","tactics":[{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : MeasurableSpace α✝\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\n⊢ ∃ c, ∀ (x : α), ↑f x = c","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : MeasurableSpace α✝\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\nhf_meas : ∀ (x : β), MeasurableSet (↑f ⁻¹' {x})\n⊢ ∃ c, ∀ (x : α), ↑f x = c","tactic":"have hf_meas := @SimpleFunc.measurableSet_fiber α _ ⊥ f","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"MeasureTheory.SimpleFunc.measurableSet_fiber","def_path":"Mathlib/MeasureTheory/Function/SimpleFunc.lean","def_pos":[69,8],"def_end_pos":[69,27]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : MeasurableSpace α✝\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\nhf_meas : ∀ (x : β), MeasurableSet (↑f ⁻¹' {x})\n⊢ ∃ c, ∀ (x : α), ↑f x = c","state_after":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : MeasurableSpace α✝\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\nhf_meas : ∀ (x : β), ↑f ⁻¹' {x} = ∅ ∨ ↑f ⁻¹' {x} = univ\n⊢ ∃ c, ∀ (x : α), ↑f x = c","tactic":"simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"MeasurableSpace.measurableSet_bot_iff","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[419,8],"def_end_pos":[419,29]}]},{"state_before":"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝¹ : MeasurableSpace α✝\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\nhf_meas : ∀ (x : β), ↑f ⁻¹' {x} = ∅ ∨ ↑f ⁻¹' {x} = univ\n⊢ ∃ c, ∀ (x : α), ↑f x = c","state_after":"no goals","tactic":"exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc ↦ congr_fun hc","premises":[{"full_name":"Exists.imp","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[193,8],"def_end_pos":[193,18]},{"full_name":"Set.exists_eq_const_of_preimage_singleton","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[122,6],"def_end_pos":[122,43]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"div_eq_of_eq_mul'","start":[885,0],"end":[887,55],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : CommGroup G\na✝ b✝ c✝ d a b c : G\nh : a = b * c\n⊢ a / b = c","state_after":"no goals","tactic":"rw [h, div_eq_mul_inv, mul_comm, inv_mul_cancel_left]","premises":[{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"inv_mul_cancel_left","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[1059,8],"def_end_pos":[1059,27]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Data/Seq/WSeq.lean","commit":"","full_name":"Stream'.WSeq.flatten_pure","start":[578,0],"end":[588,8],"file_path":"Mathlib/Data/Seq/WSeq.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ flatten (Computation.pure s) = s","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ Seq.IsBisimulation fun s1 s2 => flatten (Computation.pure s2) = s1","tactic":"refine Seq.eq_of_bisim (fun s1 s2 => flatten (Computation.pure s2) = s1) ?_ rfl","premises":[{"full_name":"Computation.pure","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[41,4],"def_end_pos":[41,8]},{"full_name":"Stream'.Seq.eq_of_bisim","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[325,8],"def_end_pos":[325,19]},{"full_name":"Stream'.WSeq.flatten","def_path":"Mathlib/Data/Seq/WSeq.lean","def_pos":[125,4],"def_end_pos":[125,11]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ Seq.IsBisimulation fun s1 s2 => flatten (Computation.pure s2) = s1","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns' s : Seq (Option α)\nh : flatten (Computation.pure s) = s'\n⊢ Seq.BisimO (fun s1 s2 => flatten (Computation.pure s2) = s1) s'.destruct s.destruct","tactic":"intro s' s h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns' s : Seq (Option α)\nh : flatten (Computation.pure s) = s'\n⊢ Seq.BisimO (fun s1 s2 => flatten (Computation.pure s2) = s1) s'.destruct s.destruct","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns' s : Seq (Option α)\nh : flatten (Computation.pure s) = s'\n⊢ Seq.BisimO (fun s1 s2 => flatten (Computation.pure s2) = s1) (Seq.destruct (flatten (Computation.pure s))) s.destruct","tactic":"rw [← h]","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns' s : Seq (Option α)\nh : flatten (Computation.pure s) = s'\n⊢ Seq.BisimO (fun s1 s2 => flatten (Computation.pure s2) = s1) (Seq.destruct (flatten (Computation.pure s))) s.destruct","state_after":"α : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns' s : Seq (Option α)\nh : flatten (Computation.pure s) = s'\n⊢ match\n    match\n      match s.destruct with\n      | none => none\n      | some (a, b) => some (a, Computation.pure b) with\n    | none => none\n    | some (a, b) =>\n      some\n        (a,\n          Seq.corec\n            (fun c =>\n              match c.destruct with\n              | Sum.inl s =>\n                match Seq.destruct s with\n                | none => none\n                | some (a, b) => some (a, Computation.pure b)\n              | Sum.inr c' => some (none, c'))\n            b),\n    s.destruct with\n  | none, none => True\n  | some (a, s), some (a', s') =>\n    a = a' ∧\n      Seq.corec\n          (fun c =>\n            match c.destruct with\n            | Sum.inl s =>\n              match Seq.destruct s with\n              | none => none\n              | some (a, b) => some (a, Computation.pure b)\n            | Sum.inr c' => some (none, c'))\n          (Computation.pure s') =\n        s\n  | x, x_1 => False","tactic":"simp only [Seq.BisimO, flatten, Seq.omap, pure_def, Seq.corec_eq, destruct_pure]","premises":[{"full_name":"Computation.destruct_pure","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[126,8],"def_end_pos":[126,21]},{"full_name":"Computation.pure_def","def_path":"Mathlib/Data/Seq/Computation.lean","def_pos":[735,8],"def_end_pos":[735,16]},{"full_name":"Stream'.Seq.BisimO","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[313,4],"def_end_pos":[313,10]},{"full_name":"Stream'.Seq.corec_eq","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[291,8],"def_end_pos":[291,16]},{"full_name":"Stream'.Seq.omap","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[115,4],"def_end_pos":[115,8]},{"full_name":"Stream'.WSeq.flatten","def_path":"Mathlib/Data/Seq/WSeq.lean","def_pos":[125,4],"def_end_pos":[125,11]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns' s : Seq (Option α)\nh : flatten (Computation.pure s) = s'\n⊢ match\n    match\n      match s.destruct with\n      | none => none\n      | some (a, b) => some (a, Computation.pure b) with\n    | none => none\n    | some (a, b) =>\n      some\n        (a,\n          Seq.corec\n            (fun c =>\n              match c.destruct with\n              | Sum.inl s =>\n                match Seq.destruct s with\n                | none => none\n                | some (a, b) => some (a, Computation.pure b)\n              | Sum.inr c' => some (none, c'))\n            b),\n    s.destruct with\n  | none, none => True\n  | some (a, s), some (a', s') =>\n    a = a' ∧\n      Seq.corec\n          (fun c =>\n            match c.destruct with\n            | Sum.inl s =>\n              match Seq.destruct s with\n              | none => none\n              | some (a, b) => some (a, Computation.pure b)\n            | Sum.inr c' => some (none, c'))\n          (Computation.pure s') =\n        s\n  | x, x_1 => False","state_after":"no goals","tactic":"cases Seq.destruct s with\n  | none => simp\n  | some val =>\n    cases' val with o s'\n    simp","premises":[{"full_name":"Option.none","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2242,4],"def_end_pos":[2242,8]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]},{"full_name":"Stream'.Seq.destruct","def_path":"Mathlib/Data/Seq/Seq.lean","def_pos":[171,4],"def_end_pos":[171,12]}]}]}
{"url":"Mathlib/GroupTheory/Nilpotent.lean","commit":"","full_name":"derived_le_lower_central","start":[608,0],"end":[612,8],"file_path":"Mathlib/GroupTheory/Nilpotent.lean","tactics":[{"state_before":"G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Normal\nn : ℕ\n⊢ derivedSeries G n ≤ lowerCentralSeries G n","state_after":"case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Normal\n⊢ derivedSeries G 0 ≤ lowerCentralSeries G 0\n\ncase succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Normal\ni : ℕ\nih : derivedSeries G i ≤ lowerCentralSeries G i\n⊢ derivedSeries G (i + 1) ≤ lowerCentralSeries G (i + 1)","tactic":"induction' n with i ih","premises":[]}]}
{"url":"Mathlib/Logic/Function/Basic.lean","commit":"","full_name":"Function.update_comm","start":[565,0],"end":[574,31],"file_path":"Mathlib/Logic/Function/Basic.lean","tactics":[{"state_before":"α✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\n⊢ update (update f a v) b w = update (update f b w) a v","state_after":"case h\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\n⊢ update (update f a v) b w c = update (update f b w) a v c","tactic":"funext c","premises":[{"full_name":"funext","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1817,8],"def_end_pos":[1817,14]}]},{"state_before":"case h\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\n⊢ update (update f a v) b w c = update (update f b w) a v c","state_after":"case h\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\n⊢ (if h : c = b then ⋯ ▸ w else if h : c = a then ⋯ ▸ v else f c) =\n    if h : c = a then ⋯ ▸ v else if h : c = b then ⋯ ▸ w else f c","tactic":"simp only [update]","premises":[{"full_name":"Function.update","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[462,4],"def_end_pos":[462,10]}]},{"state_before":"case h\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\n⊢ (if h : c = b then ⋯ ▸ w else if h : c = a then ⋯ ▸ v else f c) =\n    if h : c = a then ⋯ ▸ v else if h : c = b then ⋯ ▸ w else f c","state_after":"case pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\nh₁ : c = b\nh₂ : c = a\n⊢ (if h : c = b then ⋯ ▸ w else if h : c = a then ⋯ ▸ v else f c) =\n    if h : c = a then ⋯ ▸ v else if h : c = b then ⋯ ▸ w else f c\n\ncase neg\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\nh₁ : c = b\nh₂ : ¬c = a\n⊢ (if h : c = b then ⋯ ▸ w else if h : c = a then ⋯ ▸ v else f c) =\n    if h : c = a then ⋯ ▸ v else if h : c = b then ⋯ ▸ w else f c\n\ncase pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\nh₁ : ¬c = b\nh₂ : c = a\n⊢ (if h : c = b then ⋯ ▸ w else if h : c = a then ⋯ ▸ v else f c) =\n    if h : c = a then ⋯ ▸ v else if h : c = b then ⋯ ▸ w else f c\n\ncase neg\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g : (a : α✝) → β✝ a\na✝ : α✝\nb✝ : β✝ a✝\nα : Sort u_2\ninst✝ : DecidableEq α\nβ : α → Sort u_1\na b : α\nh : a ≠ b\nv : β a\nw : β b\nf : (a : α) → β a\nc : α\nh₁ : ¬c = b\nh₂ : ¬c = a\n⊢ (if h : c = b then ⋯ ▸ w else if h : c = a then ⋯ ▸ v else f c) =\n    if h : c = a then ⋯ ▸ v else if h : c = b then ⋯ ▸ w else f c","tactic":"by_cases h₁ : c = b <;> by_cases h₂ : c = a","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Analysis/Analytic/CPolynomial.lean","commit":"","full_name":"ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall","start":[237,0],"end":[244,26],"file_path":"Mathlib/Analysis/Analytic/CPolynomial.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nn m✝ : ℕ\ng : F →L[𝕜] G\nh : HasFiniteFPowerSeriesOnBall f p x n r\nm : ℕ\nhm : n ≤ m\n⊢ g.compFormalMultilinearSeries p m = 0","state_after":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nn m✝ : ℕ\ng : F →L[𝕜] G\nh : HasFiniteFPowerSeriesOnBall f p x n r\nm : ℕ\nhm : n ≤ m\n⊢ g.compContinuousMultilinearMap 0 = 0","tactic":"rw [compFormalMultilinearSeries_apply, h.finite m hm]","premises":[{"full_name":"ContinuousLinearMap.compFormalMultilinearSeries_apply","def_path":"Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean","def_pos":[186,8],"def_end_pos":[186,41]},{"full_name":"HasFiniteFPowerSeriesOnBall.finite","def_path":"Mathlib/Analysis/Analytic/CPolynomial.lean","def_pos":[59,2],"def_end_pos":[59,8]}]},{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nn m✝ : ℕ\ng : F →L[𝕜] G\nh : HasFiniteFPowerSeriesOnBall f p x n r\nm : ℕ\nhm : n ≤ m\n⊢ g.compContinuousMultilinearMap 0 = 0","state_after":"case H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nn m✝ : ℕ\ng : F →L[𝕜] G\nh : HasFiniteFPowerSeriesOnBall f p x n r\nm : ℕ\nhm : n ≤ m\nx✝ : Fin m → E\n⊢ (g.compContinuousMultilinearMap 0) x✝ = 0 x✝","tactic":"ext","premises":[]},{"state_before":"case H\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nn m✝ : ℕ\ng : F →L[𝕜] G\nh : HasFiniteFPowerSeriesOnBall f p x n r\nm : ℕ\nhm : n ≤ m\nx✝ : Fin m → E\n⊢ (g.compContinuousMultilinearMap 0) x✝ = 0 x✝","state_after":"no goals","tactic":"exact map_zero g","premises":[{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]}]}]}
{"url":"Mathlib/Data/List/Cycle.lean","commit":"","full_name":"Cycle.Chain.nil","start":[818,0],"end":[819,75],"file_path":"Mathlib/Data/List/Cycle.lean","tactics":[{"state_before":"α : Type u_1\nr : α → α → Prop\n⊢ Chain r Cycle.nil","state_after":"no goals","tactic":"trivial","premises":[{"full_name":"True.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[192,4],"def_end_pos":[192,9]}]}]}
{"url":"Mathlib/MeasureTheory/Constructions/Pi.lean","commit":"","full_name":"MeasureTheory.piPremeasure_pi'","start":[156,0],"end":[164,26],"file_path":"Mathlib/MeasureTheory/Constructions/Pi.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ns : (i : ι) → Set (α i)\n⊢ piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)","state_after":"case inl\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ns : (i : ι) → Set (α i)\nh✝ : IsEmpty ι\n⊢ piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)\n\ncase inr\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ns : (i : ι) → Set (α i)\nh✝ : Nonempty ι\n⊢ piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)","tactic":"cases isEmpty_or_nonempty ι","premises":[{"full_name":"isEmpty_or_nonempty","def_path":"Mathlib/Logic/IsEmpty.lean","def_pos":[195,8],"def_end_pos":[195,27]}]},{"state_before":"case inr\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ns : (i : ι) → Set (α i)\nh✝ : Nonempty ι\n⊢ piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)","state_after":"case inr.inl\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ns : (i : ι) → Set (α i)\nh✝ : Nonempty ι\nh : univ.pi s = ∅\n⊢ piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)\n\ncase inr.inr\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ns : (i : ι) → Set (α i)\nh✝ : Nonempty ι\nh : (univ.pi s).Nonempty\n⊢ piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)","tactic":"rcases (pi univ s).eq_empty_or_nonempty with h | h","premises":[{"full_name":"Set.eq_empty_or_nonempty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[506,8],"def_end_pos":[506,28]},{"full_name":"Set.pi","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[218,4],"def_end_pos":[218,6]},{"full_name":"Set.univ","def_path":"Mathlib/Init/Set.lean","def_pos":[157,4],"def_end_pos":[157,8]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","commit":"","full_name":"ContinuousLinearEquiv.comp_hasStrictFDerivAt_iff","start":[117,0],"end":[121,49],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\nx : G\nf' : G →L[𝕜] E\n⊢ HasStrictFDerivAt (⇑iso ∘ f) ((↑iso).comp f') x ↔ HasStrictFDerivAt f f' x","state_after":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\nx : G\nf' : G →L[𝕜] E\nH : HasStrictFDerivAt (⇑iso ∘ f) ((↑iso).comp f') x\n⊢ HasStrictFDerivAt f f' x","tactic":"refine ⟨fun H => ?_, fun H => iso.hasStrictFDerivAt.comp x H⟩","premises":[{"full_name":"ContinuousLinearEquiv.hasStrictFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","def_pos":[52,18],"def_end_pos":[52,35]},{"full_name":"HasStrictFDerivAt.comp","def_path":"Mathlib/Analysis/Calculus/FDeriv/Comp.lean","def_pos":[172,18],"def_end_pos":[172,40]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\nx : G\nf' : G →L[𝕜] E\nH : HasStrictFDerivAt (⇑iso ∘ f) ((↑iso).comp f') x\n⊢ HasStrictFDerivAt f f' x","state_after":"no goals","tactic":"convert iso.symm.hasStrictFDerivAt.comp x H using 1 <;>\n    ext z <;> apply (iso.symm_apply_apply _).symm","premises":[{"full_name":"ContinuousLinearEquiv.hasStrictFDerivAt","def_path":"Mathlib/Analysis/Calculus/FDeriv/Equiv.lean","def_pos":[52,18],"def_end_pos":[52,35]},{"full_name":"ContinuousLinearEquiv.symm","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1773,14],"def_end_pos":[1773,18]},{"full_name":"ContinuousLinearEquiv.symm_apply_apply","def_path":"Mathlib/Topology/Algebra/Module/Basic.lean","def_pos":[1868,8],"def_end_pos":[1868,24]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"HasStrictFDerivAt.comp","def_path":"Mathlib/Analysis/Calculus/FDeriv/Comp.lean","def_pos":[172,18],"def_end_pos":[172,40]}]}]}
{"url":"Mathlib/Topology/UniformSpace/CompactConvergence.lean","commit":"","full_name":"Filter.HasBasis.compactConvergenceUniformity","start":[179,0],"end":[185,76],"file_path":"Mathlib/Topology/UniformSpace/CompactConvergence.lean","tactics":[{"state_before":"α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nι : Type u_1\npi : ι → Prop\ns : ι → Set (β × β)\nh : (𝓤 β).HasBasis pi s\n⊢ (𝓤 C(α, β)).HasBasis (fun p => IsCompact p.1 ∧ pi p.2) fun p => {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2}","state_after":"α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nι : Type u_1\npi : ι → Prop\ns : ι → Set (β × β)\nh : (𝓤 β).HasBasis pi s\n⊢ (Filter.comap (fun x => (x.1.toUniformOnFunIsCompact, x.2.toUniformOnFunIsCompact))\n        (𝓤 (α →ᵤ[{K | IsCompact K}] β))).HasBasis\n    (fun p => IsCompact p.1 ∧ pi p.2) fun p => {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2}","tactic":"rw [← uniformEmbedding_toUniformOnFunIsCompact.comap_uniformity]","premises":[{"full_name":"ContinuousMap.uniformEmbedding_toUniformOnFunIsCompact","def_path":"Mathlib/Topology/UniformSpace/CompactConvergence.lean","def_pos":[170,8],"def_end_pos":[170,48]},{"full_name":"UniformInducing.comap_uniformity","def_path":"Mathlib/Topology/UniformSpace/UniformEmbedding.lean","def_pos":[35,2],"def_end_pos":[35,18]}]},{"state_before":"α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nι : Type u_1\npi : ι → Prop\ns : ι → Set (β × β)\nh : (𝓤 β).HasBasis pi s\n⊢ (Filter.comap (fun x => (x.1.toUniformOnFunIsCompact, x.2.toUniformOnFunIsCompact))\n        (𝓤 (α →ᵤ[{K | IsCompact K}] β))).HasBasis\n    (fun p => IsCompact p.1 ∧ pi p.2) fun p => {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2}","state_after":"no goals","tactic":"exact .comap _ <| UniformOnFun.hasBasis_uniformity_of_basis _ _ {K | IsCompact K}\n    ⟨∅, isCompact_empty⟩ (directedOn_of_sup_mem fun _ _ ↦ IsCompact.union) h","premises":[{"full_name":"EmptyCollection.emptyCollection","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[447,2],"def_end_pos":[447,17]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.HasBasis.comap","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[697,8],"def_end_pos":[697,22]},{"full_name":"IsCompact","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[254,4],"def_end_pos":[254,13]},{"full_name":"IsCompact.union","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[482,8],"def_end_pos":[482,23]},{"full_name":"UniformOnFun.hasBasis_uniformity_of_basis","def_path":"Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean","def_pos":[643,18],"def_end_pos":[643,46]},{"full_name":"directedOn_of_sup_mem","def_path":"Mathlib/Order/Directed.lean","def_pos":[97,8],"def_end_pos":[97,29]},{"full_name":"isCompact_empty","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[433,8],"def_end_pos":[433,23]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","commit":"","full_name":"Rat.normalize.reduced'","start":[31,0],"end":[34,34],"file_path":".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean","tactics":[{"state_before":"num : Int\nden g : Nat\nden_nz : den ≠ 0\ne : g = num.natAbs.gcd den\n⊢ (num / ↑g).natAbs.Coprime (den / g)","state_after":"num : Int\nden g : Nat\nden_nz : den ≠ 0\ne : g = num.natAbs.gcd den\n⊢ (num.div ↑g).natAbs.Coprime (den / g)","tactic":"rw [← Int.div_eq_ediv_of_dvd (e ▸ Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Int.div_eq_ediv_of_dvd","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[1014,8],"def_end_pos":[1014,26]},{"full_name":"Int.ofNat_dvd_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean","def_pos":[75,8],"def_end_pos":[75,22]},{"full_name":"Nat.gcd_dvd_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[86,8],"def_end_pos":[86,20]}]},{"state_before":"num : Int\nden g : Nat\nden_nz : den ≠ 0\ne : g = num.natAbs.gcd den\n⊢ (num.div ↑g).natAbs.Coprime (den / g)","state_after":"no goals","tactic":"exact normalize.reduced den_nz e","premises":[{"full_name":"Rat.normalize.reduced","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[60,8],"def_end_pos":[60,29]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Finite.lean","commit":"","full_name":"Subgroup.card_le_one_iff_eq_bot","start":[135,0],"end":[137,45],"file_path":"Mathlib/Algebra/Group/Subgroup/Finite.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\nA : Type u_2\ninst✝¹ : AddGroup A\nH K : Subgroup G\ninst✝ : Finite ↥H\nh : H = ⊥\n⊢ Nat.card ↥H ≤ 1","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/RingTheory/Norm/Basic.lean","commit":"","full_name":"IntermediateField.AdjoinSimple.norm_gen_eq_one","start":[150,0],"end":[157,67],"file_path":"Mathlib/RingTheory/Norm/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\nhx : ¬IsIntegral K x\n⊢ (norm K) (AdjoinSimple.gen K x) = 1","state_after":"case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\nhx : ¬IsIntegral K x\n⊢ ¬∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)","tactic":"rw [norm_eq_one_of_not_exists_basis]","premises":[{"full_name":"Algebra.norm_eq_one_of_not_exists_basis","def_path":"Mathlib/RingTheory/Norm/Defs.lean","def_pos":[62,8],"def_end_pos":[62,39]}]},{"state_before":"case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\nhx : ¬IsIntegral K x\n⊢ ¬∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)","state_after":"case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\nhx : ∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)\n⊢ IsIntegral K x","tactic":"contrapose! hx","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\nhx : ∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)\n⊢ IsIntegral K x","state_after":"case h.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\ns : Finset ↥K⟮x⟯\nb : Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯\n⊢ IsIntegral K x","tactic":"obtain ⟨s, ⟨b⟩⟩ := hx","premises":[]},{"state_before":"case h.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\ns : Finset ↥K⟮x⟯\nb : Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯\n⊢ IsIntegral K x","state_after":"case h.intro.intro.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\ns : Finset ↥K⟮x⟯\nb : Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯\n⊢ (Subalgebra.toSubmodule K⟮x⟯.toSubalgebra).FG\n\ncase h.intro.intro.refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ninst✝¹ : Algebra K L\ninst✝ : Algebra K F\nι : Type w\nx : L\ns : Finset ↥K⟮x⟯\nb : Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯\n⊢ x ∈ K⟮x⟯.toSubalgebra","tactic":"refine .of_mem_of_fg K⟮x⟯.toSubalgebra ?_ x ?_","premises":[{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"IsIntegral.of_mem_of_fg","def_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","def_pos":[55,8],"def_end_pos":[55,31]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]}]}
{"url":"Mathlib/Algebra/Homology/HomologySequenceLemmas.lean","commit":"","full_name":"HomologicalComplex.HomologySequence.epi_homologyMap_τ₃","start":[128,0],"end":[147,32],"file_path":"Mathlib/Algebra/Homology/HomologySequenceLemmas.lean","tactics":[{"state_before":"C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS S₁ S₂ : ShortComplex (HomologicalComplex C c)\nφ : S₁ ⟶ S₂\nhS₁ : S₁.ShortExact\nhS₂ : S₂.ShortExact\ni : ι\nh₁ : Epi (homologyMap φ.τ₂ i)\nh₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)\nh₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)\n⊢ Epi (homologyMap φ.τ₃ i)","state_after":"case pos\nC : Type u_1\nι : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS S₁ S₂ : ShortComplex (HomologicalComplex C c)\nφ : S₁ ⟶ S₂\nhS₁ : S₁.ShortExact\nhS₂ : S₂.ShortExact\ni : ι\nh₁ : Epi (homologyMap φ.τ₂ i)\nh₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)\nh₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)\nhi : ∃ j, c.Rel i j\n⊢ Epi (homologyMap φ.τ₃ i)\n\ncase neg\nC : Type u_1\nι : Type u_2\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS S₁ S₂ : ShortComplex (HomologicalComplex C c)\nφ : S₁ ⟶ S₂\nhS₁ : S₁.ShortExact\nhS₂ : S₂.ShortExact\ni : ι\nh₁ : Epi (homologyMap φ.τ₂ i)\nh₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)\nh₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)\nhi : ¬∃ j, c.Rel i j\n⊢ Epi (homologyMap φ.τ₃ i)","tactic":"by_cases hi : ∃ j, c.Rel i j","premises":[{"full_name":"ComplexShape.Rel","def_path":"Mathlib/Algebra/Homology/ComplexShape.lean","def_pos":[62,2],"def_end_pos":[62,5]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Data/Vector/Mem.lean","commit":"","full_name":"Mathlib.Vector.mem_map_succ_iff","start":[72,0],"end":[74,66],"file_path":"Mathlib/Data/Vector/Mem.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nn : ℕ\na a' : α\nb : β\nv : Vector α (n + 1)\nf : α → β\n⊢ b ∈ (map f v).toList ↔ f v.head = b ∨ ∃ a ∈ v.tail.toList, f a = b","state_after":"no goals","tactic":"rw [mem_succ_iff, head_map, tail_map, mem_map_iff, @eq_comm _ b]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Mathlib.Vector.head_map","def_path":"Mathlib/Data/Vector/Basic.lean","def_pos":[91,8],"def_end_pos":[91,16]},{"full_name":"Mathlib.Vector.mem_map_iff","def_path":"Mathlib/Data/Vector/Mem.lean","def_pos":[65,8],"def_end_pos":[65,19]},{"full_name":"Mathlib.Vector.mem_succ_iff","def_path":"Mathlib/Data/Vector/Mem.lean","def_pos":[46,8],"def_end_pos":[46,20]},{"full_name":"Mathlib.Vector.tail_map","def_path":"Mathlib/Data/Vector/Basic.lean","def_pos":[96,8],"def_end_pos":[96,16]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean","commit":"","full_name":"QuadraticForm.associated_baseChange","start":[97,0],"end":[101,11],"file_path":"Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean","tactics":[{"state_before":"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Algebra R A\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : SMulCommClass A R M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ninst✝¹ : Invertible 2\ninst✝ : Invertible 2\nQ : QuadraticForm R M₂\n⊢ associated (QuadraticForm.baseChange A Q) = BilinMap.baseChange A (associated Q)","state_after":"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Algebra R A\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : SMulCommClass A R M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ninst✝¹ : Invertible 2\ninst✝ : Invertible 2\nQ : QuadraticForm R M₂\n⊢ associated (QuadraticForm.tmul QuadraticMap.sq Q) = BilinMap.baseChange A (associated Q)","tactic":"dsimp only [QuadraticForm.baseChange, LinearMap.baseChange]","premises":[{"full_name":"LinearMap.baseChange","def_path":"Mathlib/RingTheory/TensorProduct/Basic.lean","def_pos":[65,4],"def_end_pos":[65,14]},{"full_name":"QuadraticForm.baseChange","def_path":"Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean","def_pos":[89,28],"def_end_pos":[89,38]}]},{"state_before":"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Algebra R A\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : SMulCommClass A R M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ninst✝¹ : Invertible 2\ninst✝ : Invertible 2\nQ : QuadraticForm R M₂\n⊢ associated (QuadraticForm.tmul QuadraticMap.sq Q) = BilinMap.baseChange A (associated Q)","state_after":"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Algebra R A\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : SMulCommClass A R M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ninst✝¹ : Invertible 2\ninst✝ : Invertible 2\nQ : QuadraticForm R M₂\n⊢ BilinMap.tmul (LinearMap.mul A A) (associated Q) = BilinMap.baseChange A (associated Q)","tactic":"rw [associated_tmul (QuadraticMap.sq (R := A)) Q, associated_sq]","premises":[{"full_name":"QuadraticForm.associated_tmul","def_path":"Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean","def_pos":[71,8],"def_end_pos":[71,23]},{"full_name":"QuadraticMap.associated_sq","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[933,6],"def_end_pos":[933,19]},{"full_name":"QuadraticMap.sq","def_path":"Mathlib/LinearAlgebra/QuadraticForm/Basic.lean","def_pos":[629,4],"def_end_pos":[629,6]}]},{"state_before":"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Algebra R A\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : SMulCommClass A R M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ninst✝¹ : Invertible 2\ninst✝ : Invertible 2\nQ : QuadraticForm R M₂\n⊢ BilinMap.tmul (LinearMap.mul A A) (associated Q) = BilinMap.baseChange A (associated Q)","state_after":"no goals","tactic":"exact rfl","premises":[{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Algebra/DualNumber.lean","commit":"","full_name":"DualNumber.lift_apply_eps","start":[167,0],"end":[171,86],"file_path":"Mathlib/Algebra/DualNumber.lean","tactics":[{"state_before":"R : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nfe : { fe // fe.2 * fe.2 = 0 ∧ ∀ (a : A), Commute fe.2 (fe.1 a) }\n⊢ (lift fe) ε = (↑fe).2","state_after":"no goals","tactic":"simp only [lift_apply_apply, fst_eps, map_zero, snd_eps, map_one, one_mul, zero_add]","premises":[{"full_name":"DualNumber.fst_eps","def_path":"Mathlib/Algebra/DualNumber.lean","def_pos":[62,8],"def_end_pos":[62,15]},{"full_name":"DualNumber.lift_apply_apply","def_path":"Mathlib/Algebra/DualNumber.lean","def_pos":[146,8],"def_end_pos":[146,24]},{"full_name":"DualNumber.snd_eps","def_path":"Mathlib/Algebra/DualNumber.lean","def_pos":[66,8],"def_end_pos":[66,15]},{"full_name":"map_one","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[190,8],"def_end_pos":[190,15]},{"full_name":"map_zero","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[189,2],"def_end_pos":[189,13]},{"full_name":"one_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[409,8],"def_end_pos":[409,15]},{"full_name":"zero_add","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[408,2],"def_end_pos":[408,13]}]}]}
{"url":"Mathlib/CategoryTheory/Triangulated/Yoneda.lean","commit":"","full_name":"CategoryTheory.Pretriangulated.preadditiveCoyoneda_homologySequenceδ_apply","start":[51,0],"end":[55,22],"file_path":"Mathlib/CategoryTheory/Triangulated/Yoneda.lean","tactics":[{"state_before":"C : Type u_1\ninst✝⁵ : Category.{?u.7545, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nA : Cᵒᵖ\nT : Triangle C\nn₀ n₁ : ℤ\nh : n₀ + 1 = n₁\nx : Opposite.unop A ⟶ (shiftFunctor C n₀).obj T.obj₃\n⊢ 1 + n₀ = n₁","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nA : Cᵒᵖ\nT : Triangle C\nn₀ n₁ : ℤ\nh : n₀ + 1 = n₁\nx : Opposite.unop A ⟶ (shiftFunctor C n₀).obj T.obj₃\n⊢ ((preadditiveCoyoneda.obj A).homologySequenceδ T n₀ n₁ h) x =\n    x ≫ (shiftFunctor C n₀).map T.mor₃ ≫ (shiftFunctorAdd' C 1 n₀ n₁ ⋯).inv.app T.obj₁","state_after":"no goals","tactic":"apply Category.assoc","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]}]}]}
{"url":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","commit":"","full_name":"IsPrimitiveRoot.finite_quotient_span_sub_one","start":[482,0],"end":[488,100],"file_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","tactics":[{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\n⊢ Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})","state_after":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nthis : NumberField K\n⊢ Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})","tactic":"have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K","premises":[{"full_name":"IsCyclotomicExtension.numberField","def_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","def_pos":[319,8],"def_end_pos":[319,19]},{"full_name":"NumberField","def_path":"Mathlib/NumberTheory/NumberField/Basic.lean","def_pos":[36,6],"def_end_pos":[36,17]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nthis : NumberField K\n⊢ Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})","state_after":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nthis : NumberField K\nh : Ideal.span {hζ.toInteger - 1} = ⊥\n⊢ False","tactic":"refine Fintype.finite <| Ideal.fintypeQuotientOfFreeOfNeBot _ (fun h ↦ ?_)","premises":[{"full_name":"Fintype.finite","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[374,18],"def_end_pos":[374,32]},{"full_name":"Ideal.fintypeQuotientOfFreeOfNeBot","def_path":"Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean","def_pos":[93,18],"def_end_pos":[93,46]}]},{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nthis : NumberField K\nh : Ideal.span {hζ.toInteger - 1} = ⊥\n⊢ False","state_after":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nthis : NumberField K\nh : hζ.toInteger = 1\n⊢ False","tactic":"simp only [Ideal.span_singleton_eq_bot, sub_eq_zero, ← Subtype.coe_inj] at h","premises":[{"full_name":"Ideal.span_singleton_eq_bot","def_path":"Mathlib/RingTheory/Ideal/Basic.lean","def_pos":[167,8],"def_end_pos":[167,29]},{"full_name":"Subtype.coe_inj","def_path":"Mathlib/Data/Subtype.lean","def_pos":[107,8],"def_end_pos":[107,15]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]},{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nthis : NumberField K\nh : hζ.toInteger = 1\n⊢ False","state_after":"no goals","tactic":"exact hζ.ne_one (one_lt_pow hp.1.one_lt (Nat.zero_ne_add_one k).symm) (RingOfIntegers.ext_iff.1 h)","premises":[{"full_name":"Fact.out","def_path":"Mathlib/Logic/Basic.lean","def_pos":[92,2],"def_end_pos":[92,5]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"IsPrimitiveRoot.ne_one","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[326,8],"def_end_pos":[326,14]},{"full_name":"Nat.Prime.one_lt","def_path":"Mathlib/Data/Nat/Prime/Defs.lean","def_pos":[59,8],"def_end_pos":[59,20]},{"full_name":"Nat.zero_ne_add_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[158,16],"def_end_pos":[158,31]},{"full_name":"Ne.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[704,8],"def_end_pos":[704,15]},{"full_name":"NumberField.RingOfIntegers.ext_iff","def_path":"Mathlib/NumberTheory/NumberField/Basic.lean","def_pos":[109,8],"def_end_pos":[109,15]},{"full_name":"one_lt_pow","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[95,8],"def_end_pos":[95,18]}]}]}
{"url":"Mathlib/Data/Set/Prod.lean","commit":"","full_name":"Set.range_diag","start":[406,0],"end":[409,26],"file_path":"Mathlib/Data/Set/Prod.lean","tactics":[{"state_before":"α : Type u_1\ns t : Set α\n⊢ (range fun x => (x, x)) = diagonal α","state_after":"case h.mk\nα : Type u_1\ns t : Set α\nx y : α\n⊢ ((x, y) ∈ range fun x => (x, x)) ↔ (x, y) ∈ diagonal α","tactic":"ext ⟨x, y⟩","premises":[]},{"state_before":"case h.mk\nα : Type u_1\ns t : Set α\nx y : α\n⊢ ((x, y) ∈ range fun x => (x, x)) ↔ (x, y) ∈ diagonal α","state_after":"no goals","tactic":"simp [diagonal, eq_comm]","premises":[{"full_name":"Set.diagonal","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[196,4],"def_end_pos":[196,12]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/MeasureTheory/Function/L1Space.lean","commit":"","full_name":"MeasureTheory.lintegral_nnnorm_eq_lintegral_edist","start":[65,0],"end":[66,82],"file_path":"Mathlib/MeasureTheory/Function/L1Space.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\n⊢ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ","state_after":"no goals","tactic":"simp only [edist_eq_coe_nnnorm]","premises":[{"full_name":"edist_eq_coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[694,14],"def_end_pos":[694,33]}]}]}
{"url":"Mathlib/Control/Monad/Cont.lean","commit":"","full_name":"ReaderT.goto_mkLabel","start":[226,0],"end":[227,74],"file_path":"Mathlib/Control/Monad/Cont.lean","tactics":[{"state_before":"m : Type u → Type v\nα : Type u_1\nρ β : Type u\nx : Label α m β\ni : α\n⊢ goto (mkLabel ρ x) i = monadLift (goto x i)","state_after":"case mk\nm : Type u → Type v\nα : Type u_1\nρ β : Type u\ni : α\napply✝ : α → m β\n⊢ goto (mkLabel ρ { apply := apply✝ }) i = monadLift (goto { apply := apply✝ } i)","tactic":"cases x","premises":[]},{"state_before":"case mk\nm : Type u → Type v\nα : Type u_1\nρ β : Type u\ni : α\napply✝ : α → m β\n⊢ goto (mkLabel ρ { apply := apply✝ }) i = monadLift (goto { apply := apply✝ } i)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean","commit":"","full_name":"CochainComplex.mappingCone.inr_descShortComplex","start":[48,0],"end":[50,25],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\n⊢ inr S.f ≫ descShortComplex S = S.g","state_after":"no goals","tactic":"simp [descShortComplex]","premises":[{"full_name":"CochainComplex.mappingCone.descShortComplex","def_path":"Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean","def_pos":[46,18],"def_end_pos":[46,34]}]}]}
{"url":"Mathlib/Algebra/Squarefree/Basic.lean","commit":"","full_name":"Squarefree.dvd_of_squarefree_of_mul_dvd_mul_right","start":[208,0],"end":[214,52],"file_path":"Mathlib/Algebra/Squarefree/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\n⊢ d ∣ y","state_after":"R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\n⊢ d ∣ y","tactic":"nontriviality R","premises":[]},{"state_before":"R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\n⊢ d ∣ y","state_after":"case intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nha : a ∣ x\nhb : b ∣ y\neq : d * d = a * b\n⊢ d ∣ y","tactic":"obtain ⟨a, b, ha, hb, eq⟩ := exists_dvd_and_dvd_of_dvd_mul h","premises":[{"full_name":"exists_dvd_and_dvd_of_dvd_mul","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[113,8],"def_end_pos":[113,37]}]},{"state_before":"case intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nha : a ∣ x\nhb : b ∣ y\neq : d * d = a * b\n⊢ d ∣ y","state_after":"case intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nhb : b ∣ y\neq : d * d = a * b\nha : Squarefree a\n⊢ d ∣ y","tactic":"replace ha : Squarefree a := hx.squarefree_of_dvd ha","premises":[{"full_name":"Squarefree","def_path":"Mathlib/Algebra/Squarefree/Basic.lean","def_pos":[35,4],"def_end_pos":[35,14]},{"full_name":"Squarefree.squarefree_of_dvd","def_path":"Mathlib/Algebra/Squarefree/Basic.lean","def_pos":[77,8],"def_end_pos":[77,36]}]},{"state_before":"case intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nhb : b ∣ y\neq : d * d = a * b\nha : Squarefree a\n⊢ d ∣ y","state_after":"case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nhb : b ∣ y\neq : d * d = a * b\nha : Squarefree a\nc : R\nhc : d = a * c\n⊢ d ∣ y","tactic":"obtain ⟨c, hc⟩ : a ∣ d := ha.isRadical 2 d ⟨b, by rw [sq, eq]⟩","premises":[{"full_name":"Dvd.dvd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1344,2],"def_end_pos":[1344,5]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Squarefree.isRadical","def_path":"Mathlib/Algebra/Squarefree/Basic.lean","def_pos":[165,8],"def_end_pos":[165,28]}]},{"state_before":"case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nhb : b ∣ y\neq : d * d = a * b\nha : Squarefree a\nc : R\nhc : d = a * c\n⊢ d ∣ y","state_after":"case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nhb : b ∣ y\nha : Squarefree a\nc : R\neq : c * (a * c) = b\nhc : d = a * c\n⊢ d ∣ y","tactic":"rw [hc, mul_assoc, (mul_right_injective₀ ha.ne_zero).eq_iff] at eq","premises":[{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"Squarefree.ne_zero","def_path":"Mathlib/Algebra/Squarefree/Basic.lean","def_pos":[54,8],"def_end_pos":[54,26]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"mul_right_injective₀","def_path":"Mathlib/Algebra/GroupWithZero/Defs.lean","def_pos":[51,8],"def_end_pos":[51,28]}]},{"state_before":"case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nx y p d : R\ninst✝ : DecompositionMonoid R\nhx : Squarefree x\nh : d * d ∣ x * y\na✝ : Nontrivial R\na b : R\nhb : b ∣ y\nha : Squarefree a\nc : R\neq : c * (a * c) = b\nhc : d = a * c\n⊢ d ∣ y","state_after":"no goals","tactic":"exact dvd_trans ⟨c, by rw [hc, ← eq, mul_comm]⟩ hb","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"dvd_trans","def_path":"Mathlib/Algebra/Divisibility/Basic.lean","def_pos":[65,8],"def_end_pos":[65,17]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Algebra/Group/Hom/Defs.lean","commit":"","full_name":"map_comp_div'","start":[389,0],"end":[392,27],"file_path":"Mathlib/Algebra/Group/Hom/Defs.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\nH : Type u_8\nF : Type u_9\ninst✝⁶ : MulOneClass M\ninst✝⁵ : MulOneClass N\ninst✝⁴ : FunLike F M N\ninst✝³ : FunLike F G H\ninst✝² : DivInvMonoid G\ninst✝¹ : DivInvMonoid H\ninst✝ : MonoidHomClass F G H\nf : F\nhf : ∀ (a : G), f a⁻¹ = (f a)⁻¹\ng h : ι → G\n⊢ ⇑f ∘ (g / h) = ⇑f ∘ g / ⇑f ∘ h","state_after":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\nH : Type u_8\nF : Type u_9\ninst✝⁶ : MulOneClass M\ninst✝⁵ : MulOneClass N\ninst✝⁴ : FunLike F M N\ninst✝³ : FunLike F G H\ninst✝² : DivInvMonoid G\ninst✝¹ : DivInvMonoid H\ninst✝ : MonoidHomClass F G H\nf : F\nhf : ∀ (a : G), f a⁻¹ = (f a)⁻¹\ng h : ι → G\nx✝ : ι\n⊢ (⇑f ∘ (g / h)) x✝ = (⇑f ∘ g / ⇑f ∘ h) x✝","tactic":"ext","premises":[]},{"state_before":"case h\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\nH : Type u_8\nF : Type u_9\ninst✝⁶ : MulOneClass M\ninst✝⁵ : MulOneClass N\ninst✝⁴ : FunLike F M N\ninst✝³ : FunLike F G H\ninst✝² : DivInvMonoid G\ninst✝¹ : DivInvMonoid H\ninst✝ : MonoidHomClass F G H\nf : F\nhf : ∀ (a : G), f a⁻¹ = (f a)⁻¹\ng h : ι → G\nx✝ : ι\n⊢ (⇑f ∘ (g / h)) x✝ = (⇑f ∘ g / ⇑f ∘ h) x✝","state_after":"no goals","tactic":"simp [map_div' f hf]","premises":[{"full_name":"map_div'","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[385,8],"def_end_pos":[385,16]}]}]}
{"url":"Mathlib/FieldTheory/RatFunc/Basic.lean","commit":"","full_name":"RatFunc.num_dvd","start":[975,0],"end":[985,26],"file_path":"Mathlib/FieldTheory/RatFunc/Basic.lean","tactics":[{"state_before":"K : Type u\ninst✝ : Field K\nx : RatFunc K\np : K[X]\nhp : p ≠ 0\n⊢ x.num ∣ p ↔ ∃ q, q ≠ 0 ∧ x = (algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q","state_after":"case mp\nK : Type u\ninst✝ : Field K\nx : RatFunc K\np : K[X]\nhp : p ≠ 0\n⊢ x.num ∣ p → ∃ q, q ≠ 0 ∧ x = (algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q\n\ncase mpr\nK : Type u\ninst✝ : Field K\nx : RatFunc K\np : K[X]\nhp : p ≠ 0\n⊢ (∃ q, q ≠ 0 ∧ x = (algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q) → x.num ∣ p","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Data/Rat/Cast/Lemmas.lean","commit":"","full_name":"Rat.cast_ofScientific","start":[48,0],"end":[52,84],"file_path":"Mathlib/Data/Rat/Cast/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : DivisionRing α\nK : Type u_2\ninst✝ : DivisionRing K\nm : ℕ\ns : Bool\ne : ℕ\n⊢ ↑(OfScientific.ofScientific m s e) = OfScientific.ofScientific m s e","state_after":"no goals","tactic":"rw [← NNRat.cast_ofScientific (K := K), ← NNRat.cast_ofScientific, cast_nnratCast]","premises":[{"full_name":"NNRat.cast_ofScientific","def_path":"Mathlib/Algebra/Order/Ring/Rat.lean","def_pos":[64,8],"def_end_pos":[64,38]},{"full_name":"Rat.cast_nnratCast","def_path":"Mathlib/Data/Rat/Cast/Lemmas.lean","def_pos":[39,8],"def_end_pos":[39,22]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","commit":"","full_name":"AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv","start":[72,0],"end":[98,46],"file_path":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","tactics":[{"state_before":"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\n⊢ affineSpan k (range p) = ⊤","state_after":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\n⊢ affineSpan k (range p) = ⊤","tactic":"cases nonempty_fintype ι","premises":[{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]},{"state_before":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\n⊢ affineSpan k (range p) = ⊤","state_after":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\n⊢ ∀ (i : ι), b i ∈ affineSpan k (range p)","tactic":"suffices ∀ i, b i ∈ affineSpan k (range p) by\n    rw [eq_top_iff, ← b.tot, affineSpan_le]\n    rintro q ⟨i, rfl⟩\n    exact this i","premises":[{"full_name":"AffineBasis.tot","def_path":"Mathlib/LinearAlgebra/AffineSpace/Basis.lean","def_pos":[79,8],"def_end_pos":[79,11]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"affineSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[483,4],"def_end_pos":[483,14]},{"full_name":"affineSpan_le","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[628,8],"def_end_pos":[628,28]},{"full_name":"eq_top_iff","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[116,8],"def_end_pos":[116,18]}]},{"state_before":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\n⊢ ∀ (i : ι), b i ∈ affineSpan k (range p)","state_after":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\n⊢ b i ∈ affineSpan k (range p)","tactic":"intro i","premises":[]},{"state_before":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\n⊢ b i ∈ affineSpan k (range p)","state_after":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\nhAi : ∑ j : ι', A i j = 1\n⊢ b i ∈ affineSpan k (range p)","tactic":"have hAi : ∑ j, A i j = 1 := by\n    calc\n      ∑ j, A i j = ∑ j, A i j * ∑ l, b.toMatrix p j l := by simp\n      _ = ∑ j, ∑ l, A i j * b.toMatrix p j l := by simp_rw [Finset.mul_sum]\n      _ = ∑ l, ∑ j, A i j * b.toMatrix p j l := by rw [Finset.sum_comm]\n      _ = ∑ l, (A * b.toMatrix p) i l := rfl\n      _ = 1 := by simp [hA, Matrix.one_apply, Finset.filter_eq]","premises":[{"full_name":"AffineBasis.toMatrix","def_path":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","def_pos":[36,18],"def_end_pos":[36,26]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.filter_eq","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2386,8],"def_end_pos":[2386,17]},{"full_name":"Finset.mul_sum","def_path":"Mathlib/Algebra/BigOperators/Ring.lean","def_pos":[44,6],"def_end_pos":[44,13]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_comm","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[816,2],"def_end_pos":[816,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Matrix.one_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[506,8],"def_end_pos":[506,17]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\nhAi : ∑ j : ι', A i j = 1\n⊢ b i ∈ affineSpan k (range p)","state_after":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\nhAi : ∑ j : ι', A i j = 1\nhbi : b i = (Finset.affineCombination k Finset.univ p) (A i)\n⊢ b i ∈ affineSpan k (range p)","tactic":"have hbi : b i = Finset.univ.affineCombination k p (A i) := by\n    apply b.ext_elem\n    intro j\n    rw [b.coord_apply, Finset.univ.map_affineCombination _ _ hAi,\n      Finset.univ.affineCombination_eq_linear_combination _ _ hAi]\n    change _ = (A * b.toMatrix p) i j\n    simp_rw [hA, Matrix.one_apply, @eq_comm _ i j]","premises":[{"full_name":"AffineBasis.coord_apply","def_path":"Mathlib/LinearAlgebra/AffineSpace/Basis.lean","def_pos":[163,8],"def_end_pos":[163,19]},{"full_name":"AffineBasis.ext_elem","def_path":"Mathlib/LinearAlgebra/AffineSpace/Basis.lean","def_pos":[208,8],"def_end_pos":[208,16]},{"full_name":"AffineBasis.toMatrix","def_path":"Mathlib/LinearAlgebra/AffineSpace/Matrix.lean","def_pos":[36,18],"def_end_pos":[36,26]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.affineCombination","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[339,4],"def_end_pos":[339,21]},{"full_name":"Finset.affineCombination_eq_linear_combination","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[424,8],"def_end_pos":[424,47]},{"full_name":"Finset.map_affineCombination","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[574,8],"def_end_pos":[574,29]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Matrix.one_apply","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[506,8],"def_end_pos":[506,17]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]},{"state_before":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\nhAi : ∑ j : ι', A i j = 1\nhbi : b i = (Finset.affineCombination k Finset.univ p) (A i)\n⊢ b i ∈ affineSpan k (range p)","state_after":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\nhAi : ∑ j : ι', A i j = 1\nhbi : b i = (Finset.affineCombination k Finset.univ p) (A i)\n⊢ (Finset.affineCombination k Finset.univ p) (A i) ∈ affineSpan k (range p)","tactic":"rw [hbi]","premises":[]},{"state_before":"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix p = 1\nval✝ : Fintype ι\ni : ι\nhAi : ∑ j : ι', A i j = 1\nhbi : b i = (Finset.affineCombination k Finset.univ p) (A i)\n⊢ (Finset.affineCombination k Finset.univ p) (A i) ∈ affineSpan k (range p)","state_after":"no goals","tactic":"exact affineCombination_mem_affineSpan hAi p","premises":[{"full_name":"affineCombination_mem_affineSpan","def_path":"Mathlib/LinearAlgebra/AffineSpace/Combination.lean","def_pos":[910,8],"def_end_pos":[910,40]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","commit":"","full_name":"Matrix.invOf_fromBlocks_zero₁₂_eq","start":[100,0],"end":[104,44],"file_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁹ : Fintype l\ninst✝⁸ : Fintype m\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq l\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝² : Invertible A\ninst✝¹ : Invertible D\ninst✝ : Invertible (fromBlocks A 0 C D)\n⊢ ⅟(fromBlocks A 0 C D) = fromBlocks (⅟A) 0 (-(⅟D * C * ⅟A)) ⅟D","state_after":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁹ : Fintype l\ninst✝⁸ : Fintype m\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq l\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝² : Invertible A\ninst✝¹ : Invertible D\ninst✝ : Invertible (fromBlocks A 0 C D)\nthis : Invertible (fromBlocks A 0 C D) := A.fromBlocksZero₁₂Invertible C D\n⊢ ⅟(fromBlocks A 0 C D) = fromBlocks (⅟A) 0 (-(⅟D * C * ⅟A)) ⅟D","tactic":"letI := fromBlocksZero₁₂Invertible A C D","premises":[{"full_name":"Matrix.fromBlocksZero₁₂Invertible","def_path":"Mathlib/LinearAlgebra/Matrix/SchurComplement.lean","def_pos":[85,4],"def_end_pos":[85,30]}]},{"state_before":"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁹ : Fintype l\ninst✝⁸ : Fintype m\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq l\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing α\nA : Matrix m m α\nC : Matrix n m α\nD : Matrix n n α\ninst✝² : Invertible A\ninst✝¹ : Invertible D\ninst✝ : Invertible (fromBlocks A 0 C D)\nthis : Invertible (fromBlocks A 0 C D) := A.fromBlocksZero₁₂Invertible C D\n⊢ ⅟(fromBlocks A 0 C D) = fromBlocks (⅟A) 0 (-(⅟D * C * ⅟A)) ⅟D","state_after":"no goals","tactic":"convert (rfl : ⅟ (fromBlocks A 0 C D) = _)","premises":[{"full_name":"Invertible.invOf","def_path":"Mathlib/Algebra/Group/Invertible/Defs.lean","def_pos":[86,2],"def_end_pos":[86,7]},{"full_name":"Matrix.fromBlocks","def_path":"Mathlib/Data/Matrix/Block.lean","def_pos":[41,4],"def_end_pos":[41,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/CategoryTheory/Iso.lean","commit":"","full_name":"CategoryTheory.Iso.homFromEquiv_apply","start":[218,0],"end":[224,27],"file_path":"Mathlib/CategoryTheory/Iso.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z✝ : C\nα : X ≅ Y\nZ : C\n⊢ Function.LeftInverse (fun g => α.hom ≫ g) fun f => α.inv ≫ f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z✝ : C\nα : X ≅ Y\nZ : C\n⊢ Function.RightInverse (fun g => α.hom ≫ g) fun f => α.inv ≫ f","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/Analysis/Calculus/FDeriv/Mul.lean","commit":"","full_name":"HasFDerivAt.multiset_prod","start":[705,0],"end":[716,78],"file_path":"Mathlib/Analysis/Calculus/FDeriv/Mul.lean","tactics":[{"state_before":"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g✝ : E → F\nf'✝ f₀' f₁' g'✝ e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_6\n𝔸 : Type u_7\n𝔸' : Type u_8\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedCommRing 𝔸'\ninst✝² : NormedAlgebra 𝕜 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu✝ : Finset ι\nf : ι → E → 𝔸\nf' : ι → E →L[𝕜] 𝔸\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nu : Multiset ι\nx : E\nh : ∀ i ∈ u, HasFDerivAt (fun x => g i x) (g' i) x\n⊢ HasFDerivAt (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod)\n    (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • g' i) u).sum x","state_after":"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g✝ : E → F\nf'✝ f₀' f₁' g'✝ e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_6\n𝔸 : Type u_7\n𝔸' : Type u_8\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedCommRing 𝔸'\ninst✝² : NormedAlgebra 𝕜 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu✝ : Finset ι\nf : ι → E → 𝔸\nf' : ι → E →L[𝕜] 𝔸\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nu : Multiset ι\nx : E\nh : ∀ i ∈ u, HasFDerivAt (fun x => g i x) (g' i) x\n⊢ HasFDerivAt (fun x => (Multiset.map ((fun x_1 => g x_1 x) ∘ Subtype.val) u.attach).prod)\n    (Multiset.map\n        ((fun x_1 =>\n            (Multiset.map (fun x_2 => g x_2 x) ((Multiset.map Subtype.val u.attach).erase x_1)).prod • g' x_1) ∘\n          Subtype.val)\n        u.attach).sum\n    x","tactic":"simp only [← Multiset.attach_map_val u, Multiset.map_map]","premises":[{"full_name":"Multiset.attach_map_val","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1354,8],"def_end_pos":[1354,22]},{"full_name":"Multiset.map_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1129,8],"def_end_pos":[1129,15]}]}]}
{"url":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","commit":"","full_name":"IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two'","start":[415,0],"end":[422,60],"file_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","tactics":[{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ ↑p\nhodd : p ≠ 2\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","state_after":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ ↑p\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","tactic":"have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by simpa using hcycl","premises":[{"full_name":"IsCyclotomicExtension","def_path":"Mathlib/NumberTheory/Cyclotomic/Basic.lean","def_pos":[77,6],"def_end_pos":[77,27]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ ↑p\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))","state_after":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ✝ : IsPrimitiveRoot ζ ↑p\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (↑p ^ (0 + 1))\n⊢ Prime ((Algebra.norm ℤ) (hζ✝.toInteger - 1))","tactic":"replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simpa using hζ","premises":[{"full_name":"IsPrimitiveRoot","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[259,10],"def_end_pos":[259,25]}]},{"state_before":"p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ✝ : IsPrimitiveRoot ζ ↑p\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (↑p ^ (0 + 1))\n⊢ Prime ((Algebra.norm ℤ) (hζ✝.toInteger - 1))","state_after":"no goals","tactic":"exact hζ.prime_norm_toInteger_sub_one_of_prime_ne_two hodd","premises":[{"full_name":"IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two","def_path":"Mathlib/NumberTheory/Cyclotomic/Rat.lean","def_pos":[407,6],"def_end_pos":[407,50]}]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"List.sum_toFinset","start":[2050,0],"end":[2056,83],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\nM : Type u_6\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\nx✝ : [].Nodup\n⊢ [].toFinset.prod f = (map f []).prod","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\nM : Type u_6\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\na : α\nl : List α\nhl : (a :: l).Nodup\n⊢ (a :: l).toFinset.prod f = (map f (a :: l)).prod","state_after":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\nM : Type u_6\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\na : α\nl : List α\nhl✝ : (a :: l).Nodup\nnot_mem : a ∉ l\nhl : l.Nodup\n⊢ (a :: l).toFinset.prod f = (map f (a :: l)).prod","tactic":"let ⟨not_mem, hl⟩ := List.nodup_cons.mp hl","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"List.nodup_cons","def_path":"Mathlib/Data/List/Nodup.lean","def_pos":[35,8],"def_end_pos":[35,18]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\nM : Type u_6\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\na : α\nl : List α\nhl✝ : (a :: l).Nodup\nnot_mem : a ∉ l\nhl : l.Nodup\n⊢ (a :: l).toFinset.prod f = (map f (a :: l)).prod","state_after":"no goals","tactic":"simp [Finset.prod_insert (mt List.mem_toFinset.mp not_mem), prod_toFinset _ hl]","premises":[{"full_name":"Finset.prod_insert","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[329,8],"def_end_pos":[329,19]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"List.mem_toFinset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2704,8],"def_end_pos":[2704,20]},{"full_name":"mt","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[647,8],"def_end_pos":[647,10]}]}]}
{"url":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","commit":"","full_name":"AffineSubspace.direction_top","start":[677,0],"end":[685,24],"file_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","tactics":[{"state_before":"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\n⊢ ⊤.direction = ⊤","state_after":"case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\n⊢ ⊤.direction = ⊤","tactic":"cases' S.nonempty with p","premises":[{"full_name":"AddTorsor.nonempty","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[47,3],"def_end_pos":[47,11]}]},{"state_before":"case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\n⊢ ⊤.direction = ⊤","state_after":"case intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\nv : V\n⊢ v ∈ ⊤.direction ↔ v ∈ ⊤","tactic":"ext v","premises":[]},{"state_before":"case intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\nv : V\n⊢ v ∈ ⊤.direction ↔ v ∈ ⊤","state_after":"case intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\nv : V\n_hv : v ∈ ⊤\n⊢ v ∈ ⊤.direction","tactic":"refine ⟨imp_intro Submodule.mem_top, fun _hv => ?_⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"Submodule.mem_top","def_path":"Mathlib/Algebra/Module/Submodule/Lattice.lean","def_pos":[144,8],"def_end_pos":[144,15]},{"full_name":"imp_intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1405,8],"def_end_pos":[1405,17]}]},{"state_before":"case intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\nv : V\n_hv : v ∈ ⊤\n⊢ v ∈ ⊤.direction","state_after":"case intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\nv : V\n_hv : v ∈ ⊤\nhpv : v +ᵥ p -ᵥ p ∈ ⊤.direction\n⊢ v ∈ ⊤.direction","tactic":"have hpv : (v +ᵥ p -ᵥ p : V) ∈ (⊤ : AffineSubspace k P).direction :=\n    vsub_mem_direction (mem_top k V _) (mem_top k V _)","premises":[{"full_name":"AffineSubspace","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[135,10],"def_end_pos":[135,24]},{"full_name":"AffineSubspace.direction","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[176,4],"def_end_pos":[176,13]},{"full_name":"AffineSubspace.mem_top","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[672,8],"def_end_pos":[672,15]},{"full_name":"AffineSubspace.vsub_mem_direction","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean","def_pos":[235,8],"def_end_pos":[235,26]},{"full_name":"HVAdd.hVAdd","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[61,2],"def_end_pos":[61,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"VSub.vsub","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[94,2],"def_end_pos":[94,6]}]},{"state_before":"case intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ p : P\nv : V\n_hv : v ∈ ⊤\nhpv : v +ᵥ p -ᵥ p ∈ ⊤.direction\n⊢ v ∈ ⊤.direction","state_after":"no goals","tactic":"rwa [vadd_vsub] at hpv","premises":[{"full_name":"vadd_vsub","def_path":"Mathlib/Algebra/AddTorsor.lean","def_pos":[86,8],"def_end_pos":[86,17]}]}]}
{"url":"Mathlib/Algebra/Lie/Weights/Cartan.lean","commit":"","full_name":"LieAlgebra.mem_corootSpace","start":[265,0],"end":[277,6],"file_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","tactics":[{"state_before":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\n⊢ x ∈ corootSpace α ↔ ↑x ∈ Submodule.span R {x | ∃ y ∈ rootSpace H α, ∃ z ∈ rootSpace H (-α), ⁅y, z⁆ = x}","state_after":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n⊢ x ∈ corootSpace α ↔ ↑x ∈ Submodule.span R {x | ∃ y ∈ rootSpace H α, ∃ z ∈ rootSpace H (-α), ⁅y, z⁆ = x}","tactic":"have : x ∈ corootSpace α ↔\n      (x : L) ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α) := by\n    rw [corootSpace]\n    simpa using exists_congr fun _ ↦ H.toLieSubmodule.injective_incl.eq_iff.symm","premises":[{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"Iff.symm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[813,8],"def_end_pos":[813,16]},{"full_name":"LieAlgebra.corootSpace","def_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","def_pos":[259,4],"def_end_pos":[259,15]},{"full_name":"LieSubalgebra.toLieSubmodule","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[272,4],"def_end_pos":[272,18]},{"full_name":"LieSubmodule.incl","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[602,4],"def_end_pos":[602,8]},{"full_name":"LieSubmodule.injective_incl","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[616,8],"def_end_pos":[616,22]},{"full_name":"LieSubmodule.map","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[754,4],"def_end_pos":[754,7]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"exists_congr","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[210,8],"def_end_pos":[210,20]}]},{"state_before":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n⊢ x ∈ corootSpace α ↔ ↑x ∈ Submodule.span R {x | ∃ y ∈ rootSpace H α, ∃ z ∈ rootSpace H (-α), ⁅y, z⁆ = x}","state_after":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n⊢ ↑x ∈\n      Submodule.span R\n        {x |\n          ∃ a b,\n            ↑H.toLieSubmodule.incl\n                (↑(LieModuleHom.codRestrict H.toLieSubmodule\n                      ((rootSpace H 0).incl.comp (rootSpaceProduct R L H α (-α) 0 ⋯)) ⋯)\n                  (a ⊗ₜ[R] b)) =\n              x} ↔\n    ↑x ∈ Submodule.span R {x | ∃ y ∈ ↑(rootSpace H α), ∃ z ∈ ↑(rootSpace H (-α)), ⁅y, z⁆ = x}","tactic":"simp_rw [this, corootSpace, ← LieModuleHom.map_top, ← LieSubmodule.mem_coeSubmodule,\n    LieSubmodule.coeSubmodule_map, LieSubmodule.top_coeSubmodule, ← TensorProduct.span_tmul_eq_top,\n    LinearMap.map_span, Set.image, Set.mem_setOf_eq, exists_exists_exists_and_eq]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"LieAlgebra.corootSpace","def_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","def_pos":[259,4],"def_end_pos":[259,15]},{"full_name":"LieModuleHom.map_top","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1279,8],"def_end_pos":[1279,15]},{"full_name":"LieSubmodule.coeSubmodule_map","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[764,8],"def_end_pos":[764,24]},{"full_name":"LieSubmodule.mem_coeSubmodule","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[104,8],"def_end_pos":[104,24]},{"full_name":"LieSubmodule.top_coeSubmodule","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[350,8],"def_end_pos":[350,24]},{"full_name":"Set.image","def_path":"Mathlib/Init/Set.lean","def_pos":[208,4],"def_end_pos":[208,9]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"TensorProduct.span_tmul_eq_top","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[451,8],"def_end_pos":[451,24]},{"full_name":"exists_exists_exists_and_eq","def_path":"Mathlib/Logic/Basic.lean","def_pos":[574,16],"def_end_pos":[574,43]}]},{"state_before":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n⊢ ↑x ∈\n      Submodule.span R\n        {x |\n          ∃ a b,\n            ↑H.toLieSubmodule.incl\n                (↑(LieModuleHom.codRestrict H.toLieSubmodule\n                      ((rootSpace H 0).incl.comp (rootSpaceProduct R L H α (-α) 0 ⋯)) ⋯)\n                  (a ⊗ₜ[R] b)) =\n              x} ↔\n    ↑x ∈ Submodule.span R {x | ∃ y ∈ ↑(rootSpace H α), ∃ z ∈ ↑(rootSpace H (-α)), ⁅y, z⁆ = x}","state_after":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n⊢ ↑x ∈ Submodule.span R {x | ∃ a b, ⁅↑a, ↑b⁆ = x} ↔\n    ↑x ∈ Submodule.span R {x | ∃ y ∈ ↑(rootSpace H α), ∃ z ∈ ↑(rootSpace H (-α)), ⁅y, z⁆ = x}","tactic":"change (x : L) ∈ Submodule.span R\n    {x | ∃ (a : rootSpace H α) (b : rootSpace H (-α)), ⁅(a : L), (b : L)⁆ = x} ↔ _","premises":[{"full_name":"Bracket.bracket","def_path":"Mathlib/Data/Bracket.lean","def_pos":[35,2],"def_end_pos":[35,9]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Iff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[114,10],"def_end_pos":[114,13]},{"full_name":"LieAlgebra.rootSpace","def_path":"Mathlib/Algebra/Lie/Weights/Cartan.lean","def_pos":[44,7],"def_end_pos":[44,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Submodule.span","def_path":"Mathlib/LinearAlgebra/Span.lean","def_pos":[49,4],"def_end_pos":[49,8]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"R : Type u_1\nL : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁶ : IsNilpotent R ↥H\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n⊢ ↑x ∈ Submodule.span R {x | ∃ a b, ⁅↑a, ↑b⁆ = x} ↔\n    ↑x ∈ Submodule.span R {x | ∃ y ∈ ↑(rootSpace H α), ∃ z ∈ ↑(rootSpace H (-α)), ⁅y, z⁆ = x}","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/SpecificLimits/Normed.lean","commit":"","full_name":"hasSum_geometric_of_norm_lt_one","start":[268,0],"end":[276,67],"file_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹","tactic":"have xi_ne_one : ξ ≠ 1 := by\n    contrapose! h\n    simp [h]","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹","tactic":"have A : Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹)) :=\n    ((tendsto_pow_atTop_nhds_zero_of_norm_lt_one h).sub tendsto_const_nhds).mul tendsto_const_nhds","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.mul","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[116,8],"def_end_pos":[116,26]},{"full_name":"Filter.Tendsto.sub","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[927,14],"def_end_pos":[927,17]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"tendsto_const_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[844,8],"def_end_pos":[844,26]},{"full_name":"tendsto_pow_atTop_nhds_zero_of_norm_lt_one","def_path":"Mathlib/Analysis/SpecificLimits/Normed.lean","def_pos":[247,8],"def_end_pos":[247,50]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, ξ ^ i) atTop (𝓝 (1 - ξ)⁻¹)\n\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nK : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Summable fun i => ‖ξ ^ i‖","tactic":"rw [hasSum_iff_tendsto_nat_of_summable_norm]","premises":[{"full_name":"hasSum_iff_tendsto_nat_of_summable_norm","def_path":"Mathlib/Analysis/Normed/Group/InfiniteSum.lean","def_pos":[97,8],"def_end_pos":[97,47]}]}]}
{"url":"Mathlib/Data/List/Sublists.lean","commit":"","full_name":"List.sublists'Aux_eq_array_foldl","start":[43,0],"end":[50,18],"file_path":"Mathlib/Data/List/Sublists.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na : α\n⊢ ∀ (r₁ r₂ : List (List α)),\n    sublists'Aux a r₁ r₂ = (Array.foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁) 0).toList","state_after":"α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ sublists'Aux a r₁ r₂ = (Array.foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁) 0).toList","tactic":"intro r₁ r₂","premises":[]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ sublists'Aux a r₁ r₂ = (Array.foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁) 0).toList","state_after":"α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁).data).toList","tactic":"rw [sublists'Aux, Array.foldl_eq_foldl_data]","premises":[{"full_name":"Array.foldl_eq_foldl_data","def_path":".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean","def_pos":[61,8],"def_end_pos":[61,27]},{"full_name":"List.sublists'Aux","def_path":"Mathlib/Data/List/Sublists.lean","def_pos":[40,4],"def_end_pos":[40,16]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁).data).toList","state_after":"α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\nthis :\n  foldl (fun r l => r ++ [a :: l]) (toArray r₂).toList r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) r₁).toList\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁).data).toList","tactic":"have := List.foldl_hom Array.toList (fun r l => r.push (a :: l))\n    (fun r l => r ++ [a :: l]) r₁ r₂.toArray (by simp)","premises":[{"full_name":"Array.push","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2608,4],"def_end_pos":[2608,14]},{"full_name":"Array.toList","def_path":".lake/packages/lean4/src/lean/Init/Data/Array/Basic.lean","def_pos":[498,4],"def_end_pos":[498,10]},{"full_name":"List.cons","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2289,4],"def_end_pos":[2289,8]},{"full_name":"List.foldl_hom","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[901,8],"def_end_pos":[901,17]},{"full_name":"List.nil","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2286,4],"def_end_pos":[2286,7]},{"full_name":"List.toArray","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2719,4],"def_end_pos":[2719,16]}]},{"state_before":"α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\nthis :\n  foldl (fun r l => r ++ [a :: l]) (toArray r₂).toList r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) r₁).toList\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁).data).toList","state_after":"no goals","tactic":"simpa using this","premises":[]}]}
{"url":"Mathlib/Algebra/BigOperators/Group/Finset.lean","commit":"","full_name":"Finset.card_filter","start":[1779,0],"end":[1780,67],"file_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\n⊢ (filter p s).card = ∑ a ∈ s, if p a then 1 else 0","state_after":"no goals","tactic":"simp [sum_ite]","premises":[{"full_name":"Finset.sum_ite","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1051,2],"def_end_pos":[1051,13]}]}]}
{"url":"Mathlib/NumberTheory/RamificationInertia.lean","commit":"","full_name":"Ideal.powQuotSuccInclusion_injective","start":[455,0],"end":[460,45],"file_path":"Mathlib/NumberTheory/RamificationInertia.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\n⊢ Function.Injective ⇑(powQuotSuccInclusion f p P i)","state_after":"R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\n⊢ ∀ (m : ↥(map (Quotient.mk (P ^ e)) (P ^ (i + 1)))), (powQuotSuccInclusion f p P i) m = 0 → m = 0","tactic":"rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']","premises":[{"full_name":"LinearMap.ker_eq_bot","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[185,8],"def_end_pos":[185,18]},{"full_name":"LinearMap.ker_eq_bot'","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[101,8],"def_end_pos":[101,19]}]},{"state_before":"R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\n⊢ ∀ (m : ↥(map (Quotient.mk (P ^ e)) (P ^ (i + 1)))), (powQuotSuccInclusion f p P i) m = 0 → m = 0","state_after":"case mk\nR : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\nx : S ⧸ P ^ e\nhx : x ∈ map (Quotient.mk (P ^ e)) (P ^ (i + 1))\nhx0 : (powQuotSuccInclusion f p P i) ⟨x, hx⟩ = 0\n⊢ ⟨x, hx⟩ = 0","tactic":"rintro ⟨x, hx⟩ hx0","premises":[]},{"state_before":"case mk\nR : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\nx : S ⧸ P ^ e\nhx : x ∈ map (Quotient.mk (P ^ e)) (P ^ (i + 1))\nhx0 : (powQuotSuccInclusion f p P i) ⟨x, hx⟩ = 0\n⊢ ⟨x, hx⟩ = 0","state_after":"case mk\nR : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\nx : S ⧸ P ^ e\nhx : x ∈ map (Quotient.mk (P ^ e)) (P ^ (i + 1))\nhx0 : ↑((powQuotSuccInclusion f p P i) ⟨x, hx⟩) = ↑0\n⊢ ↑⟨x, hx⟩ = ↑0","tactic":"rw [Subtype.ext_iff] at hx0 ⊢","premises":[{"full_name":"Subtype.ext_iff","def_path":"Mathlib/Data/Subtype.lean","def_pos":[62,18],"def_end_pos":[62,25]}]},{"state_before":"case mk\nR : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ni : ℕ\nx : S ⧸ P ^ e\nhx : x ∈ map (Quotient.mk (P ^ e)) (P ^ (i + 1))\nhx0 : ↑((powQuotSuccInclusion f p P i) ⟨x, hx⟩) = ↑0\n⊢ ↑⟨x, hx⟩ = ↑0","state_after":"no goals","tactic":"rwa [powQuotSuccInclusion_apply_coe] at hx0","premises":[{"full_name":"Ideal.powQuotSuccInclusion_apply_coe","def_path":"Mathlib/NumberTheory/RamificationInertia.lean","def_pos":[447,2],"def_end_pos":[447,7]}]}]}
{"url":"Mathlib/MeasureTheory/Function/LpSpace.lean","commit":"","full_name":"MeasureTheory.Lp.norm_const'","start":[906,0],"end":[909,80],"file_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","tactics":[{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\nhp_zero : p ≠ 0\nhp_top : p ≠ ⊤\n⊢ ‖(Lp.const p μ) c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)","state_after":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\nhp_zero : p ≠ 0\nhp_top : p ≠ ⊤\n⊢ (↑‖c‖₊ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)","tactic":"rw [← Memℒp.toLp_const, Lp.norm_toLp, eLpNorm_const'] <;> try assumption","premises":[{"full_name":"MeasureTheory.Lp.norm_toLp","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[259,6],"def_end_pos":[259,15]},{"full_name":"MeasureTheory.Memℒp.toLp_const","def_path":"Mathlib/MeasureTheory/Function/LpSpace.lean","def_pos":[892,6],"def_end_pos":[892,22]},{"full_name":"MeasureTheory.eLpNorm_const'","def_path":"Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean","def_pos":[395,8],"def_end_pos":[395,22]}]},{"state_before":"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\nhp_zero : p ≠ 0\nhp_top : p ≠ ⊤\n⊢ (↑‖c‖₊ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)","state_after":"no goals","tactic":"rw [ENNReal.toReal_mul, ENNReal.coe_toReal, ← ENNReal.toReal_rpow, coe_nnnorm]","premises":[{"full_name":"ENNReal.coe_toReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[234,16],"def_end_pos":[234,26]},{"full_name":"ENNReal.toReal_mul","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[378,8],"def_end_pos":[378,18]},{"full_name":"ENNReal.toReal_rpow","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[811,8],"def_end_pos":[811,19]},{"full_name":"coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[615,40],"def_end_pos":[615,50]}]}]}
{"url":"Mathlib/Analysis/Complex/AbelLimit.lean","commit":"","full_name":"Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet","start":[108,0],"end":[114,59],"file_path":"Mathlib/Analysis/Complex/AbelLimit.lean","tactics":[{"state_before":"s : ℝ\nhs : 0 < s\n⊢ ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1","state_after":"case intro.intro.intro.intro\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1","tactic":"obtain ⟨M, ε, _, hε, H⟩ := stolzCone_subset_stolzSet_aux hs","premises":[{"full_name":"Complex.stolzCone_subset_stolzSet_aux","def_path":"Mathlib/Analysis/Complex/AbelLimit.lean","def_pos":[94,6],"def_end_pos":[94,35]}]},{"state_before":"case intro.intro.intro.intro\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1","state_after":"case h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1","tactic":"use M","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1","state_after":"case h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ ∃ u, IsOpen u ∧ 1 ∈ u ∧ u ∩ stolzCone s ⊆ stolzSet M","tactic":"rw [nhdsWithin_le_iff, mem_nhdsWithin]","premises":[{"full_name":"mem_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[77,8],"def_end_pos":[77,22]},{"full_name":"nhdsWithin_le_iff","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[110,8],"def_end_pos":[110,25]}]},{"state_before":"case h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ ∃ u, IsOpen u ∧ 1 ∈ u ∧ u ∩ stolzCone s ⊆ stolzSet M","state_after":"case h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ 1 ∈ {w | 1 - ε < w.re}","tactic":"refine ⟨{w | 1 - ε < w.re}, isOpen_lt continuous_const continuous_re, ?_, H⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Complex.continuous_re","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[241,8],"def_end_pos":[241,21]},{"full_name":"Complex.re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[29,2],"def_end_pos":[29,4]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"continuous_const","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1436,8],"def_end_pos":[1436,24]},{"full_name":"isOpen_lt","def_path":"Mathlib/Topology/Order/OrderClosed.lean","def_pos":[638,8],"def_end_pos":[638,17]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]},{"state_before":"case h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nleft✝ : 0 < M\nhε : 0 < ε\nH : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M\n⊢ 1 ∈ {w | 1 - ε < w.re}","state_after":"no goals","tactic":"simp only [Set.mem_setOf_eq, one_re, sub_lt_self_iff, hε]","premises":[{"full_name":"Complex.one_re","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[146,8],"def_end_pos":[146,14]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"sub_lt_self_iff","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[269,2],"def_end_pos":[269,13]}]}]}
{"url":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","commit":"","full_name":"zpow_sub₀","start":[355,0],"end":[356,65],"file_path":"Mathlib/Algebra/GroupWithZero/Units/Basic.lean","tactics":[{"state_before":"α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝¹ : MonoidWithZero M₀\ninst✝ : GroupWithZero G₀\na b c d : G₀\nm✝ n✝ : ℕ\nha : a ≠ 0\nm n : ℤ\n⊢ a ^ (m - n) = a ^ m / a ^ n","state_after":"no goals","tactic":"rw [Int.sub_eq_add_neg, zpow_add₀ ha, zpow_neg, div_eq_mul_inv]","premises":[{"full_name":"Int.sub_eq_add_neg","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[87,18],"def_end_pos":[87,32]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]},{"full_name":"zpow_add₀","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[401,6],"def_end_pos":[401,15]},{"full_name":"zpow_neg","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[429,6],"def_end_pos":[429,14]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean","commit":"","full_name":"Basis.addHaar_eq_iff","start":[261,0],"end":[267,48],"file_path":"Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹⁰ : Fintype ι\ninst✝⁹ : Fintype ι'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nb : Basis ι ℝ E\nμ : Measure E\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsAddLeftInvariant\n⊢ b.addHaar = μ ↔ μ ↑b.parallelepiped = 1","state_after":"ι : Type u_1\nι' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹⁰ : Fintype ι\ninst✝⁹ : Fintype ι'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nb : Basis ι ℝ E\nμ : Measure E\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsAddLeftInvariant\n⊢ addHaarMeasure b.parallelepiped = μ ↔ μ ↑b.parallelepiped = 1","tactic":"rw [Basis.addHaar_def]","premises":[{"full_name":"Basis.addHaar_def","def_path":"Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean","def_pos":[249,0],"def_end_pos":[252,41]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹⁰ : Fintype ι\ninst✝⁹ : Fintype ι'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nb : Basis ι ℝ E\nμ : Measure E\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsAddLeftInvariant\n⊢ addHaarMeasure b.parallelepiped = μ ↔ μ ↑b.parallelepiped = 1","state_after":"no goals","tactic":"exact addHaarMeasure_eq_iff b.parallelepiped μ","premises":[{"full_name":"Basis.parallelepiped","def_path":"Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean","def_pos":[180,4],"def_end_pos":[180,24]},{"full_name":"MeasureTheory.Measure.addHaarMeasure_eq_iff","def_path":"Mathlib/MeasureTheory/Measure/Haar/Basic.lean","def_pos":[653,2],"def_end_pos":[653,13]}]}]}
{"url":"Mathlib/Algebra/Lie/OfAssociative.lean","commit":"","full_name":"LieModule.toEnd_module_end","start":[195,0],"end":[197,86],"file_path":"Mathlib/Algebra/Lie/OfAssociative.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ toEnd R (Module.End R M) M = LieHom.id","state_after":"case h.h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\ng : Module.End R M\nm : M\n⊢ ((toEnd R (Module.End R M) M) g) m = (LieHom.id g) m","tactic":"ext g m","premises":[]},{"state_before":"case h.h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\ng : Module.End R M\nm : M\n⊢ ((toEnd R (Module.End R M) M) g) m = (LieHom.id g) m","state_after":"no goals","tactic":"simp [lie_eq_smul]","premises":[{"full_name":"lie_eq_smul","def_path":"Mathlib/Algebra/Lie/OfAssociative.lean","def_pos":[83,8],"def_end_pos":[83,19]}]}]}
{"url":"Mathlib/Algebra/Order/BigOperators/Group/List.lean","commit":"","full_name":"List.monotone_sum_take","start":[170,0],"end":[175,72],"file_path":"Mathlib/Algebra/Order/BigOperators/Group/List.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM₀ : Type u_6\nG : Type u_7\nR : Type u_8\ninst✝ : CanonicallyOrderedCommMonoid M\nl L : List M\n⊢ Monotone fun i => (take i L).prod","state_after":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM₀ : Type u_6\nG : Type u_7\nR : Type u_8\ninst✝ : CanonicallyOrderedCommMonoid M\nl L : List M\nn : ℕ\n⊢ (take n L).prod ≤ (take (n + 1) L).prod","tactic":"refine monotone_nat_of_le_succ fun n => ?_","premises":[{"full_name":"monotone_nat_of_le_succ","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[895,8],"def_end_pos":[895,31]}]},{"state_before":"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM₀ : Type u_6\nG : Type u_7\nR : Type u_8\ninst✝ : CanonicallyOrderedCommMonoid M\nl L : List M\nn : ℕ\n⊢ (take n L).prod ≤ (take (n + 1) L).prod","state_after":"case inl\nι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM₀ : Type u_6\nG : Type u_7\nR : Type u_8\ninst✝ : CanonicallyOrderedCommMonoid M\nl L : List M\nn : ℕ\nh : n < L.length\n⊢ (take n L).prod ≤ (take (n + 1) L).prod\n\ncase inr\nι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM₀ : Type u_6\nG : Type u_7\nR : Type u_8\ninst✝ : CanonicallyOrderedCommMonoid M\nl L : List M\nn : ℕ\nh : L.length ≤ n\n⊢ (take n L).prod ≤ (take (n + 1) L).prod","tactic":"cases' lt_or_le n L.length with h h","premises":[{"full_name":"List.length","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2316,4],"def_end_pos":[2316,15]},{"full_name":"lt_or_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[287,8],"def_end_pos":[287,16]}]}]}
{"url":"Mathlib/Order/PrimeIdeal.lean","commit":"","full_name":"Order.Ideal.IsPrime.of_mem_or_mem","start":[112,0],"end":[120,32],"file_path":"Mathlib/Order/PrimeIdeal.lean","tactics":[{"state_before":"P : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ I.IsPrime","state_after":"P : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ I.IsProper ∧ IsPFilter (↑I)ᶜ","tactic":"rw [isPrime_iff]","premises":[{"full_name":"Order.Ideal.isPrime_iff","def_path":"Mathlib/Order/PrimeIdeal.lean","def_pos":[79,2],"def_end_pos":[79,8]}]},{"state_before":"P : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ I.IsProper ∧ IsPFilter (↑I)ᶜ","state_after":"case right\nP : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ IsPFilter (↑I)ᶜ","tactic":"use ‹_›","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]},{"state_before":"case right\nP : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ IsPFilter (↑I)ᶜ","state_after":"case right.refine_1\nP : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ (↑I)ᶜ.Nonempty\n\ncase right.refine_2\nP : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ DirectedOn (fun x x_1 => x ≥ x_1) (↑I)ᶜ\n\ncase right.refine_3\nP : Type u_1\ninst✝¹ : SemilatticeInf P\nx y : P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ ∀ {x y : P}, x ≤ y → x ∈ (↑I)ᶜ → y ∈ (↑I)ᶜ","tactic":"refine .of_def ?_ ?_ ?_","premises":[{"full_name":"Order.IsPFilter.of_def","def_path":"Mathlib/Order/PFilter.lean","def_pos":[50,8],"def_end_pos":[50,24]}]}]}
{"url":"Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean","commit":"","full_name":"AdicCompletion.ofTensorProduct_bijective_of_pi_of_fintype","start":[143,0],"end":[148,65],"file_path":"Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean","tactics":[{"state_before":"R : Type u_1\ninst✝⁵ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Finite ι\n⊢ Function.Bijective ⇑(ofTensorProduct I (ι → R))","state_after":"no goals","tactic":"classical\n  cases nonempty_fintype ι\n  exact EquivLike.bijective (ofTensorProductEquivOfPiFintype I ι)","premises":[{"full_name":"AdicCompletion.ofTensorProductEquivOfPiFintype","def_path":"Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean","def_pos":[133,4],"def_end_pos":[133,35]},{"full_name":"EquivLike.bijective","def_path":"Mathlib/Data/FunLike/Equiv.lean","def_pos":[171,18],"def_end_pos":[171,27]},{"full_name":"nonempty_fintype","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[390,8],"def_end_pos":[390,24]}]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","commit":"","full_name":"WeierstrassCurve.Affine.equation_neg_iff","start":[378,0],"end":[381,7],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x (W.negY x y) ↔ W.Equation x y","state_after":"R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ (-y - W.a₁ * x - W.a₃) ^ 2 + W.a₁ * x * (-y - W.a₁ * x - W.a₃) + W.a₃ * (-y - W.a₁ * x - W.a₃) =\n      x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆ ↔\n    y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆","tactic":"rw [equation_iff, equation_iff, negY]","premises":[{"full_name":"WeierstrassCurve.Affine.equation_iff","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","def_pos":[186,6],"def_end_pos":[186,18]},{"full_name":"WeierstrassCurve.Affine.negY","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean","def_pos":[309,4],"def_end_pos":[309,8]}]},{"state_before":"R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ (-y - W.a₁ * x - W.a₃) ^ 2 + W.a₁ * x * (-y - W.a₁ * x - W.a₃) + W.a₃ * (-y - W.a₁ * x - W.a₃) =\n      x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆ ↔\n    y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆","state_after":"case a.h.e'_2\nR : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ (-y - W.a₁ * x - W.a₃) ^ 2 + W.a₁ * x * (-y - W.a₁ * x - W.a₃) + W.a₃ * (-y - W.a₁ * x - W.a₃) =\n    y ^ 2 + W.a₁ * x * y + W.a₃ * y","tactic":"congr! 1","premises":[]},{"state_before":"case a.h.e'_2\nR : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ (-y - W.a₁ * x - W.a₃) ^ 2 + W.a₁ * x * (-y - W.a₁ * x - W.a₃) + W.a₃ * (-y - W.a₁ * x - W.a₃) =\n    y ^ 2 + W.a₁ * x * y + W.a₃ * y","state_after":"no goals","tactic":"ring1","premises":[]}]}
{"url":"Mathlib/NumberTheory/ADEInequality.lean","commit":"","full_name":"ADEInequality.admissible_of_one_lt_sumInv_aux'","start":[204,0],"end":[223,23],"file_path":"Mathlib/NumberTheory/ADEInequality.lean","tactics":[{"state_before":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\n⊢ Admissible {p, q, r}","state_after":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p < 3\n⊢ Admissible {p, q, r}","tactic":"have hp3 : p < 3 := lt_three hpq hqr H","premises":[{"full_name":"ADEInequality.lt_three","def_path":"Mathlib/NumberTheory/ADEInequality.lean","def_pos":[153,8],"def_end_pos":[153,16]}]},{"state_before":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p < 3\n⊢ Admissible {p, q, r}","state_after":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p ∈ Finset.Iio 3\n⊢ Admissible {p, q, r}","tactic":"replace hp3 := Finset.mem_Iio.mpr hp3","premises":[{"full_name":"Finset.mem_Iio","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[375,8],"def_end_pos":[375,15]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p ∈ Finset.Iio 3\n⊢ Admissible {p, q, r}","state_after":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p ∈ {1, 2}\n⊢ Admissible {p, q, r}","tactic":"conv at hp3 => change p ∈ ({1, 2} : Multiset ℕ+)","premises":[{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Multiset","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[28,4],"def_end_pos":[28,12]},{"full_name":"PNat","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[24,4],"def_end_pos":[24,8]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p ∈ {1, 2}\n⊢ Admissible {p, q, r}","state_after":"case head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 1 ≤ q\nH : 1 < sumInv {1, q, r}\n⊢ Admissible {1, q, r}\n\ncase tail.head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 2 ≤ q\nH : 1 < sumInv {2, q, r}\n⊢ Admissible {2, q, r}","tactic":"fin_cases hp3","premises":[]},{"state_before":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 2 ≤ q\nH : 1 < sumInv {2, q, r}\n⊢ Admissible {2, q, r}","state_after":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 2 ≤ q\nH : 1 < sumInv {2, q, r}\nhq4 : q < 4\n⊢ Admissible {2, q, r}","tactic":"have hq4 : q < 4 := lt_four hqr H","premises":[{"full_name":"ADEInequality.lt_four","def_path":"Mathlib/NumberTheory/ADEInequality.lean","def_pos":[175,8],"def_end_pos":[175,15]}]},{"state_before":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 2 ≤ q\nH : 1 < sumInv {2, q, r}\nhq4 : q < 4\n⊢ Admissible {2, q, r}","state_after":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 2 ≤ q\nH : 1 < sumInv {2, q, r}\nhq4 : q ∈ Finset.Ico 2 4\n⊢ Admissible {2, q, r}","tactic":"replace hq4 := Finset.mem_Ico.mpr ⟨hpq, hq4⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Finset.mem_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[299,8],"def_end_pos":[299,15]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nhpq : 2 ≤ q\nH : 1 < sumInv {2, q, r}\nhq4 : q ∈ Finset.Ico 2 4\n⊢ Admissible {2, q, r}","state_after":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\nhq4 : q ∈ Finset.Ico 2 4\n⊢ Admissible {2, q, r}","tactic":"clear hpq","premises":[]},{"state_before":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\nhq4 : q ∈ Finset.Ico 2 4\n⊢ Admissible {2, q, r}","state_after":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\nhq4 : q ∈ {2, 3}\n⊢ Admissible {2, q, r}","tactic":"conv at hq4 => change q ∈ ({2, 3} : Multiset ℕ+)","premises":[{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Multiset","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[28,4],"def_end_pos":[28,12]},{"full_name":"PNat","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[24,4],"def_end_pos":[24,8]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"case tail.head\nq r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\nhq4 : q ∈ {2, 3}\n⊢ Admissible {2, q, r}","state_after":"case tail.head.head\nr : ℕ+\nhqr : 2 ≤ r\nH : 1 < sumInv {2, 2, r}\n⊢ Admissible {2, 2, r}\n\ncase tail.head.tail.head\nr : ℕ+\nhqr : 3 ≤ r\nH : 1 < sumInv {2, 3, r}\n⊢ Admissible {2, 3, r}","tactic":"fin_cases hq4","premises":[]},{"state_before":"case tail.head.tail.head\nr : ℕ+\nhqr : 3 ≤ r\nH : 1 < sumInv {2, 3, r}\n⊢ Admissible {2, 3, r}","state_after":"case tail.head.tail.head\nr : ℕ+\nhqr : 3 ≤ r\nH : 1 < sumInv {2, 3, r}\nhr6 : r < 6\n⊢ Admissible {2, 3, r}","tactic":"have hr6 : r < 6 := lt_six H","premises":[{"full_name":"ADEInequality.lt_six","def_path":"Mathlib/NumberTheory/ADEInequality.lean","def_pos":[192,8],"def_end_pos":[192,14]}]},{"state_before":"case tail.head.tail.head\nr : ℕ+\nhqr : 3 ≤ r\nH : 1 < sumInv {2, 3, r}\nhr6 : r < 6\n⊢ Admissible {2, 3, r}","state_after":"case tail.head.tail.head\nr : ℕ+\nhqr : 3 ≤ r\nH : 1 < sumInv {2, 3, r}\nhr6 : r ∈ Finset.Ico 3 6\n⊢ Admissible {2, 3, r}","tactic":"replace hr6 := Finset.mem_Ico.mpr ⟨hqr, hr6⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Finset.mem_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[299,8],"def_end_pos":[299,15]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"case tail.head.tail.head\nr : ℕ+\nhqr : 3 ≤ r\nH : 1 < sumInv {2, 3, r}\nhr6 : r ∈ Finset.Ico 3 6\n⊢ Admissible {2, 3, r}","state_after":"case tail.head.tail.head\nr : ℕ+\nH : 1 < sumInv {2, 3, r}\nhr6 : r ∈ Finset.Ico 3 6\n⊢ Admissible {2, 3, r}","tactic":"clear hqr","premises":[]},{"state_before":"case tail.head.tail.head\nr : ℕ+\nH : 1 < sumInv {2, 3, r}\nhr6 : r ∈ Finset.Ico 3 6\n⊢ Admissible {2, 3, r}","state_after":"case tail.head.tail.head\nr : ℕ+\nH : 1 < sumInv {2, 3, r}\nhr6 : r ∈ {3, 4, 5}\n⊢ Admissible {2, 3, r}","tactic":"conv at hr6 => change r ∈ ({3, 4, 5} : Multiset ℕ+)","premises":[{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Multiset","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[28,4],"def_end_pos":[28,12]},{"full_name":"PNat","def_path":"Mathlib/Data/PNat/Defs.lean","def_pos":[24,4],"def_end_pos":[24,8]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"case tail.head.tail.head\nr : ℕ+\nH : 1 < sumInv {2, 3, r}\nhr6 : r ∈ {3, 4, 5}\n⊢ Admissible {2, 3, r}","state_after":"case tail.head.tail.head.head\nH : 1 < sumInv {2, 3, 3}\n⊢ Admissible {2, 3, 3}\n\ncase tail.head.tail.head.tail.head\nH : 1 < sumInv {2, 3, 4}\n⊢ Admissible {2, 3, 4}\n\ncase tail.head.tail.head.tail.tail.head\nH : 1 < sumInv {2, 3, 5}\n⊢ Admissible {2, 3, 5}","tactic":"fin_cases hr6","premises":[]}]}
{"url":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","commit":"","full_name":"IsPrimitiveRoot.pow_iff_coprime","start":[387,0],"end":[400,68],"file_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","tactics":[{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\n⊢ IsPrimitiveRoot (ζ ^ i) k ↔ i.Coprime k","state_after":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\n⊢ IsPrimitiveRoot (ζ ^ i) k → i.Coprime k","tactic":"refine ⟨?_, h.pow_of_coprime i⟩","premises":[{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]},{"full_name":"IsPrimitiveRoot.pow_of_coprime","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[368,8],"def_end_pos":[368,22]}]},{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\n⊢ IsPrimitiveRoot (ζ ^ i) k → i.Coprime k","state_after":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\n⊢ i.Coprime k","tactic":"intro hi","premises":[]},{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\n⊢ i.Coprime k","state_after":"case intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\n⊢ i.Coprime k","tactic":"obtain ⟨a, ha⟩ := i.gcd_dvd_left k","premises":[{"full_name":"Nat.gcd_dvd_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[86,8],"def_end_pos":[86,20]}]},{"state_before":"case intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\n⊢ i.Coprime k","state_after":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ i.Coprime k","tactic":"obtain ⟨b, hb⟩ := i.gcd_dvd_right k","premises":[{"full_name":"Nat.gcd_dvd_right","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[88,8],"def_end_pos":[88,21]}]},{"state_before":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ i.Coprime k","state_after":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ b = k","tactic":"suffices b = k by\n    -- Porting note: was `rwa [this, ← one_mul k, mul_left_inj' h0.ne', eq_comm] at hb`\n    rw [this, eq_comm, Nat.mul_left_eq_self_iff h0] at hb\n    rwa [Nat.Coprime]","premises":[{"full_name":"Nat.Coprime","def_path":".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean","def_pos":[20,17],"def_end_pos":[20,24]},{"full_name":"Nat.mul_left_eq_self_iff","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[330,6],"def_end_pos":[330,26]},{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]},{"state_before":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ b = k","state_after":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ b = k","tactic":"rw [ha] at hi","premises":[]},{"state_before":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ b = k","state_after":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = b * i.gcd k\n⊢ b = k","tactic":"rw [mul_comm] at hb","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"case intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = b * i.gcd k\n⊢ b = k","state_after":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = b * i.gcd k\n⊢ (ζ ^ (i.gcd k * a)) ^ b = 1","tactic":"apply Nat.dvd_antisymm ⟨i.gcd k, hb⟩ (hi.dvd_of_pow_eq_one b _)","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsPrimitiveRoot.dvd_of_pow_eq_one","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[261,2],"def_end_pos":[261,19]},{"full_name":"Nat.dvd_antisymm","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean","def_pos":[57,18],"def_end_pos":[57,30]},{"full_name":"Nat.gcd","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean","def_pos":[32,4],"def_end_pos":[32,7]}]},{"state_before":"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = b * i.gcd k\n⊢ (ζ ^ (i.gcd k * a)) ^ b = 1","state_after":"no goals","tactic":"rw [← pow_mul', ← mul_assoc, ← hb, pow_mul, h.pow_eq_one, one_pow]","premises":[{"full_name":"IsPrimitiveRoot.pow_eq_one","def_path":"Mathlib/RingTheory/RootsOfUnity/Basic.lean","def_pos":[260,2],"def_end_pos":[260,12]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"one_pow","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[593,38],"def_end_pos":[593,45]},{"full_name":"pow_mul","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[605,31],"def_end_pos":[605,38]},{"full_name":"pow_mul'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[610,6],"def_end_pos":[610,14]}]}]}
{"url":"Mathlib/Topology/Order/DenselyOrdered.lean","commit":"","full_name":"Dense.exists_countable_dense_subset_no_bot_top","start":[344,0],"end":[356,13],"file_path":"Mathlib/Topology/Order/DenselyOrdered.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\n⊢ ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t","state_after":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\n⊢ ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t","tactic":"rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩","premises":[{"full_name":"Dense.exists_countable_dense_subset","def_path":"Mathlib/Topology/Bases.lean","def_pos":[602,8],"def_end_pos":[602,43]}]},{"state_before":"case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\n⊢ ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t","state_after":"case intro.intro.intro.refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\n⊢ t \\ ({x | IsBot x} ∪ {x | IsTop x}) ⊆ s\n\ncase intro.intro.intro.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\n⊢ (t \\ ({x | IsBot x} ∪ {x | IsTop x})).Countable\n\ncase intro.intro.intro.refine_3\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\n⊢ Dense (t \\ ({x | IsBot x} ∪ {x | IsTop x}))\n\ncase intro.intro.intro.refine_4\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\nx : α\nhx : IsBot x\n⊢ x ∉ t \\ ({x | IsBot x} ∪ {x | IsTop x})\n\ncase intro.intro.intro.refine_5\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : t.Countable\nhtd : Dense t\nx : α\nhx : IsTop x\n⊢ x ∉ t \\ ({x | IsBot x} ∪ {x | IsTop x})","tactic":"refine ⟨t \\ ({ x | IsBot x } ∪ { x | IsTop x }), ?_, ?_, ?_, fun x hx => ?_, fun x hx => ?_⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"IsBot","def_path":"Mathlib/Order/Max.lean","def_pos":[167,4],"def_end_pos":[167,9]},{"full_name":"IsTop","def_path":"Mathlib/Order/Max.lean","def_pos":[174,4],"def_end_pos":[174,9]},{"full_name":"SDiff.sdiff","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[415,2],"def_end_pos":[415,7]},{"full_name":"Union.union","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[402,2],"def_end_pos":[402,7]},{"full_name":"setOf","def_path":"Mathlib/Init/Set.lean","def_pos":[56,4],"def_end_pos":[56,9]}]}]}
{"url":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","commit":"","full_name":"Polynomial.eval_mul_X_sub_C","start":[504,0],"end":[526,23],"file_path":"Mathlib/Algebra/Polynomial/AlgebraMap.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\n⊢ eval r (p * (X - C r)) = 0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\n⊢ ((p * (X - C r)).sum fun e a => a * r ^ e) = 0","tactic":"simp only [eval, eval₂_eq_sum, RingHom.id_apply]","premises":[{"full_name":"Polynomial.eval","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[266,4],"def_end_pos":[266,8]},{"full_name":"Polynomial.eval₂_eq_sum","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[44,8],"def_end_pos":[44,20]},{"full_name":"RingHom.id_apply","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[549,8],"def_end_pos":[549,16]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\n⊢ ((p * (X - C r)).sum fun e a => a * r ^ e) = 0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ((p * (X - C r)).sum fun e a => a * r ^ e) = 0","tactic":"have bound :=\n    calc\n      (p * (X - C r)).natDegree ≤ p.natDegree + (X - C r).natDegree := natDegree_mul_le\n      _ ≤ p.natDegree + 1 := add_le_add_left (natDegree_X_sub_C_le _) _\n      _ < p.natDegree + 2 := lt_add_one _","premises":[{"full_name":"Polynomial.C","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[426,4],"def_end_pos":[426,5]},{"full_name":"Polynomial.X","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[474,4],"def_end_pos":[474,5]},{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"Polynomial.natDegree_X_sub_C_le","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[1254,8],"def_end_pos":[1254,28]},{"full_name":"Polynomial.natDegree_mul_le","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[936,8],"def_end_pos":[936,24]},{"full_name":"add_le_add_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[61,31],"def_end_pos":[61,46]},{"full_name":"lt_add_one","def_path":"Mathlib/Algebra/Order/Monoid/NatCast.lean","def_pos":[18,6],"def_end_pos":[18,16]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ((p * (X - C r)).sum fun e a => a * r ^ e) = 0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ a ∈ range (p.natDegree + 2), (p * (X - C r)).coeff a * r ^ a = 0\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∀ (n : ℕ), 0 * r ^ n = 0","tactic":"rw [sum_over_range' _ _ (p.natDegree + 2) bound]","premises":[{"full_name":"Polynomial.natDegree","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[61,4],"def_end_pos":[61,13]},{"full_name":"Polynomial.sum_over_range'","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[336,8],"def_end_pos":[336,23]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ a ∈ range (p.natDegree + 2), (p * (X - C r)).coeff a * r ^ a = 0\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∀ (n : ℕ), 0 * r ^ n = 0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∀ (n : ℕ), 0 * r ^ n = 0\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ a ∈ range (p.natDegree + 2), (p * (X - C r)).coeff a * r ^ a = 0","tactic":"swap","premises":[]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ a ∈ range (p.natDegree + 2), (p * (X - C r)).coeff a * r ^ a = 0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ k ∈ range (p.natDegree + 1), (p * (X - C r)).coeff (k + 1) * r ^ (k + 1) + (p * (X - C r)).coeff 0 * r ^ 0 = 0","tactic":"rw [sum_range_succ']","premises":[{"full_name":"Finset.sum_range_succ'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1301,2],"def_end_pos":[1301,13]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ k ∈ range (p.natDegree + 1), (p * (X - C r)).coeff (k + 1) * r ^ (k + 1) + (p * (X - C r)).coeff 0 * r ^ 0 = 0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ k ∈ range (p.natDegree + 1), (p.coeff k * r ^ (k + 1) - p.coeff (k + 1) * r ^ (k + 1 + 1)) +\n      (p * (X - C r)).coeff 0 * r ^ 0 =\n    0","tactic":"conv_lhs =>\n    congr\n    arg 2\n    simp [coeff_mul_X_sub_C, sub_mul, mul_assoc, ← pow_succ']","premises":[{"full_name":"Polynomial.coeff_mul_X_sub_C","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[448,8],"def_end_pos":[448,25]},{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]},{"full_name":"pow_succ'","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[573,33],"def_end_pos":[573,42]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ ∑ k ∈ range (p.natDegree + 1), (p.coeff k * r ^ (k + 1) - p.coeff (k + 1) * r ^ (k + 1 + 1)) +\n      (p * (X - C r)).coeff 0 * r ^ 0 =\n    0","state_after":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ p.coeff 0 * r ^ (0 + 1) - p.coeff (p.natDegree + 1) * r ^ (p.natDegree + 1 + 1) + (p * (X - C r)).coeff 0 * r ^ 0 = 0","tactic":"rw [sum_range_sub']","premises":[{"full_name":"Finset.sum_range_sub'","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1440,2],"def_end_pos":[1440,13]}]},{"state_before":"R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB : Type u_2\na b : R\nn : ℕ\ninst✝ : Ring R\np : R[X]\nr : R\nbound : (p * (X - C r)).natDegree < p.natDegree + 2\n⊢ p.coeff 0 * r ^ (0 + 1) - p.coeff (p.natDegree + 1) * r ^ (p.natDegree + 1 + 1) + (p * (X - C r)).coeff 0 * r ^ 0 = 0","state_after":"no goals","tactic":"simp [coeff_monomial]","premises":[{"full_name":"Polynomial.coeff_monomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[574,8],"def_end_pos":[574,22]}]}]}
{"url":"Mathlib/Algebra/CubicDiscriminant.lean","commit":"","full_name":"Cubic.equiv_symm_apply_c","start":[236,0],"end":[249,95],"file_path":"Mathlib/Algebra/CubicDiscriminant.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP✝ Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nP : Cubic R\n⊢ (fun f => { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 })\n      ((fun P => ⟨P.toPoly, ⋯⟩) P) =\n    P","state_after":"no goals","tactic":"ext <;> simp only [Subtype.coe_mk, coeffs]","premises":[{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]},{"full_name":"_private.Mathlib.Algebra.CubicDiscriminant.0.Cubic.coeffs","def_path":"Mathlib/Algebra/CubicDiscriminant.lean","def_pos":[79,16],"def_end_pos":[79,22]}]},{"state_before":"R : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nf : { p // p.degree ≤ 3 }\n⊢ (fun P => ⟨P.toPoly, ⋯⟩)\n      ((fun f => { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }) f) =\n    f","state_after":"case a.a.succ.succ.succ.succ\nR : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nf : { p // p.degree ≤ 3 }\nn : ℕ\n⊢ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }.toPoly.coeff (4 + n) =\n    (↑f).coeff (4 + n)","tactic":"ext (_ | _ | _ | _ | n) <;> simp only [Nat.zero_eq, Nat.succ_eq_add_one] <;> ring_nf\n      <;> try simp only [coeffs]","premises":[{"full_name":"Nat.succ_eq_add_one","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[154,16],"def_end_pos":[154,31]},{"full_name":"Nat.zero_eq","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[106,16],"def_end_pos":[106,23]},{"full_name":"_private.Mathlib.Algebra.CubicDiscriminant.0.Cubic.coeffs","def_path":"Mathlib/Algebra/CubicDiscriminant.lean","def_pos":[79,16],"def_end_pos":[79,22]}]},{"state_before":"case a.a.succ.succ.succ.succ\nR : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nf : { p // p.degree ≤ 3 }\nn : ℕ\n⊢ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }.toPoly.coeff (4 + n) =\n    (↑f).coeff (4 + n)","state_after":"case a.a.succ.succ.succ.succ\nR : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nf : { p // p.degree ≤ 3 }\nn : ℕ\nh3 : 3 < 4 + n\n⊢ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }.toPoly.coeff (4 + n) =\n    (↑f).coeff (4 + n)","tactic":"have h3 : 3 < 4 + n := by linarith only","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]}]},{"state_before":"case a.a.succ.succ.succ.succ\nR : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nf : { p // p.degree ≤ 3 }\nn : ℕ\nh3 : 3 < 4 + n\n⊢ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }.toPoly.coeff (4 + n) =\n    (↑f).coeff (4 + n)","state_after":"no goals","tactic":"rw [coeff_eq_zero h3,\n      (degree_le_iff_coeff_zero (f : R[X]) 3).mp f.2 _ <| WithBot.coe_lt_coe.mpr (by exact h3)]","premises":[{"full_name":"Cubic.coeff_eq_zero","def_path":"Mathlib/Algebra/CubicDiscriminant.lean","def_pos":[89,8],"def_end_pos":[89,21]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Polynomial","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[60,10],"def_end_pos":[60,20]},{"full_name":"Polynomial.degree_le_iff_coeff_zero","def_path":"Mathlib/Algebra/Polynomial/Degree/Definitions.lean","def_pos":[1001,8],"def_end_pos":[1001,32]},{"full_name":"Subtype.property","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[590,2],"def_end_pos":[590,10]},{"full_name":"WithBot.coe_lt_coe","def_path":"Mathlib/Order/WithBot.lean","def_pos":[262,8],"def_end_pos":[262,18]}]}]}
{"url":"Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean","commit":"","full_name":"Mon_.limitConeIsLimit_lift_hom","start":[57,0],"end":[77,38],"file_path":"Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean","tactics":[{"state_before":"J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\n⊢ s.pt.mul ≫ limit.lift (F ⋙ forget C) ((forget C).mapCone s) =\n    MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n        (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n      (limitCone F).pt.mul","state_after":"J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\n⊢ s.pt.mul ≫ limit.lift (F ⋙ forget C) ((forget C).mapCone s) =\n    MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n        (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n      (limit F).mul","tactic":"dsimp","premises":[]},{"state_before":"J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\n⊢ s.pt.mul ≫ limit.lift (F ⋙ forget C) ((forget C).mapCone s) =\n    MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n        (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n      (limit F).mul","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nj✝ : J\n⊢ (s.pt.mul ≫ limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫ limit.π (F ⋙ forget C) j✝ =\n    (MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n          (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n        (limit F).mul) ≫\n      limit.π (F ⋙ forget C) j✝","tactic":"ext","premises":[]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nj✝ : J\n⊢ (s.pt.mul ≫ limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫ limit.π (F ⋙ forget C) j✝ =\n    (MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n          (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n        (limit F).mul) ≫\n      limit.π (F ⋙ forget C) j✝","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nj✝ : J\n⊢ MonoidalCategory.tensorHom (s.π.app j✝).hom (s.π.app j✝).hom ≫ (F.obj j✝).mul =\n    MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n        (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n      MonoidalCategory.tensorHom (limit.π (F ⋙ forget C) j✝) (limit.π (F ⋙ forget C) j✝) ≫\n        (MonFunctorCategoryEquivalence.inverseObj F).mul.app j✝","tactic":"simp only [Functor.comp_obj, forget_obj, Category.assoc, limit.lift_π, Functor.mapCone_pt,\n          Functor.mapCone_π_app, forget_map, Hom.mul_hom, limit_mul, Cones.postcompose_obj_pt,\n          Cones.postcompose_obj_π, NatTrans.comp_app, Functor.const_obj_obj, tensorObj_obj]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Functor.comp_obj","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[99,8],"def_end_pos":[99,11]},{"full_name":"CategoryTheory.Functor.const_obj_obj","def_path":"Mathlib/CategoryTheory/Functor/Const.lean","def_pos":[32,2],"def_end_pos":[32,7]},{"full_name":"CategoryTheory.Functor.mapCone_pt","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[662,2],"def_end_pos":[662,8]},{"full_name":"CategoryTheory.Functor.mapCone_π_app","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[662,2],"def_end_pos":[662,8]},{"full_name":"CategoryTheory.Limits.Cones.postcompose_obj_pt","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[335,2],"def_end_pos":[335,7]},{"full_name":"CategoryTheory.Limits.Cones.postcompose_obj_π","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[335,2],"def_end_pos":[335,7]},{"full_name":"CategoryTheory.Limits.limit.lift_π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[169,8],"def_end_pos":[169,20]},{"full_name":"CategoryTheory.Monoidal.tensorObj_obj","def_path":"Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean","def_pos":[97,8],"def_end_pos":[97,21]},{"full_name":"CategoryTheory.NatTrans.comp_app","def_path":"Mathlib/CategoryTheory/Functor/Category.lean","def_pos":[69,8],"def_end_pos":[69,16]},{"full_name":"Mon_.Hom.mul_hom","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[85,2],"def_end_pos":[85,9]},{"full_name":"Mon_.forget_map","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[126,2],"def_end_pos":[126,7]},{"full_name":"Mon_.forget_obj","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[126,2],"def_end_pos":[126,7]},{"full_name":"Mon_.limit_mul","def_path":"Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean","def_pos":[37,2],"def_end_pos":[37,8]}]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nj✝ : J\n⊢ MonoidalCategory.tensorHom (s.π.app j✝).hom (s.π.app j✝).hom ≫ (F.obj j✝).mul =\n    MonoidalCategory.tensorHom (limit.lift (F ⋙ forget C) ((forget C).mapCone s))\n        (limit.lift (F ⋙ forget C) ((forget C).mapCone s)) ≫\n      MonoidalCategory.tensorHom (limit.π (F ⋙ forget C) j✝) (limit.π (F ⋙ forget C) j✝) ≫\n        (MonFunctorCategoryEquivalence.inverseObj F).mul.app j✝","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nj✝ : J\n⊢ MonoidalCategory.tensorHom (s.π.app j✝).hom (s.π.app j✝).hom ≫ (F.obj j✝).mul =\n    MonoidalCategory.tensorHom (((forget C).mapCone s).π.app j✝) (((forget C).mapCone s).π.app j✝) ≫\n      (MonFunctorCategoryEquivalence.inverseObj F).mul.app j✝","tactic":"slice_rhs 1 2 => rw [← MonoidalCategory.tensor_comp, limit.lift_π]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.limit.lift_π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[169,8],"def_end_pos":[169,20]},{"full_name":"CategoryTheory.MonoidalCategory.tensor_comp","def_path":"Mathlib/CategoryTheory/Monoidal/Category.lean","def_pos":[166,2],"def_end_pos":[166,13]}]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nj✝ : J\n⊢ MonoidalCategory.tensorHom (s.π.app j✝).hom (s.π.app j✝).hom ≫ (F.obj j✝).mul =\n    MonoidalCategory.tensorHom (((forget C).mapCone s).π.app j✝) (((forget C).mapCone s).π.app j✝) ≫\n      (MonFunctorCategoryEquivalence.inverseObj F).mul.app j✝","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nh : J\n⊢ (fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s ≫\n      (limitCone F).π.app h =\n    s.π.app h","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nh : J\n⊢ ((fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s ≫\n        (limitCone F).π.app h).hom =\n    (s.π.app h).hom","tactic":"ext","premises":[]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nh : J\n⊢ ((fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s ≫\n        (limitCone F).π.app h).hom =\n    (s.π.app h).hom","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\n⊢ m = (fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\n⊢ m.hom = ((fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s).hom","tactic":"ext1","premises":[]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\n⊢ m.hom = ((fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s).hom","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ limit.π ((monFunctorCategoryEquivalence J C).inverse.obj F).X j =\n    ((fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s).hom ≫\n      limit.π ((monFunctorCategoryEquivalence J C).inverse.obj F).X j","tactic":"refine limit.hom_ext (fun j => ?_)","premises":[{"full_name":"CategoryTheory.Limits.limit.hom_ext","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[231,8],"def_end_pos":[231,21]}]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ limit.π ((monFunctorCategoryEquivalence J C).inverse.obj F).X j =\n    ((fun s => { hom := limit.lift (F ⋙ forget C) ((forget C).mapCone s), one_hom := ⋯, mul_hom := ⋯ }) s).hom ≫\n      limit.π ((monFunctorCategoryEquivalence J C).inverse.obj F).X j","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ limit.π (F ⋙ forget C) j = limit.lift (F ⋙ forget C) ((forget C).mapCone s) ≫ limit.π (F ⋙ forget C) j","tactic":"dsimp","premises":[]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ limit.π (F ⋙ forget C) j = limit.lift (F ⋙ forget C) ((forget C).mapCone s) ≫ limit.π (F ⋙ forget C) j","state_after":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ limit.π (F ⋙ forget C) j = (s.π.app j).hom","tactic":"simp only [Mon_.forget_map, limit.lift_π, Functor.mapCone_π_app]","premises":[{"full_name":"CategoryTheory.Functor.mapCone_π_app","def_path":"Mathlib/CategoryTheory/Limits/Cones.lean","def_pos":[662,2],"def_end_pos":[662,8]},{"full_name":"CategoryTheory.Limits.limit.lift_π","def_path":"Mathlib/CategoryTheory/Limits/HasLimits.lean","def_pos":[169,8],"def_end_pos":[169,20]},{"full_name":"Mon_.forget_map","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[126,2],"def_end_pos":[126,7]}]},{"state_before":"case w\nJ : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon_ C\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ limit.π (F ⋙ forget C) j = (s.π.app j).hom","state_after":"no goals","tactic":"exact congr_arg Mon_.Hom.hom (w j)","premises":[{"full_name":"Mon_.Hom.hom","def_path":"Mathlib/CategoryTheory/Monoidal/Mon_.lean","def_pos":[83,2],"def_end_pos":[83,5]}]}]}
{"url":"Mathlib/Algebra/Lie/TensorProduct.lean","commit":"","full_name":"LieModule.toModuleHom_apply","start":[171,0],"end":[174,88],"file_path":"Mathlib/Algebra/Lie/TensorProduct.lean","tactics":[{"state_before":"R : Type u\ninst✝⁶ : CommRing R\nL : Type v\nM : Type w\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx : L\nm : M\n⊢ (toModuleHom R L M) (x ⊗ₜ[R] m) = ⁅x, m⁆","state_after":"no goals","tactic":"simp only [toModuleHom, TensorProduct.LieModule.liftLie_apply, LieModuleHom.coe_mk,\n    LinearMap.coe_mk, LinearMap.coe_toAddHom, LieHom.coe_toLinearMap, toEnd_apply_apply]","premises":[{"full_name":"LieHom.coe_toLinearMap","def_path":"Mathlib/Algebra/Lie/Basic.lean","def_pos":[280,8],"def_end_pos":[280,23]},{"full_name":"LieModule.toEnd_apply_apply","def_path":"Mathlib/Algebra/Lie/OfAssociative.lean","def_pos":[177,2],"def_end_pos":[177,7]},{"full_name":"LieModule.toModuleHom","def_path":"Mathlib/Algebra/Lie/TensorProduct.lean","def_pos":[166,4],"def_end_pos":[166,15]},{"full_name":"LieModuleHom.coe_mk","def_path":"Mathlib/Algebra/Lie/Basic.lean","def_pos":[723,8],"def_end_pos":[723,14]},{"full_name":"LinearMap.coe_mk","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[253,8],"def_end_pos":[253,14]},{"full_name":"LinearMap.coe_toAddHom","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[227,8],"def_end_pos":[227,20]},{"full_name":"TensorProduct.LieModule.liftLie_apply","def_path":"Mathlib/Algebra/Lie/TensorProduct.lean","def_pos":[115,8],"def_end_pos":[115,21]}]}]}
{"url":"Mathlib/CategoryTheory/Action.lean","commit":"","full_name":"CategoryTheory.ActionCategory.cases'","start":[163,0],"end":[167,81],"file_path":"Mathlib/CategoryTheory/Action.lean","tactics":[{"state_before":"M : Type u_1\ninst✝³ : Monoid M\nX : Type u\ninst✝² : MulAction M X\nx : X\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na' b' : ActionCategory G X\nf : a' ⟶ b'\na b : X\ng : G\nha : a' = ⟨(), a⟩\nhb : b' = ⟨(), b⟩\nhg : a = g⁻¹ • b\n⊢ a' = ⟨(), g⁻¹ • b⟩","state_after":"no goals","tactic":"rw [ha, hg]","premises":[]},{"state_before":"M : Type u_1\ninst✝³ : Monoid M\nX : Type u\ninst✝² : MulAction M X\nx : X\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na' b' : ActionCategory G X\nf : a' ⟶ b'\na b : X\ng : G\nha : a' = ⟨(), a⟩\nhb : b' = ⟨(), b⟩\nhg : a = g⁻¹ • b\n⊢ ⟨(), b⟩ = b'","state_after":"no goals","tactic":"rw [hb]","premises":[]},{"state_before":"M : Type u_1\ninst✝³ : Monoid M\nX : Type u\ninst✝² : MulAction M X\nx : X\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na' b' : ActionCategory G X\nf : a' ⟶ b'\n⊢ ∃ a b g, ∃ (ha : a' = ⟨(), a⟩) (hb : b' = ⟨(), b⟩) (hg : a = g⁻¹ • b), f = eqToHom ⋯ ≫ homOfPair b g ≫ eqToHom ⋯","state_after":"M : Type u_1\ninst✝³ : Monoid M\nX : Type u\ninst✝² : MulAction M X\nx : X\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\n⊢ ∀ ⦃a' b' : ActionCategory G X⦄ (f : a' ⟶ b'),\n    ∃ a b g, ∃ (ha : a' = ⟨(), a⟩) (hb : b' = ⟨(), b⟩) (hg : a = g⁻¹ • b), f = eqToHom ⋯ ≫ homOfPair b g ≫ eqToHom ⋯","tactic":"revert a' b' f","premises":[]},{"state_before":"M : Type u_1\ninst✝³ : Monoid M\nX : Type u\ninst✝² : MulAction M X\nx : X\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\n⊢ ∀ ⦃a' b' : ActionCategory G X⦄ (f : a' ⟶ b'),\n    ∃ a b g, ∃ (ha : a' = ⟨(), a⟩) (hb : b' = ⟨(), b⟩) (hg : a = g⁻¹ • b), f = eqToHom ⋯ ≫ homOfPair b g ≫ eqToHom ⋯","state_after":"no goals","tactic":"exact ActionCategory.cases (fun t g => ⟨g⁻¹ • t, t, g, rfl, rfl, rfl, by simp⟩)","premises":[{"full_name":"CategoryTheory.ActionCategory.cases","def_path":"Mathlib/CategoryTheory/Action.lean","def_pos":[153,14],"def_end_pos":[153,19]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Basic.lean","commit":"","full_name":"MvPolynomial.coeff_monomial_mul'","start":[700,0],"end":[704,35],"file_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","tactics":[{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\n⊢ coeff m ((monomial s) r * p) = if s ≤ m then r * coeff (m - s) p else 0","state_after":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\n⊢ coeff m (p * (monomial s) r) = if s ≤ m then coeff (m - s) p * r else 0","tactic":"rw [mul_comm, mul_comm r]","premises":[{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]},{"state_before":"R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\n⊢ coeff m (p * (monomial s) r) = if s ≤ m then coeff (m - s) p * r else 0","state_after":"no goals","tactic":"exact coeff_mul_monomial' _ _ _ _","premises":[{"full_name":"MvPolynomial.coeff_mul_monomial'","def_path":"Mathlib/Algebra/MvPolynomial/Basic.lean","def_pos":[686,8],"def_end_pos":[686,27]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Basic.lean","commit":"","full_name":"Subgroup.disjoint_def","start":[2822,0],"end":[2824,97],"file_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","tactics":[{"state_before":"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nN : Type u_5\ninst✝ : Group N\nH₁ H₂ : Subgroup G\n⊢ H₁ ⊓ H₂ ≤ ⊥ ↔ ∀ {x : G}, x ∈ H₁ → x ∈ H₂ → x = 1","state_after":"no goals","tactic":"simp only [Disjoint, SetLike.le_def, mem_inf, mem_bot, and_imp]","premises":[{"full_name":"Disjoint","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[40,4],"def_end_pos":[40,12]},{"full_name":"SetLike.le_def","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[191,8],"def_end_pos":[191,14]},{"full_name":"Subgroup.mem_bot","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[656,8],"def_end_pos":[656,15]},{"full_name":"Subgroup.mem_inf","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[750,8],"def_end_pos":[750,15]},{"full_name":"and_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[115,16],"def_end_pos":[115,23]}]}]}
{"url":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","commit":"","full_name":"SimplicialObject.Split.congr_F","start":[359,0],"end":[359,92],"file_path":"Mathlib/AlgebraicTopology/SplitSimplicialObject.lean","tactics":[{"state_before":"C : Type u_1\ninst✝ : Category.{u_2, u_1} C\nS₁ S₂ : Split C\nΦ₁ Φ₂ : S₁ ⟶ S₂\nh : Φ₁ = Φ₂\n⊢ Φ₁.f = Φ₂.f","state_after":"no goals","tactic":"rw [h]","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean","commit":"","full_name":"le_or_lt_of_add_le_add","start":[82,0],"end":[87,47],"file_path":"Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂","state_after":"α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\n⊢ a₂ < a₁ ∧ b₂ ≤ b₁ → a₂ * b₂ < a₁ * b₁","tactic":"contrapose!","premises":[{"full_name":"Mathlib.Tactic.Contrapose.mtr","def_path":"Mathlib/Tactic/Contrapose.lean","def_pos":[24,6],"def_end_pos":[24,9]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\n⊢ a₂ < a₁ ∧ b₂ ≤ b₁ → a₂ * b₂ < a₁ * b₁","state_after":"no goals","tactic":"exact fun h => mul_lt_mul_of_lt_of_le h.1 h.2","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"mul_lt_mul_of_lt_of_le","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean","def_pos":[162,8],"def_end_pos":[162,30]}]}]}
{"url":"Mathlib/Order/Interval/Set/OrderEmbedding.lean","commit":"","full_name":"OrderEmbedding.preimage_uIcc","start":[42,0],"end":[44,33],"file_path":"Mathlib/Order/Interval/Set/OrderEmbedding.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Lattice β\ne : α ↪o β\nx y : α\n⊢ ⇑e ⁻¹' uIcc (e x) (e y) = uIcc x y","state_after":"no goals","tactic":"cases le_total x y <;> simp [*]","premises":[{"full_name":"le_total","def_path":"Mathlib/Order/Defs.lean","def_pos":[254,8],"def_end_pos":[254,16]}]}]}
{"url":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","commit":"","full_name":"RingHom.IsIntegralElem.of_mem_closure","start":[60,0],"end":[67,82],"file_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\n⊢ f.IsIntegralElem z","state_after":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis : Algebra R S := f.toAlgebra\n⊢ f.IsIntegralElem z","tactic":"letI : Algebra R S := f.toAlgebra","premises":[{"full_name":"Algebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[100,6],"def_end_pos":[100,13]},{"full_name":"RingHom.toAlgebra","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[198,4],"def_end_pos":[198,21]}]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis : Algebra R S := f.toAlgebra\n⊢ f.IsIntegralElem z","state_after":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis✝ : Algebra R S := f.toAlgebra\nthis : (Subalgebra.toSubmodule (Algebra.adjoin R {x}) * Subalgebra.toSubmodule (Algebra.adjoin R {y})).FG\n⊢ f.IsIntegralElem z","tactic":"have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy)","premises":[{"full_name":"IsIntegral.fg_adjoin_singleton","def_path":"Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean","def_pos":[79,8],"def_end_pos":[79,38]},{"full_name":"Submodule.FG.mul","def_path":"Mathlib/RingTheory/Finiteness.lean","def_pos":[432,8],"def_end_pos":[432,14]}]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis✝ : Algebra R S := f.toAlgebra\nthis : (Subalgebra.toSubmodule (Algebra.adjoin R {x}) * Subalgebra.toSubmodule (Algebra.adjoin R {y})).FG\n⊢ f.IsIntegralElem z","state_after":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis✝ : Algebra R S := f.toAlgebra\nthis : (Subalgebra.toSubmodule (Algebra.adjoin R {x, y})).FG\n⊢ f.IsIntegralElem z","tactic":"rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this","premises":[{"full_name":"Algebra.adjoin_union_coe_submodule","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[347,8],"def_end_pos":[347,34]},{"full_name":"Set.singleton_union","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[1056,8],"def_end_pos":[1056,23]}]},{"state_before":"R : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis✝ : Algebra R S := f.toAlgebra\nthis : (Subalgebra.toSubmodule (Algebra.adjoin R {x, y})).FG\n⊢ f.IsIntegralElem z","state_after":"no goals","tactic":"exact\n    IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z\n      (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono Set.subset_union_right hz)","premises":[{"full_name":"Algebra.adjoin","def_path":"Mathlib/Algebra/Algebra/Subalgebra/Basic.lean","def_pos":[618,4],"def_end_pos":[618,10]},{"full_name":"Algebra.mem_adjoin_iff","def_path":"Mathlib/RingTheory/Adjoin/Basic.lean","def_pos":[389,8],"def_end_pos":[389,22]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"IsIntegral.of_mem_of_fg","def_path":"Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean","def_pos":[55,8],"def_end_pos":[55,31]},{"full_name":"Set.subset_union_right","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[660,8],"def_end_pos":[660,26]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]},{"full_name":"Subring.closure_mono","def_path":"Mathlib/Algebra/Ring/Subring/Basic.lean","def_pos":[743,8],"def_end_pos":[743,20]}]}]}
{"url":"Mathlib/RingTheory/Polynomial/Basic.lean","commit":"","full_name":"Polynomial.ker_mapRingHom","start":[570,0],"end":[576,6],"file_path":"Mathlib/RingTheory/Polynomial/Basic.lean","tactics":[{"state_before":"R : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\n⊢ LinearMap.ker (mapRingHom f).toSemilinearMap = map C (RingHom.ker f)","state_after":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ x✝ ∈ LinearMap.ker (mapRingHom f).toSemilinearMap ↔ x✝ ∈ map C (RingHom.ker f)","tactic":"ext","premises":[]},{"state_before":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ x✝ ∈ LinearMap.ker (mapRingHom f).toSemilinearMap ↔ x✝ ∈ map C (RingHom.ker f)","state_after":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ (↑↑(mapRingHom f)).toFun x✝ = 0 ↔ x✝ ∈ map C (RingHom.ker f)","tactic":"simp only [LinearMap.mem_ker, RingHom.toSemilinearMap_apply, coe_mapRingHom]","premises":[{"full_name":"LinearMap.mem_ker","def_path":"Mathlib/Algebra/Module/Submodule/Ker.lean","def_pos":[62,8],"def_end_pos":[62,15]},{"full_name":"Polynomial.coe_mapRingHom","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[649,8],"def_end_pos":[649,22]},{"full_name":"RingHom.toSemilinearMap_apply","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[476,2],"def_end_pos":[476,7]}]},{"state_before":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ (↑↑(mapRingHom f)).toFun x✝ = 0 ↔ x✝ ∈ map C (RingHom.ker f)","state_after":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ (∀ (n : ℕ), ((↑↑(mapRingHom f)).toFun x✝).coeff n = coeff 0 n) ↔ ∀ (n : ℕ), x✝.coeff n ∈ RingHom.ker f","tactic":"rw [mem_map_C_iff, Polynomial.ext_iff]","premises":[{"full_name":"Ideal.mem_map_C_iff","def_path":"Mathlib/RingTheory/Polynomial/Basic.lean","def_pos":[547,8],"def_end_pos":[547,21]},{"full_name":"Polynomial.ext_iff","def_path":"Mathlib/Algebra/Polynomial/Basic.lean","def_pos":[687,8],"def_end_pos":[687,15]}]},{"state_before":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ (∀ (n : ℕ), ((↑↑(mapRingHom f)).toFun x✝).coeff n = coeff 0 n) ↔ ∀ (n : ℕ), x✝.coeff n ∈ RingHom.ker f","state_after":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ (∀ (n : ℕ), ((↑↑(mapRingHom f)).toFun x✝).coeff n = coeff 0 n) ↔ ∀ (n : ℕ), f (x✝.coeff n) = 0","tactic":"simp_rw [RingHom.mem_ker f]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"RingHom.mem_ker","def_path":"Mathlib/RingTheory/Ideal/Maps.lean","def_pos":[524,8],"def_end_pos":[524,15]}]},{"state_before":"case h\nR : Type u\nS : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nf : R →+* S\nx✝ : R[X]\n⊢ (∀ (n : ℕ), ((↑↑(mapRingHom f)).toFun x✝).coeff n = coeff 0 n) ↔ ∀ (n : ℕ), f (x✝.coeff n) = 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Algebra/Polynomial/Eval.lean","commit":"","full_name":"Polynomial.eval₂_at_one","start":[287,0],"end":[290,6],"file_path":"Mathlib/Algebra/Polynomial/Eval.lean","tactics":[{"state_before":"R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\n⊢ eval₂ f 1 p = f (eval 1 p)","state_after":"case h.e'_2.h.e'_6\nR : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\n⊢ 1 = f 1","tactic":"convert eval₂_at_apply (p := p) f 1","premises":[{"full_name":"Polynomial.eval₂_at_apply","def_path":"Mathlib/Algebra/Polynomial/Eval.lean","def_pos":[282,8],"def_end_pos":[282,22]}]},{"state_before":"case h.e'_2.h.e'_6\nR : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\n⊢ 1 = f 1","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Data/Finset/NoncommProd.lean","commit":"","full_name":"Finset.noncommSum_eq_sum","start":[356,0],"end":[361,17],"file_path":"Mathlib/Data/Finset/NoncommProd.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\n⊢ s.noncommProd f ⋯ = s.prod f","state_after":"case h₁\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\nf : α → β\n⊢ ∅.noncommProd f ⋯ = ∅.prod f\n\ncase h₂\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\nf : α → β\na : α\ns : Finset α\nha : a ∉ s\nIH : s.noncommProd f ⋯ = s.prod f\n⊢ (cons a s ha).noncommProd f ⋯ = (cons a s ha).prod f","tactic":"induction' s using Finset.cons_induction_on with a s ha IH","premises":[{"full_name":"Finset.cons_induction_on","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[1061,8],"def_end_pos":[1061,25]}]}]}
{"url":"Mathlib/Analysis/Calculus/Deriv/Inv.lean","commit":"","full_name":"hasStrictDerivAt_inv","start":[46,0],"end":[59,90],"file_path":"Mathlib/Analysis/Calculus/Deriv/Inv.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ (fun p => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p => (p.1 - p.2) * 1","tactic":"suffices\n    (fun p : 𝕜 × 𝕜 => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p =>\n      (p.1 - p.2) * 1 by\n    refine this.congr' ?_ (eventually_of_forall fun _ => mul_one _)\n    refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds ⟨hx, hx⟩) ?_\n    rintro ⟨y, z⟩ ⟨hy, hz⟩\n    simp only [mem_setOf_eq] at hy hz\n    -- hy : y ≠ 0, hz : z ≠ 0\n    field_simp [hx, hy, hz]\n    ring","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Asymptotics.IsLittleO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[156,16],"def_end_pos":[156,25]},{"full_name":"Asymptotics.IsLittleO.congr'","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[332,8],"def_end_pos":[332,24]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]},{"full_name":"Filter.eventually_of_forall","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[979,8],"def_end_pos":[979,28]},{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"IsOpen.prod","def_path":"Mathlib/Topology/Constructions.lean","def_pos":[472,8],"def_end_pos":[472,19]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"isOpen_ne","def_path":"Mathlib/Topology/Separation.lean","def_pos":[530,8],"def_end_pos":[530,17]},{"full_name":"mul_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[413,8],"def_end_pos":[413,15]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ (fun p => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p => (p.1 - p.2) * 1","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.1 * p.2)⁻¹) (𝓝 (x, x)) (𝓝 0)","tactic":"refine (isBigO_refl (fun p : 𝕜 × 𝕜 => p.1 - p.2) _).mul_isLittleO ((isLittleO_one_iff 𝕜).2 ?_)","premises":[{"full_name":"Asymptotics.IsBigO.mul_isLittleO","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1341,8],"def_end_pos":[1341,28]},{"full_name":"Asymptotics.isBigO_refl","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[515,8],"def_end_pos":[515,19]},{"full_name":"Asymptotics.isLittleO_one_iff","def_path":"Mathlib/Analysis/Asymptotics/Asymptotics.lean","def_pos":[1114,8],"def_end_pos":[1114,25]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Prod","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[479,10],"def_end_pos":[479,14]},{"full_name":"Prod.fst","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[483,2],"def_end_pos":[483,5]},{"full_name":"Prod.snd","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[485,2],"def_end_pos":[485,5]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.1 * p.2)⁻¹) (𝓝 (x, x)) (𝓝 0)","state_after":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.1 * p.2)⁻¹) (𝓝 (x, x)) (𝓝 ((x * x)⁻¹ - (x * x)⁻¹))","tactic":"rw [← sub_self (x * x)⁻¹]","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"sub_self","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[695,29],"def_end_pos":[695,37]}]},{"state_before":"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhx : x ≠ 0\n⊢ Tendsto (fun p => (x * x)⁻¹ - (p.1 * p.2)⁻¹) (𝓝 (x, x)) (𝓝 ((x * x)⁻¹ - (x * x)⁻¹))","state_after":"no goals","tactic":"exact tendsto_const_nhds.sub ((continuous_mul.tendsto (x, x)).inv₀ <| mul_ne_zero hx hx)","premises":[{"full_name":"Continuous.tendsto","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1413,8],"def_end_pos":[1413,26]},{"full_name":"Filter.Tendsto.inv₀","def_path":"Mathlib/Topology/Algebra/GroupWithZero.lean","def_pos":[104,8],"def_end_pos":[104,27]},{"full_name":"Filter.Tendsto.sub","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[927,14],"def_end_pos":[927,17]},{"full_name":"Prod.mk","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[481,2],"def_end_pos":[481,4]},{"full_name":"continuous_mul","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[64,8],"def_end_pos":[64,22]},{"full_name":"mul_ne_zero","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[80,8],"def_end_pos":[80,19]},{"full_name":"tendsto_const_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[844,8],"def_end_pos":[844,26]}]}]}
{"url":"Mathlib/FieldTheory/Finite/Polynomial.lean","commit":"","full_name":"MvPolynomial.range_evalᵢ","start":[217,0],"end":[219,34],"file_path":"Mathlib/FieldTheory/Finite/Polynomial.lean","tactics":[{"state_before":"σ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\n⊢ range (evalᵢ σ K) = ⊤","state_after":"σ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\n⊢ Submodule.map (evalₗ K σ) (restrictDegree σ K (Fintype.card K - 1)) = ⊤","tactic":"rw [evalᵢ, LinearMap.range_comp, range_subtype]","premises":[{"full_name":"LinearMap.range_comp","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[81,8],"def_end_pos":[81,18]},{"full_name":"MvPolynomial.evalᵢ","def_path":"Mathlib/FieldTheory/Finite/Polynomial.lean","def_pos":[173,4],"def_end_pos":[173,9]},{"full_name":"Submodule.range_subtype","def_path":"Mathlib/Algebra/Module/Submodule/Range.lean","def_pos":[262,8],"def_end_pos":[262,21]}]},{"state_before":"σ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\n⊢ Submodule.map (evalₗ K σ) (restrictDegree σ K (Fintype.card K - 1)) = ⊤","state_after":"no goals","tactic":"exact map_restrict_dom_evalₗ K σ","premises":[{"full_name":"MvPolynomial.map_restrict_dom_evalₗ","def_path":"Mathlib/FieldTheory/Finite/Polynomial.lean","def_pos":[126,8],"def_end_pos":[126,30]}]}]}
{"url":"Mathlib/RingTheory/Discriminant.lean","commit":"","full_name":"Algebra.discr_eq_discr_of_algEquiv","start":[72,0],"end":[76,96],"file_path":"Mathlib/RingTheory/Discriminant.lean","tactics":[{"state_before":"A : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\n⊢ discr A b = discr A (⇑f ∘ b)","state_after":"A : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\n⊢ (traceMatrix A b).det = discr A (⇑f ∘ b)","tactic":"rw [discr_def]","premises":[{"full_name":"Algebra.discr_def","def_path":"Mathlib/RingTheory/Discriminant.lean","def_pos":[69,8],"def_end_pos":[69,17]}]},{"state_before":"A : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\n⊢ (traceMatrix A b).det = discr A (⇑f ∘ b)","state_after":"case e_M\nA : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\n⊢ traceMatrix A b = traceMatrix A (⇑f ∘ b)","tactic":"congr","premises":[]},{"state_before":"case e_M\nA : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\n⊢ traceMatrix A b = traceMatrix A (⇑f ∘ b)","state_after":"case e_M.a\nA : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\ni✝ j✝ : ι\n⊢ traceMatrix A b i✝ j✝ = traceMatrix A (⇑f ∘ b) i✝ j✝","tactic":"ext","premises":[]},{"state_before":"case e_M.a\nA : Type u\nB : Type v\nC : Type z\nι : Type w\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Fintype ι\nb : ι → B\nf : B ≃ₐ[A] C\ni✝ j✝ : ι\n⊢ traceMatrix A b i✝ j✝ = traceMatrix A (⇑f ∘ b) i✝ j✝","state_after":"no goals","tactic":"simp_rw [traceMatrix_apply, traceForm_apply, Function.comp, ← map_mul f, trace_eq_of_algEquiv]","premises":[{"full_name":"Algebra.traceForm_apply","def_path":"Mathlib/RingTheory/Trace/Defs.lean","def_pos":[163,8],"def_end_pos":[163,23]},{"full_name":"Algebra.traceMatrix_apply","def_path":"Mathlib/RingTheory/Trace/Basic.lean","def_pos":[290,8],"def_end_pos":[290,25]},{"full_name":"Algebra.trace_eq_of_algEquiv","def_path":"Mathlib/RingTheory/Trace/Basic.lean","def_pos":[160,6],"def_end_pos":[160,34]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"map_mul","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[281,8],"def_end_pos":[281,15]}]}]}
{"url":"Mathlib/Topology/Order/UpperLowerSetTopology.lean","commit":"","full_name":"Topology.IsUpperSet.upperSet_le_upper","start":[279,0],"end":[282,39],"file_path":"Mathlib/Topology/Order/UpperLowerSetTopology.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Preorder α\ninst✝² : Preorder β\nt₁ t₂ : TopologicalSpace α\ninst✝¹ : Topology.IsUpperSet α\ninst✝ : IsUpper α\ns : Set α\nhs : IsOpen s\n⊢ IsOpen s","state_after":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Preorder α\ninst✝² : Preorder β\nt₁ t₂ : TopologicalSpace α\ninst✝¹ : Topology.IsUpperSet α\ninst✝ : IsUpper α\ns : Set α\nhs : IsOpen s\n⊢ IsUpperSet s","tactic":"rw [@isOpen_iff_isUpperSet α _ t₁]","premises":[{"full_name":"Topology.IsUpperSet.isOpen_iff_isUpperSet","def_path":"Mathlib/Topology/Order/UpperLowerSetTopology.lean","def_pos":[225,6],"def_end_pos":[225,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Preorder α\ninst✝² : Preorder β\nt₁ t₂ : TopologicalSpace α\ninst✝¹ : Topology.IsUpperSet α\ninst✝ : IsUpper α\ns : Set α\nhs : IsOpen s\n⊢ IsUpperSet s","state_after":"no goals","tactic":"exact IsUpper.isUpperSet_of_isOpen hs","premises":[{"full_name":"Topology.IsUpper.isUpperSet_of_isOpen","def_path":"Mathlib/Topology/Order/LowerUpperTopology.lean","def_pos":[324,8],"def_end_pos":[324,28]}]}]}
{"url":"Mathlib/GroupTheory/HNNExtension.lean","commit":"","full_name":"HNNExtension.NormalWord.unitsSMulEquiv_symm_apply","start":[450,0],"end":[456,64],"file_path":"Mathlib/GroupTheory/HNNExtension.lean","tactics":[{"state_before":"G : Type u_1\ninst✝³ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝² : Group H\nM : Type u_3\ninst✝¹ : Monoid M\nd : TransversalPair G A B\ninst✝ : DecidableEq G\nx✝ : NormalWord d\n⊢ unitsSMul φ (-1) (unitsSMul φ 1 x✝) = x✝","state_after":"no goals","tactic":"rw [unitsSMul_neg]","premises":[{"full_name":"HNNExtension.NormalWord.unitsSMul_neg","def_path":"Mathlib/GroupTheory/HNNExtension.lean","def_pos":[412,8],"def_end_pos":[412,21]}]},{"state_before":"G : Type u_1\ninst✝³ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝² : Group H\nM : Type u_3\ninst✝¹ : Monoid M\nd : TransversalPair G A B\ninst✝ : DecidableEq G\nw : NormalWord d\n⊢ unitsSMul φ 1 (unitsSMul φ (-1) w) = w","state_after":"case h.e'_2.h.e'_7\nG : Type u_1\ninst✝³ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝² : Group H\nM : Type u_3\ninst✝¹ : Monoid M\nd : TransversalPair G A B\ninst✝ : DecidableEq G\nw : NormalWord d\n⊢ 1 = - -1","tactic":"convert unitsSMul_neg _ _ w","premises":[{"full_name":"HNNExtension.NormalWord.unitsSMul_neg","def_path":"Mathlib/GroupTheory/HNNExtension.lean","def_pos":[412,8],"def_end_pos":[412,21]}]},{"state_before":"case h.e'_2.h.e'_7\nG : Type u_1\ninst✝³ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝² : Group H\nM : Type u_3\ninst✝¹ : Monoid M\nd : TransversalPair G A B\ninst✝ : DecidableEq G\nw : NormalWord d\n⊢ 1 = - -1","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","commit":"","full_name":"LinearMap.rTensor_comp_map","start":[1241,0],"end":[1244,60],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type u_2\ninst✝¹⁵ : Monoid R'\nR'' : Type u_3\ninst✝¹⁴ : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : AddCommMonoid T\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module R P\ninst✝⁴ : Module R Q\ninst✝³ : Module R S\ninst✝² : Module R T\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\ng✝ : P →ₗ[R] Q\nf✝ : N →ₗ[R] P\nf' : P →ₗ[R] S\nf : M →ₗ[R] P\ng : N →ₗ[R] Q\n⊢ rTensor Q f' ∘ₗ map f g = map (f' ∘ₗ f) g","state_after":"no goals","tactic":"simp only [lTensor, rTensor, ← map_comp, id_comp, comp_id]","premises":[{"full_name":"LinearMap.comp_id","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[512,8],"def_end_pos":[512,15]},{"full_name":"LinearMap.id_comp","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[516,8],"def_end_pos":[516,15]},{"full_name":"LinearMap.lTensor","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1055,4],"def_end_pos":[1055,11]},{"full_name":"LinearMap.rTensor","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1060,4],"def_end_pos":[1060,11]},{"full_name":"TensorProduct.map_comp","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[807,8],"def_end_pos":[807,16]}]}]}
{"url":"Mathlib/Data/Fin/Tuple/Basic.lean","commit":"","full_name":"Fin.find_spec","start":[909,0],"end":[925,77],"file_path":"Mathlib/Data/Fin/Tuple/Basic.lean","tactics":[{"state_before":"m n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi : i ∈ find p\n⊢ p i","state_after":"m n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n  i ∈\n    Option.casesOn (find fun i => p (i.castLT ⋯)) (if x : p (last n) then some (last n) else none) fun i =>\n      some (i.castLT ⋯)\n⊢ p i","tactic":"rw [find] at hi","premises":[{"full_name":"Fin.find","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[901,4],"def_end_pos":[901,8]}]},{"state_before":"m n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n  i ∈\n    Option.casesOn (find fun i => p (i.castLT ⋯)) (if x : p (last n) then some (last n) else none) fun i =>\n      some (i.castLT ⋯)\n⊢ p i","state_after":"case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n  i ∈\n    Option.casesOn (find fun i => p (i.castLT ⋯)) (if x : p (last n) then some (last n) else none) fun i =>\n      some (i.castLT ⋯)\nh : (find fun i => p (i.castLT ⋯)) = none\n⊢ p i\n\ncase some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n  i ∈\n    Option.casesOn (find fun i => p (i.castLT ⋯)) (if x : p (last n) then some (last n) else none) fun i =>\n      some (i.castLT ⋯)\nj : Fin n\nh : (find fun i => p (i.castLT ⋯)) = some j\n⊢ p i","tactic":"cases' h : find fun i : Fin n ↦ p (i.castLT (Nat.lt_succ_of_lt i.2)) with j","premises":[{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Fin.castLT","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Basic.lean","def_pos":[168,14],"def_end_pos":[168,20]},{"full_name":"Fin.find","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[901,4],"def_end_pos":[901,8]},{"full_name":"Fin.isLt","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1838,2],"def_end_pos":[1838,6]},{"full_name":"Nat.lt_succ_of_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[622,8],"def_end_pos":[622,21]}]}]}
{"url":"Mathlib/CategoryTheory/Shift/Basic.lean","commit":"","full_name":"CategoryTheory.shiftFunctorZero_hom_app_shift","start":[553,0],"end":[558,94],"file_path":"Mathlib/CategoryTheory/Shift/Basic.lean","tactics":[{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\nn : A\n⊢ (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =\n    (shiftFunctorComm C n 0).hom.app X ≫ (shiftFunctor C n).map ((shiftFunctorZero C A).hom.app X)","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\nn : A\n⊢ (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =\n    ((shiftFunctorAdd' C n 0 n ⋯).symm ≪≫ shiftFunctorAdd' C 0 n n ⋯).hom.app X ≫ (shiftFunctorAdd' C 0 n n ⋯).inv.app X","tactic":"rw [← shiftFunctorAdd'_zero_add_inv_app n X, shiftFunctorComm_eq C n 0 n (add_zero n)]","premises":[{"full_name":"CategoryTheory.shiftFunctorAdd'_zero_add_inv_app","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[282,6],"def_end_pos":[282,39]},{"full_name":"CategoryTheory.shiftFunctorComm_eq","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[508,6],"def_end_pos":[508,25]},{"full_name":"add_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[412,2],"def_end_pos":[412,13]}]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\nn : A\n⊢ (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =\n    ((shiftFunctorAdd' C n 0 n ⋯).symm ≪≫ shiftFunctorAdd' C 0 n n ⋯).hom.app X ≫ (shiftFunctorAdd' C 0 n n ⋯).inv.app X","state_after":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\nn : A\n⊢ (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =\n    ((shiftFunctorAdd' C n 0 n ⋯).inv.app X ≫ (shiftFunctorAdd' C 0 n n ⋯).hom.app X) ≫\n      (shiftFunctorAdd' C 0 n n ⋯).inv.app X","tactic":"dsimp","premises":[]},{"state_before":"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\nn : A\n⊢ (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =\n    ((shiftFunctorAdd' C n 0 n ⋯).inv.app X ≫ (shiftFunctorAdd' C 0 n n ⋯).hom.app X) ≫\n      (shiftFunctorAdd' C 0 n n ⋯).inv.app X","state_after":"no goals","tactic":"rw [Category.assoc, Iso.hom_inv_id_app, Category.comp_id, shiftFunctorAdd'_add_zero_inv_app]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Iso.hom_inv_id_app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[59,8],"def_end_pos":[59,22]},{"full_name":"CategoryTheory.shiftFunctorAdd'_add_zero_inv_app","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[300,6],"def_end_pos":[300,39]}]}]}
{"url":"Mathlib/Order/MinMax.lean","commit":"","full_name":"min_cases","start":[133,0],"end":[141,62],"file_path":"Mathlib/Order/MinMax.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\ns : Set α\na✝ b✝ c d a b : α\n⊢ min a b = a ∧ a ≤ b ∨ min a b = b ∧ b < a","state_after":"case pos\nα : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\ns : Set α\na✝ b✝ c d a b : α\nh : a ≤ b\n⊢ min a b = a ∧ a ≤ b ∨ min a b = b ∧ b < a\n\ncase neg\nα : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\ns : Set α\na✝ b✝ c d a b : α\nh : ¬a ≤ b\n⊢ min a b = a ∧ a ≤ b ∨ min a b = b ∧ b < a","tactic":"by_cases h : a ≤ b","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]}]}
{"url":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","commit":"","full_name":"ContDiffWithinAt.eventually","start":[625,0],"end":[631,50],"file_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","tactics":[{"state_before":"𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑n) f s y","state_after":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nleft✝ : u ⊆ insert x s\nhd : ContDiffOn 𝕜 (↑n) f u\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑n) f s y","tactic":"rcases h.contDiffOn le_rfl with ⟨u, hu, _, hd⟩","premises":[{"full_name":"ContDiffWithinAt.contDiffOn","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[620,8],"def_end_pos":[620,35]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nleft✝ : u ⊆ insert x s\nhd : ContDiffOn 𝕜 (↑n) f u\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑n) f s y","state_after":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nleft✝ : u ⊆ insert x s\nhd : ContDiffOn 𝕜 (↑n) f u\nthis : ∀ᶠ (y : E) in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑n) f s y","tactic":"have : ∀ᶠ y : E in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u :=\n    (eventually_nhdsWithin_nhdsWithin.2 hu).and hu","premises":[{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Eventually.and","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[973,18],"def_end_pos":[973,32]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Insert.insert","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[458,2],"def_end_pos":[458,8]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"eventually_nhdsWithin_nhdsWithin","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[56,8],"def_end_pos":[56,40]},{"full_name":"nhdsWithin","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[121,4],"def_end_pos":[121,14]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nleft✝ : u ⊆ insert x s\nhd : ContDiffOn 𝕜 (↑n) f u\nthis : ∀ᶠ (y : E) in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑n) f s y","state_after":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nleft✝ : u ⊆ insert x s\nhd : ContDiffOn 𝕜 (↑n) f u\nthis : ∀ᶠ (y : E) in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u\ny : E\nhy : u ∈ 𝓝[insert x s] y ∧ y ∈ u\n⊢ u ∈ 𝓝[s] y","tactic":"refine this.mono fun y hy => (hd y hy.2).mono_of_mem ?_","premises":[{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"ContDiffWithinAt.mono_of_mem","def_path":"Mathlib/Analysis/Calculus/ContDiff/Defs.lean","def_pos":[460,8],"def_end_pos":[460,36]},{"full_name":"Filter.Eventually.mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1002,8],"def_end_pos":[1002,23]}]},{"state_before":"case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nleft✝ : u ⊆ insert x s\nhd : ContDiffOn 𝕜 (↑n) f u\nthis : ∀ᶠ (y : E) in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u\ny : E\nhy : u ∈ 𝓝[insert x s] y ∧ y ∈ u\n⊢ u ∈ 𝓝[s] y","state_after":"no goals","tactic":"exact nhdsWithin_mono y (subset_insert _ _) hy.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"Set.subset_insert","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[877,8],"def_end_pos":[877,21]},{"full_name":"nhdsWithin_mono","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1091,8],"def_end_pos":[1091,23]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/Periodic.lean","commit":"","full_name":"isAddFundamentalDomain_Ioc'","start":[44,0],"end":[50,51],"file_path":"Mathlib/MeasureTheory/Integral/Periodic.lean","tactics":[{"state_before":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\n⊢ IsAddFundamentalDomain (↥(zmultiples T).op) (Ioc t (t + T)) μ","state_after":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\nx : ℝ\n⊢ ∃! g, g +ᵥ x ∈ Ioc t (t + T)","tactic":"refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_","premises":[{"full_name":"MeasurableSet.nullMeasurableSet","def_path":"Mathlib/MeasureTheory/Measure/NullMeasurable.lean","def_pos":[92,8],"def_end_pos":[92,46]},{"full_name":"MeasureTheory.IsAddFundamentalDomain.mk'","def_path":"Mathlib/MeasureTheory/Group/FundamentalDomain.lean","def_pos":[82,2],"def_end_pos":[82,13]},{"full_name":"measurableSet_Ioc","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[178,8],"def_end_pos":[178,25]}]},{"state_before":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\nx : ℝ\n⊢ ∃! g, g +ᵥ x ∈ Ioc t (t + T)","state_after":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\nx : ℝ\nthis : Bijective (codRestrict (fun n => n • T) ↑(zmultiples T) ⋯)\n⊢ ∃! g, g +ᵥ x ∈ Ioc t (t + T)","tactic":"have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=\n    (Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strictMono_left hT).injective).bijective","premises":[{"full_name":"AddSubgroup.zmultiples","def_path":"Mathlib/Algebra/Group/Subgroup/ZPowers.lean","def_pos":[77,4],"def_end_pos":[77,14]},{"full_name":"Equiv.bijective","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[179,18],"def_end_pos":[179,27]},{"full_name":"Equiv.ofInjective","def_path":"Mathlib/Logic/Equiv/Set.lean","def_pos":[529,18],"def_end_pos":[529,29]},{"full_name":"Function.Bijective","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[119,4],"def_end_pos":[119,13]},{"full_name":"Int","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[40,10],"def_end_pos":[40,13]},{"full_name":"Set.codRestrict","def_path":"Mathlib/Data/Set/Function.lean","def_pos":[129,4],"def_end_pos":[129,15]},{"full_name":"StrictMono.injective","def_path":"Mathlib/Order/Monotone/Basic.lean","def_pos":[756,8],"def_end_pos":[756,28]},{"full_name":"zsmul_strictMono_left","def_path":"Mathlib/Algebra/Order/Group/Basic.lean","def_pos":[30,14],"def_end_pos":[30,35]}]},{"state_before":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\nx : ℝ\nthis : Bijective (codRestrict (fun n => n • T) ↑(zmultiples T) ⋯)\n⊢ ∃! g, g +ᵥ x ∈ Ioc t (t + T)","state_after":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\nx : ℝ\nthis : Bijective (codRestrict (fun n => n • T) ↑(zmultiples T) ⋯)\n⊢ ∃! x_1, (⇑(zmultiples T).equivOp ∘ codRestrict (fun n => n • T) ↑(zmultiples T) ⋯) x_1 +ᵥ x ∈ Ioc t (t + T)","tactic":"refine (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2 ?_","premises":[{"full_name":"AddSubgroup.equivOp","def_path":"Mathlib/Algebra/Group/Subgroup/MulOpposite.lean","def_pos":[159,2],"def_end_pos":[159,13]},{"full_name":"Equiv.bijective","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[179,18],"def_end_pos":[179,27]},{"full_name":"Function.Bijective.comp","def_path":"Mathlib/Logic/Function/Defs.lean","def_pos":[122,8],"def_end_pos":[122,22]},{"full_name":"Function.Bijective.existsUnique_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[216,8],"def_end_pos":[216,34]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : autoParam (Measure ℝ) _auto✝\nx : ℝ\nthis : Bijective (codRestrict (fun n => n • T) ↑(zmultiples T) ⋯)\n⊢ ∃! x_1, (⇑(zmultiples T).equivOp ∘ codRestrict (fun n => n • T) ↑(zmultiples T) ⋯) x_1 +ᵥ x ∈ Ioc t (t + T)","state_after":"no goals","tactic":"simpa using existsUnique_add_zsmul_mem_Ioc hT x t","premises":[{"full_name":"existsUnique_add_zsmul_mem_Ioc","def_path":"Mathlib/Algebra/Order/Archimedean.lean","def_pos":[95,8],"def_end_pos":[95,38]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean","commit":"","full_name":"jacobiTheta_two_add","start":[30,0],"end":[31,73],"file_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean","tactics":[{"state_before":"τ : ℂ\n⊢ jacobiTheta (2 + τ) = jacobiTheta τ","state_after":"no goals","tactic":"simp_rw [jacobiTheta_eq_jacobiTheta₂, add_comm, jacobiTheta₂_add_right]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"add_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[293,2],"def_end_pos":[293,13]},{"full_name":"jacobiTheta_eq_jacobiTheta₂","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean","def_pos":[27,6],"def_end_pos":[27,33]},{"full_name":"jacobiTheta₂_add_right","def_path":"Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean","def_pos":[380,6],"def_end_pos":[380,28]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","commit":"","full_name":"Complex.Gamma_nat_eq_factorial","start":[326,0],"end":[330,70],"file_path":"Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean","tactics":[{"state_before":"n : ℕ\n⊢ Gamma (↑n + 1) = ↑n !","state_after":"case zero\n\n⊢ Gamma (↑0 + 1) = ↑0!\n\ncase succ\nn : ℕ\nhn : Gamma (↑n + 1) = ↑n !\n⊢ Gamma (↑(n + 1) + 1) = ↑(n + 1)!","tactic":"induction' n with n hn","premises":[]}]}
{"url":"Mathlib/Algebra/Order/Floor.lean","commit":"","full_name":"Int.fract_natCast","start":[872,0],"end":[873,68],"file_path":"Mathlib/Algebra/Order/Floor.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\nn : ℕ\n⊢ fract ↑n = 0","state_after":"no goals","tactic":"simp [fract]","premises":[{"full_name":"Int.fract","def_path":"Mathlib/Algebra/Order/Floor.lean","def_pos":[571,4],"def_end_pos":[571,9]}]}]}
{"url":"Mathlib/FieldTheory/RatFunc/AsPolynomial.lean","commit":"","full_name":"RatFunc.smul_eq_C_mul","start":[62,0],"end":[63,40],"file_path":"Mathlib/FieldTheory/RatFunc/AsPolynomial.lean","tactics":[{"state_before":"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nr : K\nx : RatFunc K\n⊢ r • x = C r * x","state_after":"no goals","tactic":"rw [Algebra.smul_def, algebraMap_eq_C]","premises":[{"full_name":"Algebra.smul_def","def_path":"Mathlib/Algebra/Algebra/Defs.lean","def_pos":[270,8],"def_end_pos":[270,16]},{"full_name":"RatFunc.algebraMap_eq_C","def_path":"Mathlib/FieldTheory/RatFunc/AsPolynomial.lean","def_pos":[51,8],"def_end_pos":[51,23]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean","commit":"","full_name":"HomotopyCategory.Pretriangulated.rotate_distinguished_triangle","start":[457,0],"end":[463,42],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{u_3, u_1} C\ninst✝⁶ : Category.{?u.447213, u_2} D\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasBinaryBiproducts C\ninst✝² : Preadditive D\ninst✝¹ : HasZeroObject D\ninst✝ : HasBinaryBiproducts D\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\n⊢ T ∈ distinguishedTriangles C ↔ T.rotate ∈ distinguishedTriangles C","state_after":"case mp\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{u_3, u_1} C\ninst✝⁶ : Category.{?u.447213, u_2} D\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasBinaryBiproducts C\ninst✝² : Preadditive D\ninst✝¹ : HasZeroObject D\ninst✝ : HasBinaryBiproducts D\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\n⊢ T ∈ distinguishedTriangles C → T.rotate ∈ distinguishedTriangles C\n\ncase mpr\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{u_3, u_1} C\ninst✝⁶ : Category.{?u.447213, u_2} D\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasBinaryBiproducts C\ninst✝² : Preadditive D\ninst✝¹ : HasZeroObject D\ninst✝ : HasBinaryBiproducts D\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\n⊢ T.rotate ∈ distinguishedTriangles C → T ∈ distinguishedTriangles C","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/Analysis/Normed/Group/Basic.lean","commit":"","full_name":"tendsto_norm_one","start":[756,0],"end":[758,43],"file_path":"Mathlib/Analysis/Normed/Group/Basic.lean","tactics":[{"state_before":"𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : FunLike 𝓕 E F\n⊢ Tendsto (fun a => ‖a‖) (𝓝 1) (𝓝 0)","state_after":"no goals","tactic":"simpa using tendsto_norm_div_self (1 : E)","premises":[{"full_name":"tendsto_norm_div_self","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[748,8],"def_end_pos":[748,29]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Monic.lean","commit":"","full_name":"Polynomial.isUnit_leadingCoeff_mul_right_eq_zero_iff","start":[466,0],"end":[479,8],"file_path":"Mathlib/Algebra/Polynomial/Monic.lean","tactics":[{"state_before":"R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit p.leadingCoeff\nq : R[X]\n⊢ p * q = 0 ↔ q = 0","state_after":"case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit p.leadingCoeff\nq : R[X]\n⊢ p * q = 0 → q = 0\n\ncase mpr\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit p.leadingCoeff\nq : R[X]\n⊢ q = 0 → p * q = 0","tactic":"constructor","premises":[]}]}
{"url":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","commit":"","full_name":"LinearMap.lTensor_neg","start":[1470,0],"end":[1473,59],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝¹⁰ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : AddCommGroup S\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\ninst✝¹ : Module R Q\ninst✝ : Module R S\nf : N →ₗ[R] P\n⊢ lTensor M (-f) = -lTensor M f","state_after":"R : Type u_1\ninst✝¹⁰ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : AddCommGroup S\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\ninst✝¹ : Module R Q\ninst✝ : Module R S\nf : N →ₗ[R] P\n⊢ (lTensorHom M) (-f) = -(lTensorHom M) f","tactic":"simp only [← coe_lTensorHom]","premises":[{"full_name":"LinearMap.coe_lTensorHom","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1135,8],"def_end_pos":[1135,22]}]},{"state_before":"R : Type u_1\ninst✝¹⁰ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : AddCommGroup S\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\ninst✝¹ : Module R Q\ninst✝ : Module R S\nf : N →ₗ[R] P\n⊢ (lTensorHom M) (-f) = -(lTensorHom M) f","state_after":"no goals","tactic":"exact (lTensorHom (R := R) (N := N) (P := P) M).map_neg f","premises":[{"full_name":"LinearMap.lTensorHom","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[1113,4],"def_end_pos":[1113,14]},{"full_name":"LinearMap.map_neg","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[572,18],"def_end_pos":[572,25]}]}]}
{"url":"Mathlib/Data/List/Dedup.lean","commit":"","full_name":"List.replicate_dedup","start":[152,0],"end":[157,37],"file_path":"Mathlib/Data/List/Dedup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\nx : α\nn : ℕ\nx✝ : n + 2 ≠ 0\n⊢ (replicate (n + 2) x).dedup = [x]","state_after":"no goals","tactic":"rw [replicate_succ, dedup_cons_of_mem (mem_replicate.2 ⟨n.succ_ne_zero, rfl⟩),\n      replicate_dedup n.succ_ne_zero]","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"List.dedup_cons_of_mem","def_path":"Mathlib/Data/List/Dedup.lean","def_pos":[47,8],"def_end_pos":[47,25]},{"full_name":"List.mem_replicate","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1302,16],"def_end_pos":[1302,29]},{"full_name":"List.replicate_succ","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[573,8],"def_end_pos":[573,22]},{"full_name":"Nat.succ_ne_zero","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[708,16],"def_end_pos":[708,28]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Topology/Algebra/WithZeroTopology.lean","commit":"","full_name":"WithZeroTopology.singleton_mem_nhds_of_ne_zero","start":[92,0],"end":[94,94],"file_path":"Mathlib/Topology/Algebra/WithZeroTopology.lean","tactics":[{"state_before":"α : Type u_1\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nγ γ₁ γ₂ : Γ₀\nl : Filter α\nf : α → Γ₀\nh : γ ≠ 0\n⊢ {γ} ∈ 𝓝 γ","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/Algebra/MvPolynomial/Variables.lean","commit":"","full_name":"MvPolynomial.vars_rename","start":[296,0],"end":[301,59],"file_path":"Mathlib/Algebra/MvPolynomial/Variables.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\ninst✝¹ : CommSemiring S\ninst✝ : DecidableEq τ\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ((rename f) φ).vars ⊆ Finset.image f φ.vars","state_after":"no goals","tactic":"classical\n  intro i hi\n  simp only [vars_def, exists_prop, Multiset.mem_toFinset, Finset.mem_image] at hi ⊢\n  simpa only [Multiset.mem_map] using degrees_rename _ _ hi","premises":[{"full_name":"Finset.mem_image","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[303,8],"def_end_pos":[303,17]},{"full_name":"Multiset.mem_map","def_path":"Mathlib/Data/Multiset/Basic.lean","def_pos":[1081,8],"def_end_pos":[1081,15]},{"full_name":"Multiset.mem_toFinset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2586,8],"def_end_pos":[2586,20]},{"full_name":"MvPolynomial.degrees_rename","def_path":"Mathlib/Algebra/MvPolynomial/Degrees.lean","def_pos":[173,8],"def_end_pos":[173,22]},{"full_name":"MvPolynomial.vars_def","def_path":"Mathlib/Algebra/MvPolynomial/Variables.lean","def_pos":[68,8],"def_end_pos":[68,16]},{"full_name":"exists_prop","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[307,16],"def_end_pos":[307,27]}]}]}
{"url":"Mathlib/RingTheory/HahnSeries/Addition.lean","commit":"","full_name":"HahnSeries.addOppositeEquiv_support","start":[88,0],"end":[92,31],"file_path":"Mathlib/RingTheory/HahnSeries/Addition.lean","tactics":[{"state_before":"Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx : HahnSeries Γ Rᵃᵒᵖ\n⊢ (AddOpposite.unop (addOppositeEquiv x)).support = x.support","state_after":"case h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx : HahnSeries Γ Rᵃᵒᵖ\nx✝ : Γ\n⊢ x✝ ∈ (AddOpposite.unop (addOppositeEquiv x)).support ↔ x✝ ∈ x.support","tactic":"ext","premises":[]},{"state_before":"case h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx : HahnSeries Γ Rᵃᵒᵖ\nx✝ : Γ\n⊢ x✝ ∈ (AddOpposite.unop (addOppositeEquiv x)).support ↔ x✝ ∈ x.support","state_after":"no goals","tactic":"simp [addOppositeEquiv_apply]","premises":[{"full_name":"HahnSeries.addOppositeEquiv_apply","def_path":"Mathlib/RingTheory/HahnSeries/Addition.lean","def_pos":[74,2],"def_end_pos":[74,7]}]}]}
{"url":"Mathlib/LinearAlgebra/Semisimple.lean","commit":"","full_name":"Module.End.isSemisimple_id","start":[73,0],"end":[75,47],"file_path":"Mathlib/LinearAlgebra/Semisimple.lean","tactics":[{"state_before":"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nf g : End R M\ninst✝ : IsSemisimpleModule R M\n⊢ IsSemisimple LinearMap.id","state_after":"no goals","tactic":"simpa [isSemisimple_iff] using exists_isCompl","premises":[{"full_name":"ComplementedLattice.exists_isCompl","def_path":"Mathlib/Order/Disjoint.lean","def_pos":[583,2],"def_end_pos":[583,16]},{"full_name":"Module.End.isSemisimple_iff","def_path":"Mathlib/LinearAlgebra/Semisimple.lean","def_pos":[61,6],"def_end_pos":[61,22]}]}]}
{"url":"Mathlib/Algebra/Regular/Basic.lean","commit":"","full_name":"IsUnit.isRegular","start":[299,0],"end":[303,25],"file_path":"Mathlib/Algebra/Regular/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : Monoid R\na b : R\nua : IsUnit a\n⊢ IsRegular a","state_after":"case intro\nR : Type u_1\ninst✝ : Monoid R\nb : R\na : Rˣ\n⊢ IsRegular ↑a","tactic":"rcases ua with ⟨a, rfl⟩","premises":[]},{"state_before":"case intro\nR : Type u_1\ninst✝ : Monoid R\nb : R\na : Rˣ\n⊢ IsRegular ↑a","state_after":"no goals","tactic":"exact Units.isRegular a","premises":[{"full_name":"Units.isRegular","def_path":"Mathlib/Algebra/Regular/Basic.lean","def_pos":[296,8],"def_end_pos":[296,23]}]}]}
{"url":"Mathlib/Data/DFinsupp/Basic.lean","commit":"","full_name":"DFinsupp.mk_neg","start":[907,0],"end":[910,85],"file_path":"Mathlib/Data/DFinsupp/Basic.lean","tactics":[{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddGroup (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (mk s (-x)) i = (-mk s x) i","state_after":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddGroup (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (if H : i ∈ s then (-x) ⟨i, H⟩ else 0) = -if H : i ∈ s then x ⟨i, H⟩ else 0","tactic":"simp only [neg_apply, mk_apply]","premises":[{"full_name":"DFinsupp.mk_apply","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[501,8],"def_end_pos":[501,16]},{"full_name":"DFinsupp.neg_apply","def_path":"Mathlib/Data/DFinsupp/Basic.lean","def_pos":[265,8],"def_end_pos":[265,17]}]},{"state_before":"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddGroup (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (if H : i ∈ s then (-x) ⟨i, H⟩ else 0) = -if H : i ∈ s then x ⟨i, H⟩ else 0","state_after":"no goals","tactic":"split_ifs <;> [rfl; rw [neg_zero]]","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]},{"full_name":"neg_zero","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[950,2],"def_end_pos":[950,13]}]}]}
{"url":"Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean","commit":"","full_name":"CategoryTheory.TwoSquare.GuitartExact.whiskerVertical","start":[40,0],"end":[53,16],"file_path":"Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean","tactics":[{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\n⊢ (w.whiskerVertical α.hom β.inv).GuitartExact","state_after":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\n⊢ ∀ (X₂ : D₁), ((w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂).Initial","tactic":"rw [guitartExact_iff_initial]","premises":[{"full_name":"CategoryTheory.TwoSquare.guitartExact_iff_initial","def_path":"Mathlib/CategoryTheory/GuitartExact/Basic.lean","def_pos":[264,6],"def_end_pos":[264,30]}]},{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\n⊢ ∀ (X₂ : D₁), ((w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂).Initial","state_after":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\nX₂ : D₁\n⊢ ((w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂).Initial","tactic":"intro X₂","premises":[]},{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\nX₂ : D₁\n⊢ ((w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂).Initial","state_after":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\nX₂ : D₁\ne : (w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂ ≅\n  w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂)).functor :=\n  NatIso.ofComponents (fun f => StructuredArrow.isoMk (α.symm.app f.right) ⋯) ⋯\n⊢ ((w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂).Initial","tactic":"let e : structuredArrowDownwards (w.whiskerVertical α.hom β.inv) X₂ ≅\n      w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂) ).functor :=\n    NatIso.ofComponents (fun f => StructuredArrow.isoMk (α.symm.app f.right) (by\n      dsimp\n      simp only [NatTrans.naturality_assoc, assoc, NatIso.cancel_natIso_inv_left, ← B.map_comp,\n        Iso.hom_inv_id_app, B.map_id, comp_id]))","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Category.comp_id","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[153,2],"def_end_pos":[153,9]},{"full_name":"CategoryTheory.Comma.right","def_path":"Mathlib/CategoryTheory/Comma/Basic.lean","def_pos":[63,2],"def_end_pos":[63,7]},{"full_name":"CategoryTheory.Equivalence.functor","def_path":"Mathlib/CategoryTheory/Equivalence.lean","def_pos":[81,2],"def_end_pos":[81,9]},{"full_name":"CategoryTheory.Functor.comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[100,4],"def_end_pos":[100,8]},{"full_name":"CategoryTheory.Functor.map_comp","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[41,2],"def_end_pos":[41,10]},{"full_name":"CategoryTheory.Functor.map_id","def_path":"Mathlib/CategoryTheory/Functor/Basic.lean","def_pos":[39,2],"def_end_pos":[39,8]},{"full_name":"CategoryTheory.Iso","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[49,10],"def_end_pos":[49,13]},{"full_name":"CategoryTheory.Iso.app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[51,4],"def_end_pos":[51,7]},{"full_name":"CategoryTheory.Iso.hom","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[51,2],"def_end_pos":[51,5]},{"full_name":"CategoryTheory.Iso.hom_inv_id_app","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[59,8],"def_end_pos":[59,22]},{"full_name":"CategoryTheory.Iso.inv","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[53,2],"def_end_pos":[53,5]},{"full_name":"CategoryTheory.Iso.symm","def_path":"Mathlib/CategoryTheory/Iso.lean","def_pos":[84,4],"def_end_pos":[84,8]},{"full_name":"CategoryTheory.NatIso.cancel_natIso_inv_left","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[116,8],"def_end_pos":[116,30]},{"full_name":"CategoryTheory.NatIso.ofComponents","def_path":"Mathlib/CategoryTheory/NatIso.lean","def_pos":[186,4],"def_end_pos":[186,16]},{"full_name":"CategoryTheory.StructuredArrow.isoMk","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[161,4],"def_end_pos":[161,9]},{"full_name":"CategoryTheory.StructuredArrow.mapIso","def_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","def_pos":[241,4],"def_end_pos":[241,10]},{"full_name":"CategoryTheory.TwoSquare.structuredArrowDownwards","def_path":"Mathlib/CategoryTheory/GuitartExact/Basic.lean","def_pos":[88,4],"def_end_pos":[88,28]},{"full_name":"CategoryTheory.TwoSquare.whiskerVertical","def_path":"Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean","def_pos":[34,4],"def_end_pos":[34,19]}]},{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\nX₂ : D₁\ne : (w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂ ≅\n  w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂)).functor :=\n  NatIso.ofComponents (fun f => StructuredArrow.isoMk (α.symm.app f.right) ⋯) ⋯\n⊢ ((w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂).Initial","state_after":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\nX₂ : D₁\ne : (w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂ ≅\n  w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂)).functor :=\n  NatIso.ofComponents (fun f => StructuredArrow.isoMk (α.symm.app f.right) ⋯) ⋯\n⊢ (w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂)).functor).Initial","tactic":"rw [Functor.initial_natIso_iff e]","premises":[{"full_name":"CategoryTheory.Functor.initial_natIso_iff","def_path":"Mathlib/CategoryTheory/Limits/Final.lean","def_pos":[160,8],"def_end_pos":[160,26]}]},{"state_before":"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD₁ : Type u_4\nD₂ : Type u_5\nD₃ : Type u_6\ninst✝⁶ : Category.{u_7, u_1} C₁\ninst✝⁵ : Category.{u_9, u_2} C₂\ninst✝⁴ : Category.{?u.3150, u_3} C₃\ninst✝³ : Category.{u_8, u_4} D₁\ninst✝² : Category.{u_10, u_5} D₂\ninst✝¹ : Category.{?u.3162, u_6} D₃\nT : C₁ ⥤ D₁\nL : C₁ ⥤ C₂\nR : D₁ ⥤ D₂\nB : C₂ ⥤ D₂\nw : TwoSquare T L R B\nL' : C₁ ⥤ C₂\nR' : D₁ ⥤ D₂\ninst✝ : w.GuitartExact\nα : L ≅ L'\nβ : R ≅ R'\nX₂ : D₁\ne : (w.whiskerVertical α.hom β.inv).structuredArrowDownwards X₂ ≅\n  w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂)).functor :=\n  NatIso.ofComponents (fun f => StructuredArrow.isoMk (α.symm.app f.right) ⋯) ⋯\n⊢ (w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (β.app X₂)).functor).Initial","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/Combinatorics/Additive/Corner/Defs.lean","commit":"","full_name":"IsCorner.image","start":[68,0],"end":[73,78],"file_path":"Mathlib/Combinatorics/Additive/Corner/Defs.lean","tactics":[{"state_before":"G : Type u_1\nH : Type u_2\ninst✝¹ : AddCommMonoid G\ninst✝ : AddCommMonoid H\nA B : Set (G × G)\ns : Set G\nt : Set H\nf : G → H\na b c x₁ y₁ x₂ y₂ : G\nhf : IsAddFreimanHom 2 s t f\nhAs : A ⊆ s ×ˢ s\nhA : IsCorner A x₁ y₁ x₂ y₂\n⊢ IsCorner (Prod.map f f '' A) (f x₁) (f y₁) (f x₂) (f y₂)","state_after":"case mk\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddCommMonoid G\ninst✝ : AddCommMonoid H\nA B : Set (G × G)\ns : Set G\nt : Set H\nf : G → H\na b c x₁ y₁ x₂ y₂ : G\nhf : IsAddFreimanHom 2 s t f\nhAs : A ⊆ s ×ˢ s\nhx₁y₁ : (x₁, y₁) ∈ A\nhx₁y₂ : (x₁, y₂) ∈ A\nhx₂y₁ : (x₂, y₁) ∈ A\nhxy : x₁ + y₂ = x₂ + y₁\n⊢ IsCorner (Prod.map f f '' A) (f x₁) (f y₁) (f x₂) (f y₂)","tactic":"obtain ⟨hx₁y₁, hx₁y₂, hx₂y₁, hxy⟩ := hA","premises":[]},{"state_before":"case mk\nG : Type u_1\nH : Type u_2\ninst✝¹ : AddCommMonoid G\ninst✝ : AddCommMonoid H\nA B : Set (G × G)\ns : Set G\nt : Set H\nf : G → H\na b c x₁ y₁ x₂ y₂ : G\nhf : IsAddFreimanHom 2 s t f\nhAs : A ⊆ s ×ˢ s\nhx₁y₁ : (x₁, y₁) ∈ A\nhx₁y₂ : (x₁, y₂) ∈ A\nhx₂y₁ : (x₂, y₁) ∈ A\nhxy : x₁ + y₂ = x₂ + y₁\n⊢ IsCorner (Prod.map f f '' A) (f x₁) (f y₁) (f x₂) (f y₂)","state_after":"no goals","tactic":"exact ⟨mem_image_of_mem _ hx₁y₁, mem_image_of_mem _ hx₁y₂, mem_image_of_mem _ hx₂y₁,\n    hf.add_eq_add (hAs hx₁y₁).1 (hAs hx₁y₂).2 (hAs hx₂y₁).1 (hAs hx₁y₁).2 hxy⟩","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"IsAddFreimanHom.add_eq_add","def_path":"Mathlib/Combinatorics/Additive/FreimanHom.lean","def_pos":[105,2],"def_end_pos":[105,13]},{"full_name":"Set.mem_image_of_mem","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[122,8],"def_end_pos":[122,24]}]}]}
{"url":"Mathlib/Analysis/MeanInequalities.lean","commit":"","full_name":"NNReal.inner_le_Lp_mul_Lq","start":[401,0],"end":[430,33],"file_path":"Mathlib/Analysis/MeanInequalities.lean","tactics":[{"state_before":"ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","state_after":"case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ∑ i ∈ s, f i ^ p = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)\n\ncase neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","tactic":"by_cases hF_zero : ∑ i ∈ s, f i ^ p = 0","premises":[{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","state_after":"case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ∑ i ∈ s, g i ^ q = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)\n\ncase neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","tactic":"by_cases hG_zero : ∑ i ∈ s, g i ^ q = 0","premises":[{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","state_after":"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\nf' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","tactic":"let f' i := f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)","premises":[{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]}]},{"state_before":"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\nf' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","state_after":"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\nf' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)\ng' : ι → ℝ≥0 := fun i => g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)","tactic":"let g' i := g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)","premises":[{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]}]},{"state_before":"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\nf' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)\ng' : ι → ℝ≥0 := fun i => g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)\n⊢ ∑ i ∈ s, f' i * g' i ≤ 1","state_after":"case neg.refine_1\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\nf' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)\ng' : ι → ℝ≥0 := fun i => g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)\n⊢ ∑ i ∈ s, f' i ^ p = 1\n\ncase neg.refine_2\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.IsConjExponent q\nhF_zero : ¬∑ i ∈ s, f i ^ p = 0\nhG_zero : ¬∑ i ∈ s, g i ^ q = 0\nf' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)\ng' : ι → ℝ≥0 := fun i => g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)\n⊢ ∑ i ∈ s, g' i ^ q = 1","tactic":"refine inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq ?_) (le_of_eq ?_)","premises":[{"full_name":"_private.Mathlib.Analysis.MeanInequalities.0.NNReal.inner_le_Lp_mul_Lp_of_norm_le_one","def_path":"Mathlib/Analysis/MeanInequalities.lean","def_pos":[375,16],"def_end_pos":[375,49]},{"full_name":"le_of_eq","def_path":"Mathlib/Order/Defs.lean","def_pos":[60,8],"def_end_pos":[60,16]}]}]}
{"url":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","commit":"","full_name":"EisensteinSeries.r1_eq","start":[52,0],"end":[53,60],"file_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","tactics":[{"state_before":"z : ℍ\n⊢ r1 z = 1 / ((z.re / z.im) ^ 2 + 1)","state_after":"no goals","tactic":"rw [div_pow, div_add_one (by positivity), one_div_div, r1]","premises":[{"full_name":"EisensteinSeries.r1","def_path":"Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean","def_pos":[50,4],"def_end_pos":[50,6]},{"full_name":"div_add_one","def_path":"Mathlib/Algebra/Field/Basic.lean","def_pos":[40,8],"def_end_pos":[40,19]},{"full_name":"div_pow","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[589,6],"def_end_pos":[589,13]},{"full_name":"one_div_div","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[403,8],"def_end_pos":[403,19]}]}]}
{"url":"Mathlib/CategoryTheory/DifferentialObject.lean","commit":"","full_name":"CategoryTheory.DifferentialObject.eqToHom_f","start":[92,0],"end":[97,5],"file_path":"Mathlib/CategoryTheory/DifferentialObject.lean","tactics":[{"state_before":"S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX Y : DifferentialObject S C\nh : X = Y\n⊢ (eqToHom h).f = eqToHom ⋯","state_after":"S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX : DifferentialObject S C\n⊢ (eqToHom ⋯).f = eqToHom ⋯","tactic":"subst h","premises":[]},{"state_before":"S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX : DifferentialObject S C\n⊢ (eqToHom ⋯).f = eqToHom ⋯","state_after":"S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX : DifferentialObject S C\n⊢ (𝟙 X).f = 𝟙 X.obj","tactic":"rw [eqToHom_refl, eqToHom_refl]","premises":[{"full_name":"CategoryTheory.eqToHom_refl","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[44,8],"def_end_pos":[44,20]}]},{"state_before":"S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX : DifferentialObject S C\n⊢ (𝟙 X).f = 𝟙 X.obj","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/FieldTheory/RatFunc/Basic.lean","commit":"","full_name":"RatFunc.num_eq_zero_iff","start":[942,0],"end":[944,97],"file_path":"Mathlib/FieldTheory/RatFunc/Basic.lean","tactics":[{"state_before":"K : Type u\ninst✝ : Field K\nx : RatFunc K\nh : x.num = 0\n⊢ x = 0","state_after":"no goals","tactic":"rw [← num_div_denom x, h, RingHom.map_zero, zero_div]","premises":[{"full_name":"RatFunc.num_div_denom","def_path":"Mathlib/FieldTheory/RatFunc/Basic.lean","def_pos":[921,8],"def_end_pos":[921,21]},{"full_name":"RingHom.map_zero","def_path":"Mathlib/Algebra/Ring/Hom/Defs.lean","def_pos":[472,18],"def_end_pos":[472,26]},{"full_name":"zero_div","def_path":"Mathlib/Algebra/GroupWithZero/Basic.lean","def_pos":[298,8],"def_end_pos":[298,16]}]}]}
{"url":"Mathlib/NumberTheory/DirichletCharacter/Basic.lean","commit":"","full_name":"DirichletCharacter.eq_one_iff_conductor_eq_one","start":[207,0],"end":[210,58],"file_path":"Mathlib/NumberTheory/DirichletCharacter/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n ≠ 0\n⊢ χ = 1 ↔ χ.conductor = 1","state_after":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n ≠ 0\nhχ : χ.conductor = 1\n⊢ χ = 1","tactic":"refine ⟨fun h ↦ h ▸ conductor_one hn, fun hχ ↦ ?_⟩","premises":[{"full_name":"DirichletCharacter.conductor_one","def_path":"Mathlib/NumberTheory/DirichletCharacter/Basic.lean","def_pos":[198,6],"def_end_pos":[198,19]},{"full_name":"Iff.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[116,2],"def_end_pos":[116,7]}]},{"state_before":"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n ≠ 0\nhχ : χ.conductor = 1\n⊢ χ = 1","state_after":"case intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n ≠ 0\nhχ : χ.conductor = 1\nh' : χ.conductor ∣ n\nχ₀ : DirichletCharacter R χ.conductor\nh : χ = (changeLevel h') χ₀\n⊢ χ = 1","tactic":"obtain ⟨h', χ₀, h⟩ := factorsThrough_conductor χ","premises":[{"full_name":"DirichletCharacter.factorsThrough_conductor","def_path":"Mathlib/NumberTheory/DirichletCharacter/Basic.lean","def_pos":[192,6],"def_end_pos":[192,30]}]},{"state_before":"case intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n ≠ 0\nhχ : χ.conductor = 1\nh' : χ.conductor ∣ n\nχ₀ : DirichletCharacter R χ.conductor\nh : χ = (changeLevel h') χ₀\n⊢ χ = 1","state_after":"no goals","tactic":"exact (level_one' χ₀ hχ ▸ h).trans <| changeLevel_one h'","premises":[{"full_name":"DirichletCharacter.changeLevel_one","def_path":"Mathlib/NumberTheory/DirichletCharacter/Basic.lean","def_pos":[162,6],"def_end_pos":[162,21]},{"full_name":"DirichletCharacter.level_one'","def_path":"Mathlib/NumberTheory/DirichletCharacter/Basic.lean","def_pos":[152,6],"def_end_pos":[152,16]},{"full_name":"Eq.trans","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[335,8],"def_end_pos":[335,16]}]}]}
{"url":"Mathlib/GroupTheory/NoncommPiCoprod.lean","commit":"","full_name":"AddSubgroup.noncommPiCoprod_single","start":[278,0],"end":[281,96],"file_path":"Mathlib/GroupTheory/NoncommPiCoprod.lean","tactics":[{"state_before":"G : Type u_1\ninst✝ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Subgroup G\nf g : (i : ι) → ↥(H i)\nhcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y\ni : ι\ny : ↥(H i)\n⊢ (noncommPiCoprod hcomm) (Pi.mulSingle i y) = ↑y","state_after":"no goals","tactic":"apply MonoidHom.noncommPiCoprod_mulSingle","premises":[{"full_name":"MonoidHom.noncommPiCoprod_mulSingle","def_path":"Mathlib/GroupTheory/NoncommPiCoprod.lean","def_pos":[117,8],"def_end_pos":[117,33]}]}]}
{"url":"Mathlib/RingTheory/Coprime/Basic.lean","commit":"","full_name":"IsCoprime.of_add_mul_left_right","start":[177,0],"end":[179,30],"file_path":"Mathlib/RingTheory/Coprime/Basic.lean","tactics":[{"state_before":"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n⊢ IsCoprime x y","state_after":"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (y + x * z) x\n⊢ IsCoprime y x","tactic":"rw [isCoprime_comm] at h ⊢","premises":[{"full_name":"isCoprime_comm","def_path":"Mathlib/RingTheory/Coprime/Basic.lean","def_pos":[48,8],"def_end_pos":[48,22]}]},{"state_before":"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (y + x * z) x\n⊢ IsCoprime y x","state_after":"no goals","tactic":"exact h.of_add_mul_left_left","premises":[{"full_name":"IsCoprime.of_add_mul_left_left","def_path":"Mathlib/RingTheory/Coprime/Basic.lean","def_pos":[167,8],"def_end_pos":[167,38]}]}]}
{"url":"Mathlib/FieldTheory/PurelyInseparable.lean","commit":"","full_name":"separableClosure.isPurelyInseparable","start":[569,0],"end":[579,87],"file_path":"Mathlib/FieldTheory/PurelyInseparable.lean","tactics":[{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\n⊢ IsIntegral (↥(separableClosure F E)) x ∧\n    (IsSeparable (↥(separableClosure F E)) x → x ∈ (algebraMap (↥(separableClosure F E)) E).range)","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\n⊢ IsIntegral (↥L) x ∧ (IsSeparable (↥L) x → x ∈ (algebraMap (↥L) E).range)","tactic":"set L := separableClosure F E","premises":[{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]},{"full_name":"separableClosure","def_path":"Mathlib/FieldTheory/SeparableClosure.lean","def_pos":[78,4],"def_end_pos":[78,20]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\n⊢ IsIntegral (↥L) x ∧ (IsSeparable (↥L) x → x ∈ (algebraMap (↥L) E).range)","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\n⊢ x ∈ (algebraMap (↥L) E).range","tactic":"refine ⟨(IsAlgebraic.tower_top L (Algebra.IsAlgebraic.isAlgebraic (R := F) x)).isIntegral,\n    fun h ↦ ?_⟩","premises":[{"full_name":"Algebra.IsAlgebraic.isAlgebraic","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[51,2],"def_end_pos":[51,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"IsAlgebraic.tower_top","def_path":"Mathlib/RingTheory/Algebraic.lean","def_pos":[266,8],"def_end_pos":[266,29]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\n⊢ x ∈ (algebraMap (↥L) E).range","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\nthis : Algebra.IsSeparable ↥L ↥(↥L)⟮x⟯\n⊢ x ∈ (algebraMap (↥L) E).range","tactic":"haveI := (isSeparable_adjoin_simple_iff_isSeparable L E).2 h","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IntermediateField.isSeparable_adjoin_simple_iff_isSeparable","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[748,8],"def_end_pos":[748,67]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\nthis : Algebra.IsSeparable ↥L ↥(↥L)⟮x⟯\n⊢ x ∈ (algebraMap (↥L) E).range","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\nthis✝ : Algebra.IsSeparable ↥L ↥(↥L)⟮x⟯\nthis : Algebra.IsSeparable F ↥(restrictScalars F (↥L)⟮x⟯)\n⊢ x ∈ (algebraMap (↥L) E).range","tactic":"haveI : Algebra.IsSeparable F (restrictScalars F L⟮x⟯) := Algebra.IsSeparable.trans F L L⟮x⟯","premises":[{"full_name":"Algebra.IsSeparable","def_path":"Mathlib/FieldTheory/Separable.lean","def_pos":[519,42],"def_end_pos":[519,61]},{"full_name":"Algebra.IsSeparable.trans","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[790,8],"def_end_pos":[790,33]},{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"IntermediateField.restrictScalars","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[573,4],"def_end_pos":[573,19]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\nthis✝ : Algebra.IsSeparable ↥L ↥(↥L)⟮x⟯\nthis : Algebra.IsSeparable F ↥(restrictScalars F (↥L)⟮x⟯)\n⊢ x ∈ (algebraMap (↥L) E).range","state_after":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\nthis✝ : Algebra.IsSeparable ↥L ↥(↥L)⟮x⟯\nthis : Algebra.IsSeparable F ↥(restrictScalars F (↥L)⟮x⟯)\nhx : x ∈ restrictScalars F (↥L)⟮x⟯\n⊢ x ∈ (algebraMap (↥L) E).range","tactic":"have hx : x ∈ restrictScalars F L⟮x⟯ := mem_adjoin_simple_self _ x","premises":[{"full_name":"IntermediateField.adjoin","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[42,4],"def_end_pos":[42,10]},{"full_name":"IntermediateField.mem_adjoin_simple_self","def_path":"Mathlib/FieldTheory/Adjoin.lean","def_pos":[529,8],"def_end_pos":[529,30]},{"full_name":"IntermediateField.restrictScalars","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[573,4],"def_end_pos":[573,19]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Singleton.singleton","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[467,2],"def_end_pos":[467,11]}]},{"state_before":"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : Algebra.IsAlgebraic F E\nx : E\nL : IntermediateField F E := separableClosure F E\nh : IsSeparable (↥L) x\nthis✝ : Algebra.IsSeparable ↥L ↥(↥L)⟮x⟯\nthis : Algebra.IsSeparable F ↥(restrictScalars F (↥L)⟮x⟯)\nhx : x ∈ restrictScalars F (↥L)⟮x⟯\n⊢ x ∈ (algebraMap (↥L) E).range","state_after":"no goals","tactic":"exact ⟨⟨x, mem_separableClosure_iff.2 <| isSeparable_of_mem_isSeparable F E hx⟩, rfl⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"IntermediateField.isSeparable_of_mem_isSeparable","def_path":"Mathlib/FieldTheory/SeparableDegree.lean","def_pos":[742,6],"def_end_pos":[742,54]},{"full_name":"mem_separableClosure_iff","def_path":"Mathlib/FieldTheory/SeparableClosure.lean","def_pos":[89,8],"def_end_pos":[89,32]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","commit":"","full_name":"CategoryTheory.Subgroupoid.sInf_isNormal","start":[316,0],"end":[318,97],"file_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\ns : Set (Subgroupoid C)\nsn : ∀ S ∈ s, S.IsNormal\n⊢ ∀ (c : C), 𝟙 c ∈ (sInf s).arrows c c","state_after":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\ns : Set (Subgroupoid C)\nsn : ∀ S ∈ s, S.IsNormal\n⊢ ∀ (c : C), ∀ i ∈ s, 𝟙 c ∈ i.arrows c c","tactic":"simp_rw [sInf, mem_iInter₂]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"InfSet.sInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[48,2],"def_end_pos":[48,6]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_iInter₂","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[69,8],"def_end_pos":[69,19]}]},{"state_before":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\ns : Set (Subgroupoid C)\nsn : ∀ S ∈ s, S.IsNormal\n⊢ ∀ (c : C), ∀ i ∈ s, 𝟙 c ∈ i.arrows c c","state_after":"no goals","tactic":"exact fun c S Ss => (sn S Ss).wide c","premises":[{"full_name":"CategoryTheory.Subgroupoid.IsWide.wide","def_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","def_pos":[277,2],"def_end_pos":[277,6]}]},{"state_before":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\ns : Set (Subgroupoid C)\nsn : ∀ S ∈ s, S.IsNormal\n⊢ ∀ {c d : C} (p : c ⟶ d) {γ : c ⟶ c}, γ ∈ (sInf s).arrows c c → Groupoid.inv p ≫ γ ≫ p ∈ (sInf s).arrows d d","state_after":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\ns : Set (Subgroupoid C)\nsn : ∀ S ∈ s, S.IsNormal\n⊢ ∀ {c d : C} (p : c ⟶ d) {γ : c ⟶ c}, (∀ i ∈ s, γ ∈ i.arrows c c) → ∀ i ∈ s, Groupoid.inv p ≫ γ ≫ p ∈ i.arrows d d","tactic":"simp_rw [sInf, mem_iInter₂]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"InfSet.sInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[48,2],"def_end_pos":[48,6]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Set.mem_iInter₂","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[69,8],"def_end_pos":[69,19]}]},{"state_before":"C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\ns : Set (Subgroupoid C)\nsn : ∀ S ∈ s, S.IsNormal\n⊢ ∀ {c d : C} (p : c ⟶ d) {γ : c ⟶ c}, (∀ i ∈ s, γ ∈ i.arrows c c) → ∀ i ∈ s, Groupoid.inv p ≫ γ ≫ p ∈ i.arrows d d","state_after":"no goals","tactic":"exact fun p γ hγ S Ss => (sn S Ss).conj p (hγ S Ss)","premises":[{"full_name":"CategoryTheory.Subgroupoid.IsNormal.conj","def_path":"Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean","def_pos":[297,2],"def_end_pos":[297,6]}]}]}
{"url":"Mathlib/Dynamics/Ergodic/AddCircle.lean","commit":"","full_name":"AddCircle.ergodic_zsmul","start":[101,0],"end":[117,85],"file_path":"Mathlib/Dynamics/Ergodic/AddCircle.lean","tactics":[{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","tactic":"let u : ℕ → AddCircle T := fun j => ↑((↑1 : ℝ) / ↑(n.natAbs ^ j) * T)","premises":[{"full_name":"AddCircle","def_path":"Mathlib/Topology/Instances/AddCircle.lean","def_pos":[115,7],"def_end_pos":[115,16]},{"full_name":"Int.natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[262,4],"def_end_pos":[262,10]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]},{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","tactic":"replace hn : 1 < n.natAbs := by rwa [Int.abs_eq_natAbs, Nat.one_lt_cast] at hn","premises":[{"full_name":"Int.abs_eq_natAbs","def_path":"Mathlib/Algebra/Order/Group/Int.lean","def_pos":[46,8],"def_end_pos":[46,21]},{"full_name":"Int.natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[262,4],"def_end_pos":[262,10]},{"full_name":"Nat.one_lt_cast","def_path":"Mathlib/Data/Nat/Cast/Order/Basic.lean","def_pos":[86,8],"def_end_pos":[86,19]}]},{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","tactic":"have hu₀ : ∀ j, addOrderOf (u j) = n.natAbs ^ j := fun j => by\n        convert addOrderOf_div_of_gcd_eq_one (p := T) (m := 1)\n          (pow_pos (pos_of_gt hn) j) (gcd_one_left _)\n        norm_cast","premises":[{"full_name":"AddCircle.addOrderOf_div_of_gcd_eq_one","def_path":"Mathlib/Topology/Instances/AddCircle.lean","def_pos":[390,8],"def_end_pos":[390,36]},{"full_name":"Int.natAbs","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean","def_pos":[262,4],"def_end_pos":[262,10]},{"full_name":"addOrderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[126,2],"def_end_pos":[126,13]},{"full_name":"gcd_one_left","def_path":"Mathlib/Algebra/GCDMonoid/Basic.lean","def_pos":[367,8],"def_end_pos":[367,20]},{"full_name":"pos_of_gt","def_path":"Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean","def_pos":[170,8],"def_end_pos":[170,17]},{"full_name":"pow_pos","def_path":"Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean","def_pos":[385,8],"def_end_pos":[385,15]}]},{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\nhnu : ∀ (j : ℕ), n ^ j • u j = 0\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","tactic":"have hnu : ∀ j, n ^ j • u j = 0 := fun j => by\n        rw [← addOrderOf_dvd_iff_zsmul_eq_zero, hu₀, Int.natCast_pow, Int.natCast_natAbs, ← abs_pow,\n          abs_dvd]","premises":[{"full_name":"Int.natCast_natAbs","def_path":"Mathlib/Algebra/Order/Group/Int.lean","def_pos":[50,25],"def_end_pos":[50,39]},{"full_name":"Int.natCast_pow","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Pow.lean","def_pos":[38,8],"def_end_pos":[38,19]},{"full_name":"abs_dvd","def_path":"Mathlib/Algebra/Order/Ring/Abs.lean","def_pos":[157,8],"def_end_pos":[157,15]},{"full_name":"abs_pow","def_path":"Mathlib/Algebra/Order/Ring/Abs.lean","def_pos":[56,6],"def_end_pos":[56,13]},{"full_name":"addOrderOf_dvd_iff_zsmul_eq_zero","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[547,2],"def_end_pos":[547,13]}]},{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\nhnu : ∀ (j : ℕ), n ^ j • u j = 0\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\nhnu : ∀ (j : ℕ), n ^ j • u j = 0\nhu₁ : ∀ (j : ℕ), u j +ᵥ s =ᶠ[ae volume] s\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","tactic":"have hu₁ : ∀ j, (u j +ᵥ s : Set _) =ᵐ[volume] s := fun j => by\n        rw [vadd_eq_self_of_preimage_zsmul_eq_self hs' (hnu j)]","premises":[{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"HVAdd.hVAdd","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[61,2],"def_end_pos":[61,7]},{"full_name":"MeasureTheory.MeasureSpace.volume","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[326,2],"def_end_pos":[326,8]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"vadd_eq_self_of_preimage_zsmul_eq_self","def_path":"Mathlib/Data/Set/Pointwise/Iterate.lean","def_pos":[23,2],"def_end_pos":[23,13]}]},{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\nhnu : ∀ (j : ℕ), n ^ j • u j = 0\nhu₁ : ∀ (j : ℕ), u j +ᵥ s =ᶠ[ae volume] s\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\nhnu : ∀ (j : ℕ), n ^ j • u j = 0\nhu₁ : ∀ (j : ℕ), u j +ᵥ s =ᶠ[ae volume] s\nhu₂ : Tendsto (fun j => addOrderOf (u j)) atTop atTop\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","tactic":"have hu₂ : Tendsto (fun j => addOrderOf <| u j) atTop atTop := by\n        simp_rw [hu₀]; exact Nat.tendsto_pow_atTop_atTop_of_one_lt hn","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Nat.tendsto_pow_atTop_atTop_of_one_lt","def_path":"Mathlib/Analysis/SpecificLimits/Basic.lean","def_pos":[129,8],"def_end_pos":[129,45]},{"full_name":"addOrderOf","def_path":"Mathlib/GroupTheory/OrderOfElement.lean","def_pos":[126,2],"def_end_pos":[126,13]}]},{"state_before":"T : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn✝ : 1 < |n|\ns : Set (AddCircle T)\nhs : MeasurableSet s\nhs' : (fun y => n • y) ⁻¹' s = s\nu : ℕ → AddCircle T := fun j => ↑(1 / ↑(n.natAbs ^ j) * T)\nhn : 1 < n.natAbs\nhu₀ : ∀ (j : ℕ), addOrderOf (u j) = n.natAbs ^ j\nhnu : ∀ (j : ℕ), n ^ j • u j = 0\nhu₁ : ∀ (j : ℕ), u j +ᵥ s =ᶠ[ae volume] s\nhu₂ : Tendsto (fun j => addOrderOf (u j)) atTop atTop\n⊢ s =ᶠ[ae volume] ∅ ∨ s =ᶠ[ae volume] univ","state_after":"no goals","tactic":"exact ae_empty_or_univ_of_forall_vadd_ae_eq_self hs.nullMeasurableSet hu₁ hu₂","premises":[{"full_name":"AddCircle.ae_empty_or_univ_of_forall_vadd_ae_eq_self","def_path":"Mathlib/Dynamics/Ergodic/AddCircle.lean","def_pos":[43,8],"def_end_pos":[43,50]},{"full_name":"MeasurableSet.nullMeasurableSet","def_path":"Mathlib/MeasureTheory/Measure/NullMeasurable.lean","def_pos":[92,8],"def_end_pos":[92,46]}]}]}
{"url":"Mathlib/Topology/Compactness/Compact.lean","commit":"","full_name":"IsCompact.nhdsSet_prod_eq_biSup","start":[377,0],"end":[380,63],"file_path":"Mathlib/Topology/Compactness/Compact.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns✝ t K : Set X\nhK : IsCompact K\nl : Filter Y\ns : Set (X × Y)\nhs : s ∈ ⨆ x ∈ K, 𝓝 x ×ˢ l\n⊢ ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l","state_after":"no goals","tactic":"simpa using hs","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","commit":"","full_name":"WeierstrassCurve.Jacobian.negMap_of_Z_eq_zero","start":[1135,0],"end":[1138,87],"file_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","tactics":[{"state_before":"R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhP : W.Nonsingular P\nhPz : P z = 0\n⊢ W.negMap ⟦P⟧ = ⟦![1, 1, 0]⟧","state_after":"no goals","tactic":"rw [negMap_eq, neg_of_Z_eq_zero hP hPz,\n    smul_eq _ ((isUnit_Y_of_Z_eq_zero hP hPz).div <| isUnit_X_of_Z_eq_zero hP hPz).neg]","premises":[{"full_name":"IsUnit.div","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[824,21],"def_end_pos":[824,24]},{"full_name":"IsUnit.neg","def_path":"Mathlib/Algebra/Ring/Units.lean","def_pos":[94,8],"def_end_pos":[94,18]},{"full_name":"WeierstrassCurve.Jacobian.isUnit_X_of_Z_eq_zero","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[421,6],"def_end_pos":[421,27]},{"full_name":"WeierstrassCurve.Jacobian.isUnit_Y_of_Z_eq_zero","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[431,6],"def_end_pos":[431,27]},{"full_name":"WeierstrassCurve.Jacobian.negMap_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[1132,6],"def_end_pos":[1132,15]},{"full_name":"WeierstrassCurve.Jacobian.neg_of_Z_eq_zero","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[1083,6],"def_end_pos":[1083,22]},{"full_name":"WeierstrassCurve.Jacobian.smul_eq","def_path":"Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean","def_pos":[159,6],"def_end_pos":[159,13]}]}]}
{"url":"Mathlib/Analysis/Convex/Strict.lean","commit":"","full_name":"StrictConvex.affine_preimage","start":[321,0],"end":[327,41],"file_path":"Mathlib/Analysis/Convex/Strict.lean","tactics":[{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx y : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\n⊢ StrictConvex 𝕜 (⇑f ⁻¹' s)","state_after":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\nx : E\nhx : x ∈ ⇑f ⁻¹' s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior (⇑f ⁻¹' s)","tactic":"intro x hx y hy hxy a b ha hb hab","premises":[]},{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\nx : E\nhx : x ∈ ⇑f ⁻¹' s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior (⇑f ⁻¹' s)","state_after":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\nx : E\nhx : x ∈ ⇑f ⁻¹' s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ ⇑f ⁻¹' interior s","tactic":"refine preimage_interior_subset_interior_preimage hf ?_","premises":[{"full_name":"preimage_interior_subset_interior_preimage","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1377,8],"def_end_pos":[1377,50]}]},{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\nx : E\nhx : x ∈ ⇑f ⁻¹' s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ ⇑f ⁻¹' interior s","state_after":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\nx : E\nhx : x ∈ ⇑f ⁻¹' s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • f x + b • f y ∈ interior s","tactic":"rw [mem_preimage, Convex.combo_affine_apply hab]","premises":[{"full_name":"Convex.combo_affine_apply","def_path":"Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean","def_pos":[830,8],"def_end_pos":[830,33]},{"full_name":"Set.mem_preimage","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[112,8],"def_end_pos":[112,20]}]},{"state_before":"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\nF : Type u_4\nβ : Type u_5\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ t : Set E\nx✝ y✝ : E\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj : Injective ⇑f\nx : E\nhx : x ∈ ⇑f ⁻¹' s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • f x + b • f y ∈ interior s","state_after":"no goals","tactic":"exact hs hx hy (hfinj.ne hxy) ha hb hab","premises":[{"full_name":"Function.Injective.ne","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[79,8],"def_end_pos":[79,20]}]}]}
{"url":"Mathlib/AlgebraicGeometry/Cover/Open.lean","commit":"","full_name":"AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_obj","start":[214,0],"end":[234,44],"file_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","tactics":[{"state_before":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\n⊢ X.OpenCover","state_after":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\nthis : ∃ t, ⋃ x ∈ t, (fun x => Set.range ⇑(𝒰.map (𝒰.f x)).val.base) x = ⊤\n⊢ X.OpenCover","tactic":"have :=\n    @CompactSpace.elim_nhds_subcover _ _ H (fun x : X => Set.range (𝒰.map (𝒰.f x)).1.base)\n      fun x => (IsOpenImmersion.isOpen_range (𝒰.map (𝒰.f x))).mem_nhds (𝒰.covers x)","premises":[{"full_name":"AlgebraicGeometry.IsOpenImmersion.isOpen_range","def_path":"Mathlib/AlgebraicGeometry/OpenImmersion.lean","def_pos":[63,8],"def_end_pos":[63,36]},{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.covers","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[52,2],"def_end_pos":[52,8]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.f","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"CompactSpace.elim_nhds_subcover","def_path":"Mathlib/Topology/Compactness/Compact.lean","def_pos":[741,8],"def_end_pos":[741,39]},{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]}]},{"state_before":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\nthis : ∃ t, ⋃ x ∈ t, (fun x => Set.range ⇑(𝒰.map (𝒰.f x)).val.base) x = ⊤\n⊢ X.OpenCover","state_after":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\nthis : ∃ t, ⋃ x ∈ t, (fun x => Set.range ⇑(𝒰.map (𝒰.f x)).val.base) x = ⊤\nt : Finset ↑↑X.toPresheafedSpace := this.choose\n⊢ X.OpenCover","tactic":"let t := this.choose","premises":[{"full_name":"Exists.choose","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[174,31],"def_end_pos":[174,44]}]},{"state_before":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\nthis : ∃ t, ⋃ x ∈ t, (fun x => Set.range ⇑(𝒰.map (𝒰.f x)).val.base) x = ⊤\nt : Finset ↑↑X.toPresheafedSpace := this.choose\n⊢ X.OpenCover","state_after":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\nthis : ∃ t, ⋃ x ∈ t, (fun x => Set.range ⇑(𝒰.map (𝒰.f x)).val.base) x = ⊤\nt : Finset ↑↑X.toPresheafedSpace := this.choose\nh : ∀ (x : ↑↑X.toPresheafedSpace), ∃ y, x ∈ Set.range ⇑(𝒰.map (𝒰.f ↑y)).val.base\n⊢ X.OpenCover","tactic":"have h : ∀ x : X, ∃ y : t, x ∈ Set.range (𝒰.map (𝒰.f y)).1.base := by\n    intro x\n    have h' : x ∈ (⊤ : Set X) := trivial\n    rw [← Classical.choose_spec this, Set.mem_iUnion] at h'\n    rcases h' with ⟨y, _, ⟨hy, rfl⟩, hy'⟩\n    exact ⟨⟨y, hy⟩, hy'⟩","premises":[{"full_name":"AlgebraicGeometry.LocallyRingedSpace.Hom.val","def_path":"Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean","def_pos":[75,2],"def_end_pos":[75,5]},{"full_name":"AlgebraicGeometry.PresheafedSpace.Hom.base","def_path":"Mathlib/Geometry/RingedSpace/PresheafedSpace.lean","def_pos":[90,2],"def_end_pos":[90,6]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.f","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"Classical.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[28,8],"def_end_pos":[28,19]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"Set.mem_iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[254,8],"def_end_pos":[254,18]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"Top.top","def_path":"Mathlib/Order/Notation.lean","def_pos":[94,2],"def_end_pos":[94,5]},{"full_name":"trivial","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[645,34],"def_end_pos":[645,41]}]},{"state_before":"X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↑↑X.toPresheafedSpace\nthis : ∃ t, ⋃ x ∈ t, (fun x => Set.range ⇑(𝒰.map (𝒰.f x)).val.base) x = ⊤\nt : Finset ↑↑X.toPresheafedSpace := this.choose\nh : ∀ (x : ↑↑X.toPresheafedSpace), ∃ y, x ∈ Set.range ⇑(𝒰.map (𝒰.f ↑y)).val.base\n⊢ X.OpenCover","state_after":"no goals","tactic":"exact\n    { J := t\n      obj := fun x => 𝒰.obj (𝒰.f x.1)\n      map := fun x => 𝒰.map (𝒰.f x.1)\n      f := fun x => (h x).choose\n      covers := fun x => (h x).choose_spec }","premises":[{"full_name":"AlgebraicGeometry.Scheme.OpenCover.f","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[50,2],"def_end_pos":[50,3]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.map","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[48,2],"def_end_pos":[48,5]},{"full_name":"AlgebraicGeometry.Scheme.OpenCover.obj","def_path":"Mathlib/AlgebraicGeometry/Cover/Open.lean","def_pos":[46,2],"def_end_pos":[46,5]},{"full_name":"Exists.choose","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[174,31],"def_end_pos":[174,44]},{"full_name":"Exists.choose_spec","def_path":".lake/packages/lean4/src/lean/Init/Classical.lean","def_pos":[177,8],"def_end_pos":[177,26]},{"full_name":"Subtype.val","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[587,2],"def_end_pos":[587,5]}]}]}
{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"intervalIntegral.norm_integral_le_of_norm_le_const_ae","start":[510,0],"end":[515,54],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ : Measure ℝ\na b C : ℝ\nf : ℝ → E\nh : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C\n⊢ ‖∫ (x : ℝ) in a..b, f x‖ ≤ C * |b - a|","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ : Measure ℝ\na b C : ℝ\nf : ℝ → E\nh : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C\n⊢ ‖∫ (x : ℝ) in Ι a b, f x ∂volume‖ ≤ C * |b - a|","tactic":"rw [norm_integral_eq_norm_integral_Ioc]","premises":[{"full_name":"intervalIntegral.norm_integral_eq_norm_integral_Ioc","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[483,8],"def_end_pos":[483,42]}]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ : Measure ℝ\na b C : ℝ\nf : ℝ → E\nh : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C\n⊢ ‖∫ (x : ℝ) in Ι a b, f x ∂volume‖ ≤ C * |b - a|","state_after":"case h.e'_4\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ : Measure ℝ\na b C : ℝ\nf : ℝ → E\nh : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C\n⊢ C * |b - a| = C * (volume (Ioc (min a b) (max a b))).toReal\n\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ : Measure ℝ\na b C : ℝ\nf : ℝ → E\nh : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C\n⊢ volume (Ioc (min a b) (max a b)) < ⊤","tactic":"convert norm_setIntegral_le_of_norm_le_const_ae'' _ measurableSet_Ioc h using 1","premises":[{"full_name":"MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae''","def_path":"Mathlib/MeasureTheory/Integral/SetIntegral.lean","def_pos":[596,8],"def_end_pos":[596,49]},{"full_name":"measurableSet_Ioc","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[178,8],"def_end_pos":[178,25]}]}]}
{"url":"Mathlib/Order/Minimal.lean","commit":"","full_name":"map_mem_maximals_iff","start":[299,0],"end":[300,61],"file_path":"Mathlib/Order/Minimal.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nr✝ r₁ r₂ : α → α → Prop\ns✝ t : Set α\na✝ b x✝² : α\nf : α → β\nr : α → α → Prop\ns : β → β → Prop\nx : Set α\nhf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))\na : α\nha : a ∈ x\nx✝¹ x✝ : α\nh₁ : x✝¹ ∈ x\nh₂ : x✝ ∈ x\n⊢ r x✝ x✝¹ ↔ s (f x✝) (f x✝¹)","state_after":"no goals","tactic":"exact hf h₂ h₁","premises":[]}]}
{"url":"Mathlib/Topology/Compactness/LocallyCompact.lean","commit":"","full_name":"ClosedEmbedding.locallyCompactSpace","start":[161,0],"end":[166,60],"file_path":"Mathlib/Topology/Compactness/LocallyCompact.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : LocallyCompactSpace Y\nf : X → Y\nhf : ClosedEmbedding f\nx : X\n⊢ (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) fun x => f ⁻¹' x","state_after":"X : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : LocallyCompactSpace Y\nf : X → Y\nhf : ClosedEmbedding f\nx : X\n⊢ (comap f (𝓝 (f x))).HasBasis (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) fun x => f ⁻¹' x","tactic":"rw [hf.toInducing.nhds_eq_comap]","premises":[{"full_name":"Inducing.nhds_eq_comap","def_path":"Mathlib/Topology/Maps/Basic.lean","def_pos":[82,8],"def_end_pos":[82,21]}]},{"state_before":"X : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : LocallyCompactSpace Y\nf : X → Y\nhf : ClosedEmbedding f\nx : X\n⊢ (comap f (𝓝 (f x))).HasBasis (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) fun x => f ⁻¹' x","state_after":"no goals","tactic":"exact (compact_basis_nhds _).comap _","premises":[{"full_name":"Filter.HasBasis.comap","def_path":"Mathlib/Order/Filter/Bases.lean","def_pos":[697,8],"def_end_pos":[697,22]},{"full_name":"compact_basis_nhds","def_path":"Mathlib/Topology/Compactness/LocallyCompact.lean","def_pos":[54,8],"def_end_pos":[54,26]}]}]}
{"url":"Mathlib/Data/Nat/Lattice.lean","commit":"","full_name":"Nat.nonempty_of_pos_sInf","start":[80,0],"end":[87,12],"file_path":"Mathlib/Data/Nat/Lattice.lean","tactics":[{"state_before":"s : Set ℕ\nh : 0 < sInf s\n⊢ s.Nonempty","state_after":"s : Set ℕ\nh : 0 < sInf s\ncontra : ¬s.Nonempty\n⊢ False","tactic":"by_contra contra","premises":[{"full_name":"Decidable.byContradiction","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[880,8],"def_end_pos":[880,23]},{"full_name":"Not","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[224,4],"def_end_pos":[224,7]}]},{"state_before":"s : Set ℕ\nh : 0 < sInf s\ncontra : ¬s.Nonempty\n⊢ False","state_after":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\n⊢ False","tactic":"rw [Set.not_nonempty_iff_eq_empty] at contra","premises":[{"full_name":"Set.not_nonempty_iff_eq_empty","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[481,8],"def_end_pos":[481,33]}]},{"state_before":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\n⊢ False","state_after":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ False","tactic":"have h' : sInf s ≠ 0 := ne_of_gt h","premises":[{"full_name":"InfSet.sInf","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[48,2],"def_end_pos":[48,6]},{"full_name":"Nat.ne_of_gt","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean","def_pos":[454,8],"def_end_pos":[454,16]},{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]}]},{"state_before":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ False","state_after":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ sInf s = 0","tactic":"apply h'","premises":[]},{"state_before":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ sInf s = 0","state_after":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ 0 ∈ s ∨ s = ∅","tactic":"rw [Nat.sInf_eq_zero]","premises":[{"full_name":"Nat.sInf_eq_zero","def_path":"Mathlib/Data/Nat/Lattice.lean","def_pos":[45,8],"def_end_pos":[45,20]}]},{"state_before":"s : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ 0 ∈ s ∨ s = ∅","state_after":"case h\ns : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ s = ∅","tactic":"right","premises":[]},{"state_before":"case h\ns : Set ℕ\nh : 0 < sInf s\ncontra : s = ∅\nh' : sInf s ≠ 0\n⊢ s = ∅","state_after":"no goals","tactic":"assumption","premises":[]}]}
{"url":"Mathlib/Topology/Algebra/InfiniteSum/Order.lean","commit":"","full_name":"Finite.of_summable_const","start":[268,0],"end":[277,16],"file_path":"Mathlib/Topology/Algebra/InfiniteSum/Order.lean","tactics":[{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\n⊢ Finite ι","state_after":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\n⊢ Finite ι","tactic":"have H : ∀ s : Finset ι, s.card • b ≤ ∑' _ : ι, b := fun s ↦ by\n    simpa using sum_le_hasSum s (fun a _ ↦ hb.le) hf.hasSum","premises":[{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"Summable.hasSum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[152,2],"def_end_pos":[152,13]},{"full_name":"sum_le_hasSum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Order.lean","def_pos":[86,2],"def_end_pos":[86,13]},{"full_name":"tsum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[94,2],"def_end_pos":[94,13]}]},{"state_before":"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\n⊢ Finite ι","state_after":"case intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b\n⊢ Finite ι","tactic":"obtain ⟨n, hn⟩ := Archimedean.arch (∑' _ : ι, b) hb","premises":[{"full_name":"Archimedean.arch","def_path":"Mathlib/Algebra/Order/Archimedean.lean","def_pos":[39,2],"def_end_pos":[39,6]},{"full_name":"tsum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[94,2],"def_end_pos":[94,13]}]},{"state_before":"case intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b\n⊢ Finite ι","state_after":"case intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b\nthis : ∀ (s : Finset ι), s.card ≤ n\n⊢ Finite ι","tactic":"have : ∀ s : Finset ι, s.card ≤ n := fun s ↦ by\n    simpa [nsmul_le_nsmul_iff_left hb] using (H s).trans hn","premises":[{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card","def_path":"Mathlib/Data/Finset/Card.lean","def_pos":[40,4],"def_end_pos":[40,8]},{"full_name":"nsmul_le_nsmul_iff_left","def_path":"Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean","def_pos":[212,14],"def_end_pos":[212,37]}]},{"state_before":"case intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b\nthis : ∀ (s : Finset ι), s.card ≤ n\n⊢ Finite ι","state_after":"case intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b\nthis✝ : ∀ (s : Finset ι), s.card ≤ n\nthis : Fintype ι\n⊢ Finite ι","tactic":"have : Fintype ι := fintypeOfFinsetCardLe n this","premises":[{"full_name":"Fintype","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[55,6],"def_end_pos":[55,13]},{"full_name":"fintypeOfFinsetCardLe","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[1010,18],"def_end_pos":[1010,39]}]},{"state_before":"case intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x => b\nH : ∀ (s : Finset ι), s.card • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b\nthis✝ : ∀ (s : Finset ι), s.card ≤ n\nthis : Fintype ι\n⊢ Finite ι","state_after":"no goals","tactic":"infer_instance","premises":[{"full_name":"inferInstance","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[99,7],"def_end_pos":[99,20]}]}]}
{"url":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","commit":"","full_name":"ValuationSubring.valuation_eq_one_iff","start":[180,0],"end":[187,61],"file_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","tactics":[{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\n⊢ IsUnit a","state_after":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\n⊢ IsUnit a","tactic":"have ha : (a : K) ≠ 0 := by\n      intro c\n      rw [c, A.valuation.map_zero] at h\n      exact zero_ne_one h","premises":[{"full_name":"Ne","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[689,17],"def_end_pos":[689,19]},{"full_name":"Valuation.map_zero","def_path":"Mathlib/RingTheory/Valuation/Basic.lean","def_pos":[147,8],"def_end_pos":[147,16]},{"full_name":"ValuationSubring.valuation","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[154,4],"def_end_pos":[154,13]},{"full_name":"zero_ne_one","def_path":"Mathlib/Algebra/NeZero.lean","def_pos":[47,14],"def_end_pos":[47,25]}]},{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\n⊢ IsUnit a","state_after":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\nha' : (↑a)⁻¹ ∈ A\n⊢ IsUnit a","tactic":"have ha' : (a : K)⁻¹ ∈ A := by rw [← valuation_le_one_iff, map_inv₀, h, inv_one]","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"ValuationSubring.valuation_le_one_iff","def_path":"Mathlib/RingTheory/Valuation/ValuationSubring.lean","def_pos":[166,8],"def_end_pos":[166,28]},{"full_name":"inv_one","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[951,8],"def_end_pos":[951,15]},{"full_name":"map_inv₀","def_path":"Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean","def_pos":[59,8],"def_end_pos":[59,16]}]},{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\nha' : (↑a)⁻¹ ∈ A\n⊢ IsUnit a","state_after":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\nha' : (↑a)⁻¹ ∈ A\n⊢ a * ⟨(↑a)⁻¹, ha'⟩ = 1","tactic":"apply isUnit_of_mul_eq_one a ⟨a⁻¹, ha'⟩","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"isUnit_of_mul_eq_one","def_path":"Mathlib/Algebra/Group/Units.lean","def_pos":[563,8],"def_end_pos":[563,28]}]},{"state_before":"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\nha' : (↑a)⁻¹ ∈ A\n⊢ a * ⟨(↑a)⁻¹, ha'⟩ = 1","state_after":"case a\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\nha' : (↑a)⁻¹ ∈ A\n⊢ ↑(a * ⟨(↑a)⁻¹, ha'⟩) = ↑1","tactic":"ext","premises":[]},{"state_before":"case a\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nha : ↑a ≠ 0\nha' : (↑a)⁻¹ ∈ A\n⊢ ↑(a * ⟨(↑a)⁻¹, ha'⟩) = ↑1","state_after":"no goals","tactic":"field_simp","premises":[]}]}
{"url":"Mathlib/Data/PFun.lean","commit":"","full_name":"PFun.comp_assoc","start":[523,0],"end":[525,58],"file_path":"Mathlib/Data/PFun.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : γ →. δ\ng : β →. γ\nh : α →. β\nx✝¹ : α\nx✝ : δ\n⊢ x✝ ∈ (f.comp g).comp h x✝¹ ↔ x✝ ∈ f.comp (g.comp h) x✝¹","state_after":"no goals","tactic":"simp only [comp_apply, Part.bind_comp]","premises":[{"full_name":"PFun.Part.bind_comp","def_path":"Mathlib/Data/PFun.lean","def_pos":[516,8],"def_end_pos":[516,22]},{"full_name":"PFun.comp_apply","def_path":"Mathlib/Data/PFun.lean","def_pos":[489,8],"def_end_pos":[489,18]}]}]}
{"url":"Mathlib/Data/Fin/Tuple/Basic.lean","commit":"","full_name":"Fin.init_snoc","start":[477,0],"end":[481,27],"file_path":"Mathlib/Data/Fin/Tuple/Basic.lean","tactics":[{"state_before":"m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α i.castSucc\ni : Fin n\ny : α i.castSucc\nz : α (last n)\n⊢ init (snoc p x) = p","state_after":"case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α i.castSucc\ni✝ : Fin n\ny : α i✝.castSucc\nz : α (last n)\ni : Fin n\n⊢ init (snoc p x) i = p i","tactic":"ext i","premises":[]},{"state_before":"case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α i.castSucc\ni✝ : Fin n\ny : α i✝.castSucc\nz : α (last n)\ni : Fin n\n⊢ init (snoc p x) i = p i","state_after":"case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α i.castSucc\ni✝ : Fin n\ny : α i✝.castSucc\nz : α (last n)\ni : Fin n\n⊢ p (i.castSucc.castLT ⋯) = p i","tactic":"simp only [init, snoc, coe_castSucc, is_lt, cast_eq, dite_true]","premises":[{"full_name":"Fin.coe_castSucc","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[370,16],"def_end_pos":[370,28]},{"full_name":"Fin.init","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[464,4],"def_end_pos":[464,8]},{"full_name":"Fin.is_lt","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[31,16],"def_end_pos":[31,21]},{"full_name":"Fin.snoc","def_path":"Mathlib/Data/Fin/Tuple/Basic.lean","def_pos":[473,4],"def_end_pos":[473,8]},{"full_name":"cast_eq","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[682,16],"def_end_pos":[682,23]},{"full_name":"dite_true","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[93,16],"def_end_pos":[93,25]}]},{"state_before":"case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α i.castSucc\ni✝ : Fin n\ny : α i✝.castSucc\nz : α (last n)\ni : Fin n\n⊢ p (i.castSucc.castLT ⋯) = p i","state_after":"no goals","tactic":"convert cast_eq rfl (p i)","premises":[{"full_name":"cast_eq","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[682,16],"def_end_pos":[682,23]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]}]}
{"url":"Mathlib/Algebra/Polynomial/Derivative.lean","commit":"","full_name":"Polynomial.derivative_comp_one_sub_X","start":[543,0],"end":[544,89],"file_path":"Mathlib/Algebra/Polynomial/Derivative.lean","tactics":[{"state_before":"R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\n⊢ derivative (p.comp (1 - X)) = -(derivative p).comp (1 - X)","state_after":"no goals","tactic":"simp [derivative_comp]","premises":[{"full_name":"Polynomial.derivative_comp","def_path":"Mathlib/Algebra/Polynomial/Derivative.lean","def_pos":[455,8],"def_end_pos":[455,23]}]}]}
{"url":"Mathlib/Computability/Language.lean","commit":"","full_name":"Language.map_map","start":[154,0],"end":[156,25],"file_path":"Mathlib/Computability/Language.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\ng : β → γ\nf : α → β\nl : Language α\n⊢ (map g) ((map f) l) = (map (g ∘ f)) l","state_after":"no goals","tactic":"simp [map, image_image]","premises":[{"full_name":"Language.map","def_path":"Mathlib/Computability/Language.lean","def_pos":[144,4],"def_end_pos":[144,7]},{"full_name":"Set.image_image","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[232,8],"def_end_pos":[232,19]}]}]}
{"url":"Mathlib/Topology/NhdsSet.lean","commit":"","full_name":"eventually_nhdsSet_iUnion₂","start":[192,0],"end":[194,45],"file_path":"Mathlib/Topology/NhdsSet.lean","tactics":[{"state_before":"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : Filter X\ns✝ t s₁ s₂ t₁ t₂ : Set X\nx : X\nι : Sort u_3\np : ι → Prop\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, ⋃ (_ : p i), s i), P x) ↔ ∀ (i : ι), p i → ∀ᶠ (x : X) in 𝓝ˢ (s i), P x","state_after":"no goals","tactic":"simp only [nhdsSet_iUnion, eventually_iSup]","premises":[{"full_name":"Filter.eventually_iSup","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1078,8],"def_end_pos":[1078,23]},{"full_name":"nhdsSet_iUnion","def_path":"Mathlib/Topology/NhdsSet.lean","def_pos":[189,8],"def_end_pos":[189,22]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean","commit":"","full_name":"CategoryTheory.ShortComplex.SnakeInput.naturality_δ","start":[490,0],"end":[493,91],"file_path":"Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nS S₁ S₂ S₃ : SnakeInput C\nf : S₁ ⟶ S₂\n⊢ S₁.δ ≫ f.f₃.τ₁ = f.f₀.τ₃ ≫ S₂.δ","state_after":"no goals","tactic":"rw [← cancel_epi (pullback.snd _ _ : S₁.P ⟶ _), S₁.snd_δ_assoc, ← comp_τ₁, ← f.comm₂₃,\n    comp_τ₁, naturality_φ₁_assoc, ← S₂.snd_δ, functorP_map, pullback.lift_snd_assoc, assoc]","premises":[{"full_name":"CategoryTheory.Category.assoc","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[155,2],"def_end_pos":[155,7]},{"full_name":"CategoryTheory.Limits.pullback.snd","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean","def_pos":[112,7],"def_end_pos":[112,19]},{"full_name":"CategoryTheory.ShortComplex.SnakeInput.Hom.comm₂₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean","def_pos":[395,2],"def_end_pos":[395,8]},{"full_name":"CategoryTheory.ShortComplex.SnakeInput.P","def_path":"Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean","def_pos":[221,18],"def_end_pos":[221,19]},{"full_name":"CategoryTheory.ShortComplex.SnakeInput.functorP_map","def_path":"Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean","def_pos":[473,2],"def_end_pos":[473,7]},{"full_name":"CategoryTheory.ShortComplex.SnakeInput.snd_δ","def_path":"Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean","def_pos":[270,6],"def_end_pos":[270,11]},{"full_name":"CategoryTheory.ShortComplex.comp_τ₁","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[102,17],"def_end_pos":[102,24]},{"full_name":"CategoryTheory.cancel_epi","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[260,8],"def_end_pos":[260,18]},{"full_name":"Quiver.Hom","def_path":"Mathlib/Combinatorics/Quiver/Basic.lean","def_pos":[43,2],"def_end_pos":[43,5]}]}]}
{"url":"Mathlib/Order/Filter/Basic.lean","commit":"","full_name":"Filter.frequently_or_distrib","start":[1182,0],"end":[1185,69],"file_path":"Mathlib/Order/Filter/Basic.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nf : Filter α\np q : α → Prop\n⊢ (∃ᶠ (x : α) in f, p x ∨ q x) ↔ (∃ᶠ (x : α) in f, p x) ∨ ∃ᶠ (x : α) in f, q x","state_after":"no goals","tactic":"simp only [Filter.Frequently, ← not_and_or, not_or, eventually_and]","premises":[{"full_name":"Filter.Frequently","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1102,14],"def_end_pos":[1102,24]},{"full_name":"Filter.eventually_and","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1011,8],"def_end_pos":[1011,22]},{"full_name":"not_and_or","def_path":"Mathlib/Logic/Basic.lean","def_pos":[339,8],"def_end_pos":[339,18]},{"full_name":"not_or","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[134,16],"def_end_pos":[134,22]}]}]}
{"url":"Mathlib/GroupTheory/FreeGroup/Basic.lean","commit":"","full_name":"FreeGroup.lift.range_eq_closure","start":[626,0],"end":[631,48],"file_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","tactics":[{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ (lift f).range = Subgroup.closure (Set.range f)","state_after":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Subgroup.closure (Set.range f) ≤ (lift f).range","tactic":"apply le_antisymm (lift.range_le Subgroup.subset_closure)","premises":[{"full_name":"FreeGroup.lift.range_le","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[620,8],"def_end_pos":[620,21]},{"full_name":"Subgroup.subset_closure","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[853,8],"def_end_pos":[853,22]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Subgroup.closure (Set.range f) ≤ (lift f).range","state_after":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Set.range f ⊆ ↑(lift f).range","tactic":"rw [Subgroup.closure_le]","premises":[{"full_name":"Subgroup.closure_le","def_path":"Mathlib/Algebra/Group/Subgroup/Basic.lean","def_pos":[864,8],"def_end_pos":[864,18]}]},{"state_before":"α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Set.range f ⊆ ↑(lift f).range","state_after":"case intro\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\na : α\n⊢ f a ∈ ↑(lift f).range","tactic":"rintro _ ⟨a, rfl⟩","premises":[]},{"state_before":"case intro\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\na : α\n⊢ f a ∈ ↑(lift f).range","state_after":"no goals","tactic":"exact ⟨FreeGroup.of a, by simp only [lift.of]⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"FreeGroup.lift.of","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[594,8],"def_end_pos":[594,15]},{"full_name":"FreeGroup.of","def_path":"Mathlib/GroupTheory/FreeGroup/Basic.lean","def_pos":[535,4],"def_end_pos":[535,6]}]}]}
{"url":"Mathlib/NumberTheory/LucasLehmer.lean","commit":"","full_name":"LucasLehmer.X.ωb_mul_ω","start":[356,0],"end":[357,25],"file_path":"Mathlib/NumberTheory/LucasLehmer.lean","tactics":[{"state_before":"q✝ q : ℕ+\n⊢ ωb * ω = 1","state_after":"no goals","tactic":"rw [mul_comm, ω_mul_ωb]","premises":[{"full_name":"LucasLehmer.X.ω_mul_ωb","def_path":"Mathlib/NumberTheory/LucasLehmer.lean","def_pos":[352,8],"def_end_pos":[352,16]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Data/Finset/Sups.lean","commit":"","full_name":"Finset.disjSups_union_left","start":[454,0],"end":[455,44],"file_path":"Mathlib/Data/Finset/Sups.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : DecidableEq α\ninst✝³ : DecidableEq β\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ (s₁ ∪ s₂) ○ t = s₁ ○ t ∪ s₂ ○ t","state_after":"no goals","tactic":"simp [disjSups, filter_union, image_union]","premises":[{"full_name":"Finset.disjSups","def_path":"Mathlib/Data/Finset/Sups.lean","def_pos":[392,4],"def_end_pos":[392,12]},{"full_name":"Finset.filter_union","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2297,8],"def_end_pos":[2297,20]},{"full_name":"Finset.image_union","def_path":"Mathlib/Data/Finset/Image.lean","def_pos":[437,8],"def_end_pos":[437,19]}]}]}
{"url":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","commit":"","full_name":"Finset.ruzsa_triangle_inequality_mul_div_mul","start":[59,0],"end":[64,51],"file_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A * C).card * B.card ≤ (A / B).card * (B * C).card","state_after":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A / C⁻¹).card * B.card ≤ (A / B).card * (B / C⁻¹).card","tactic":"rw [← div_inv_eq_mul, ← div_inv_eq_mul B]","premises":[{"full_name":"div_inv_eq_mul","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[501,8],"def_end_pos":[501,22]}]},{"state_before":"α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ (A / C⁻¹).card * B.card ≤ (A / B).card * (B / C⁻¹).card","state_after":"no goals","tactic":"exact ruzsa_triangle_inequality_div_div_div _ _ _","premises":[{"full_name":"Finset.ruzsa_triangle_inequality_div_div_div","def_path":"Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean","def_pos":[39,8],"def_end_pos":[39,45]}]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","commit":"","full_name":"NNReal.rpow_self_rpow_inv","start":[98,0],"end":[99,25],"file_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","tactics":[{"state_before":"w x✝ y✝ z y : ℝ\nhy : y ≠ 0\nx : ℝ≥0\n⊢ (x ^ (1 / y)) ^ y = x","state_after":"no goals","tactic":"field_simp [← rpow_mul]","premises":[{"full_name":"NNReal.rpow_mul","def_path":"Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean","def_pos":[81,8],"def_end_pos":[81,16]}]}]}
{"url":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","commit":"","full_name":"Equiv.Perm.IsThreeCycle.isThreeCycle_sq","start":[577,0],"end":[580,10],"file_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ g : Perm α\nht : g.IsThreeCycle\n⊢ (g * g).IsThreeCycle","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ g : Perm α\nht : g.IsThreeCycle\n⊢ Nat.Coprime 2 (_root_.orderOf g)","tactic":"rw [← pow_two, ← card_support_eq_three_iff, support_pow_coprime, ht.card_support]","premises":[{"full_name":"Equiv.Perm.IsThreeCycle.card_support","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[541,8],"def_end_pos":[541,20]},{"full_name":"Equiv.Perm.support_pow_coprime","def_path":"Mathlib/GroupTheory/Perm/Finite.lean","def_pos":[226,8],"def_end_pos":[226,27]},{"full_name":"card_support_eq_three_iff","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[544,8],"def_end_pos":[544,40]},{"full_name":"pow_two","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[581,31],"def_end_pos":[581,38]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ g : Perm α\nht : g.IsThreeCycle\n⊢ Nat.Coprime 2 (_root_.orderOf g)","state_after":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ g : Perm α\nht : g.IsThreeCycle\n⊢ Nat.Coprime 2 3","tactic":"rw [ht.orderOf]","premises":[{"full_name":"Equiv.Perm.IsThreeCycle.orderOf","def_path":"Mathlib/GroupTheory/Perm/Cycle/Type.lean","def_pos":[574,8],"def_end_pos":[574,15]}]},{"state_before":"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ g : Perm α\nht : g.IsThreeCycle\n⊢ Nat.Coprime 2 3","state_after":"no goals","tactic":"norm_num","premises":[]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","commit":"","full_name":"CategoryTheory.ShortComplex.opMap_τ₂","start":[224,0],"end":[235,35],"file_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","tactics":[{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.59011, u_1} C\ninst✝² : Category.{?u.59015, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nφ : S₁ ⟶ S₂\n⊢ φ.τ₃.op ≫ S₁.op.f = S₂.op.f ≫ φ.τ₂.op","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.59011, u_1} C\ninst✝² : Category.{?u.59015, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nφ : S₁ ⟶ S₂\n⊢ φ.τ₃.op ≫ S₁.g.op = S₂.g.op ≫ φ.τ₂.op","tactic":"dsimp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.59011, u_1} C\ninst✝² : Category.{?u.59015, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nφ : S₁ ⟶ S₂\n⊢ φ.τ₃.op ≫ S₁.g.op = S₂.g.op ≫ φ.τ₂.op","state_after":"no goals","tactic":"simp only [← op_comp, φ.comm₂₃]","premises":[{"full_name":"CategoryTheory.ShortComplex.Hom.comm₂₃","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[62,2],"def_end_pos":[62,8]},{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]}]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.59011, u_1} C\ninst✝² : Category.{?u.59015, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nφ : S₁ ⟶ S₂\n⊢ φ.τ₂.op ≫ S₁.op.g = S₂.op.g ≫ φ.τ₁.op","state_after":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.59011, u_1} C\ninst✝² : Category.{?u.59015, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nφ : S₁ ⟶ S₂\n⊢ φ.τ₂.op ≫ S₁.f.op = S₂.f.op ≫ φ.τ₁.op","tactic":"dsimp","premises":[]},{"state_before":"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{?u.59011, u_1} C\ninst✝² : Category.{?u.59015, u_2} D\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : HasZeroMorphisms D\nφ : S₁ ⟶ S₂\n⊢ φ.τ₂.op ≫ S₁.f.op = S₂.f.op ≫ φ.τ₁.op","state_after":"no goals","tactic":"simp only [← op_comp, φ.comm₁₂]","premises":[{"full_name":"CategoryTheory.ShortComplex.Hom.comm₁₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[60,2],"def_end_pos":[60,8]},{"full_name":"CategoryTheory.op_comp","def_path":"Mathlib/CategoryTheory/Opposites.lean","def_pos":[72,8],"def_end_pos":[72,15]}]}]}
{"url":"Mathlib/Analysis/NormedSpace/Pointwise.lean","commit":"","full_name":"infDist_smul₀","start":[63,0],"end":[66,16],"file_path":"Mathlib/Analysis/NormedSpace/Pointwise.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\nhc : c ≠ 0\ns : Set E\nx : E\n⊢ infDist (c • x) (c • s) = ‖c‖ * infDist x s","state_after":"no goals","tactic":"simp_rw [Metric.infDist, infEdist_smul₀ hc s, ENNReal.toReal_smul, NNReal.smul_def, coe_nnnorm,\n    smul_eq_mul]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"ENNReal.toReal_smul","def_path":"Mathlib/Data/ENNReal/Real.lean","def_pos":[408,8],"def_end_pos":[408,19]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Metric.infDist","def_path":"Mathlib/Topology/MetricSpace/HausdorffDistance.lean","def_pos":[433,4],"def_end_pos":[433,11]},{"full_name":"NNReal.smul_def","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[211,8],"def_end_pos":[211,16]},{"full_name":"coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[615,40],"def_end_pos":[615,50]},{"full_name":"infEdist_smul₀","def_path":"Mathlib/Analysis/NormedSpace/Pointwise.lean","def_pos":[52,8],"def_end_pos":[52,22]},{"full_name":"smul_eq_mul","def_path":"Mathlib/Algebra/Group/Action/Defs.lean","def_pos":[79,6],"def_end_pos":[79,17]}]}]}
{"url":"Mathlib/Analysis/Convolution.lean","commit":"","full_name":"MeasureTheory.convolution_neg_of_neg_eq","start":[664,0],"end":[674,43],"file_path":"Mathlib/Analysis/Convolution.lean","tactics":[{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\n⊢ ∫ (t : G), (L (f t)) (g (-x - t)) ∂μ = ∫ (t : G), (L (f (-t))) (g (x + t)) ∂μ","state_after":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\n⊢ (fun a => (L (f a)) (g (-x - a))) =ᶠ[ae μ] fun a => (L (f (-a))) (g (x + a))","tactic":"apply integral_congr_ae","premises":[{"full_name":"MeasureTheory.integral_congr_ae","def_path":"Mathlib/MeasureTheory/Integral/Bochner.lean","def_pos":[843,8],"def_end_pos":[843,25]}]},{"state_before":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\n⊢ (fun a => (L (f a)) (g (-x - a))) =ᶠ[ae μ] fun a => (L (f (-a))) (g (x + a))","state_after":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\nt : G\nht : f (-t) = f t\nh't : g (-(x + t)) = g (x + t)\n⊢ (L (f t)) (g (-x - t)) = (L (f (-t))) (g (x + t))","tactic":"filter_upwards [h1, (eventually_add_left_iff μ x).2 h2] with t ht h't","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Filter.mp_mem","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[159,8],"def_end_pos":[159,14]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.eventually_add_left_iff","def_path":"Mathlib/MeasureTheory/Group/Measure.lean","def_pos":[296,2],"def_end_pos":[296,13]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]}]},{"state_before":"case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\nt : G\nht : f (-t) = f t\nh't : g (-(x + t)) = g (x + t)\n⊢ (L (f t)) (g (-x - t)) = (L (f (-t))) (g (x + t))","state_after":"no goals","tactic":"simp_rw [ht, ← h't, neg_add']","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"neg_add'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[527,2],"def_end_pos":[527,13]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\n⊢ ∫ (t : G), (L (f (-t))) (g (x + t)) ∂μ = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ","state_after":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\n⊢ ∫ (x_1 : G), (L (f (- -x_1))) (g (x + -x_1)) ∂μ = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ","tactic":"rw [← integral_neg_eq_self]","premises":[{"full_name":"MeasureTheory.integral_neg_eq_self","def_path":"Mathlib/MeasureTheory/Group/Integral.lean","def_pos":[35,2],"def_end_pos":[35,13]}]},{"state_before":"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedAddCommGroup E'\ninst✝¹³ : NormedAddCommGroup E''\ninst✝¹² : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx x' : G\ny y' : E\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 E'\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝⁶ : MeasurableSpace G\nμ ν : Measure G\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : AddCommGroup G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : μ.IsNegInvariant\ninst✝¹ : MeasurableNeg G\ninst✝ : MeasurableAdd G\nh1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x\nh2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x\n⊢ ∫ (x_1 : G), (L (f (- -x_1))) (g (x + -x_1)) ∂μ = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ","state_after":"no goals","tactic":"simp only [neg_neg, ← sub_eq_add_neg]","premises":[{"full_name":"neg_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[733,2],"def_end_pos":[733,13]},{"full_name":"sub_eq_add_neg","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[905,2],"def_end_pos":[905,13]}]}]}
{"url":"Mathlib/Algebra/Order/Group/Basic.lean","commit":"","full_name":"zsmul_strictMono_left","start":[30,0],"end":[35,49],"file_path":"Mathlib/Algebra/Order/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 < a\nm n : ℤ\nh : m < n\n⊢ a ^ m * a ^ (n - m) = a ^ n","state_after":"no goals","tactic":"simp [← zpow_add, m.add_comm]","premises":[{"full_name":"Int.add_comm","def_path":".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean","def_pos":[135,18],"def_end_pos":[135,26]},{"full_name":"zpow_add","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[816,6],"def_end_pos":[816,14]}]}]}
{"url":"Mathlib/Data/Rat/Cast/Defs.lean","commit":"","full_name":"NNRat.cast_mul_of_ne_zero","start":[81,0],"end":[89,27],"file_path":"Mathlib/Data/Rat/Cast/Defs.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝ : DivisionSemiring α\nq r : ℚ≥0\nhq : ↑q.den ≠ 0\nhr : ↑r.den ≠ 0\n⊢ ↑(q * r) = ↑q * ↑r","state_after":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝ : DivisionSemiring α\nq r : ℚ≥0\nhq : ↑q.den ≠ 0\nhr : ↑r.den ≠ 0\n⊢ ↑(q.num * r.num) / ↑(q.den * r.den) = ↑q.num * ↑r.num / (↑q.den * ↑r.den)\n\ncase hb\nF : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝ : DivisionSemiring α\nq r : ℚ≥0\nhq : ↑q.den ≠ 0\nhr : ↑r.den ≠ 0\n⊢ ↑(q.den * r.den) ≠ 0","tactic":"rw [mul_def, cast_divNat_of_ne_zero, cast_def, cast_def,\n    (Nat.commute_cast _ _).div_mul_div_comm (Nat.commute_cast _ _)]","premises":[{"full_name":"Commute.div_mul_div_comm","def_path":"Mathlib/Algebra/Group/Commute/Basic.lean","def_pos":[34,18],"def_end_pos":[34,34]},{"full_name":"NNRat.cast_def","def_path":"Mathlib/Algebra/Field/Defs.lean","def_pos":[186,6],"def_end_pos":[186,14]},{"full_name":"NNRat.cast_divNat_of_ne_zero","def_path":"Mathlib/Data/Rat/Cast/Defs.lean","def_pos":[51,19],"def_end_pos":[51,41]},{"full_name":"NNRat.mul_def","def_path":"Mathlib/Data/NNRat/Defs.lean","def_pos":[367,6],"def_end_pos":[367,13]},{"full_name":"Nat.commute_cast","def_path":"Mathlib/Data/Nat/Cast/Commute.lean","def_pos":[33,8],"def_end_pos":[33,20]}]}]}
{"url":"Mathlib/Tactic/CategoryTheory/Elementwise.lean","commit":"","full_name":"Tactic.Elementwise.hom_elementwise","start":[52,0],"end":[53,72],"file_path":"Mathlib/Tactic/CategoryTheory/Elementwise.lean","tactics":[{"state_before":"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : ConcreteCategory C\nX Y : C\nf g : X ⟶ Y\nh : f = g\nx : (forget C).obj X\n⊢ f x = g x","state_after":"no goals","tactic":"rw [h]","premises":[]}]}
{"url":"Mathlib/Algebra/Group/Even.lean","commit":"","full_name":"Even.zsmul","start":[135,0],"end":[136,61],"file_path":"Mathlib/Algebra/Group/Even.lean","tactics":[{"state_before":"F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝ : DivisionMonoid α\na : α\nn : ℤ\n⊢ IsSquare a → IsSquare (a ^ n)","state_after":"case intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝ : DivisionMonoid α\nn : ℤ\na : α\n⊢ IsSquare ((a * a) ^ n)","tactic":"rintro ⟨a, rfl⟩","premises":[]},{"state_before":"case intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝ : DivisionMonoid α\nn : ℤ\na : α\n⊢ IsSquare ((a * a) ^ n)","state_after":"no goals","tactic":"exact ⟨a ^ n, (Commute.refl _).mul_zpow _⟩","premises":[{"full_name":"Commute.mul_zpow","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[193,16],"def_end_pos":[193,24]},{"full_name":"Commute.refl","def_path":"Mathlib/Algebra/Group/Commute/Defs.lean","def_pos":[57,18],"def_end_pos":[57,22]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","commit":"","full_name":"TensorProduct.map_id","start":[822,0],"end":[825,65],"file_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","tactics":[{"state_before":"R : Type u_1\ninst✝²⁰ : CommSemiring R\nR' : Type u_2\ninst✝¹⁹ : Monoid R'\nR'' : Type u_3\ninst✝¹⁸ : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : AddCommMonoid P\ninst✝¹⁴ : AddCommMonoid Q\ninst✝¹³ : AddCommMonoid S\ninst✝¹² : AddCommMonoid T\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\ninst✝⁹ : Module R P\ninst✝⁸ : Module R Q\ninst✝⁷ : Module R S\ninst✝⁶ : Module R T\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\n⊢ map LinearMap.id LinearMap.id = LinearMap.id","state_after":"case H.h.h\nR : Type u_1\ninst✝²⁰ : CommSemiring R\nR' : Type u_2\ninst✝¹⁹ : Monoid R'\nR'' : Type u_3\ninst✝¹⁸ : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : AddCommMonoid P\ninst✝¹⁴ : AddCommMonoid Q\ninst✝¹³ : AddCommMonoid S\ninst✝¹² : AddCommMonoid T\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\ninst✝⁹ : Module R P\ninst✝⁸ : Module R Q\ninst✝⁷ : Module R S\ninst✝⁶ : Module R T\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nx✝¹ : M\nx✝ : N\n⊢ (((mk R M N).compr₂ (map LinearMap.id LinearMap.id)) x✝¹) x✝ = (((mk R M N).compr₂ LinearMap.id) x✝¹) x✝","tactic":"ext","premises":[]},{"state_before":"case H.h.h\nR : Type u_1\ninst✝²⁰ : CommSemiring R\nR' : Type u_2\ninst✝¹⁹ : Monoid R'\nR'' : Type u_3\ninst✝¹⁸ : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : AddCommMonoid P\ninst✝¹⁴ : AddCommMonoid Q\ninst✝¹³ : AddCommMonoid S\ninst✝¹² : AddCommMonoid T\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\ninst✝⁹ : Module R P\ninst✝⁸ : Module R Q\ninst✝⁷ : Module R S\ninst✝⁶ : Module R T\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nx✝¹ : M\nx✝ : N\n⊢ (((mk R M N).compr₂ (map LinearMap.id LinearMap.id)) x✝¹) x✝ = (((mk R M N).compr₂ LinearMap.id) x✝¹) x✝","state_after":"no goals","tactic":"simp only [mk_apply, id_coe, compr₂_apply, _root_.id, map_tmul]","premises":[{"full_name":"LinearMap.compr₂_apply","def_path":"Mathlib/LinearAlgebra/BilinearMap.lean","def_pos":[347,8],"def_end_pos":[347,20]},{"full_name":"LinearMap.id_coe","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[282,8],"def_end_pos":[282,14]},{"full_name":"TensorProduct.map_tmul","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[733,8],"def_end_pos":[733,16]},{"full_name":"TensorProduct.mk_apply","def_path":"Mathlib/LinearAlgebra/TensorProduct/Basic.lean","def_pos":[421,8],"def_end_pos":[421,16]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]}]}]}
{"url":"Mathlib/CategoryTheory/Filtered/Basic.lean","commit":"","full_name":"CategoryTheory.IsCofilteredOrEmpty.of_left_adjoint","start":[636,0],"end":[645,100],"file_path":"Mathlib/CategoryTheory/Filtered/Basic.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : IsCofilteredOrEmpty C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX Y : D\nf g : X ⟶ Y\n⊢ (h.homEquiv (eq (R.map f) (R.map g)) X).symm (eqHom (R.map f) (R.map g)) ≫ f =\n    (h.homEquiv (eq (R.map f) (R.map g)) X).symm (eqHom (R.map f) (R.map g)) ≫ g","state_after":"no goals","tactic":"rw [← h.homEquiv_naturality_right_symm, ← h.homEquiv_naturality_right_symm, eq_condition]","premises":[{"full_name":"CategoryTheory.Adjunction.homEquiv_naturality_right_symm","def_path":"Mathlib/CategoryTheory/Adjunction/Basic.lean","def_pos":[160,8],"def_end_pos":[160,38]},{"full_name":"CategoryTheory.IsCofiltered.eq_condition","def_path":"Mathlib/CategoryTheory/Filtered/Basic.lean","def_pos":[609,8],"def_end_pos":[609,20]}]}]}
{"url":"Mathlib/Tactic/Ring/Basic.lean","commit":"","full_name":"Mathlib.Tactic.Ring.atom_pf","start":[913,0],"end":[913,90],"file_path":"Mathlib/Tactic/Ring/Basic.lean","tactics":[{"state_before":"u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR : Type u_1\ninst✝ : CommSemiring R\na✝ a' a₁ a₂ a₃ b b' b₁ b₂ b₃ c c₁ c₂ a : R\n⊢ a = a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean","commit":"","full_name":"ProjectiveSpectrum.basicOpen_one","start":[347,0],"end":[349,39],"file_path":"Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean","tactics":[{"state_before":"R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ ↑(basicOpen 𝒜 1) = ↑⊤","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean","commit":"","full_name":"Real.arccos_le_pi_div_four","start":[366,0],"end":[371,14],"file_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean","tactics":[{"state_before":"x✝ y x : ℝ\n⊢ arccos x ≤ π / 4 ↔ √2 / 2 ≤ x","state_after":"x✝ y x : ℝ\n⊢ π / 2 - arcsin x ≤ π / 4 ↔ π / 4 ≤ arcsin x","tactic":"rw [arccos, ← pi_div_four_le_arcsin]","premises":[{"full_name":"Real.arccos","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean","def_pos":[284,18],"def_end_pos":[284,24]},{"full_name":"Real.pi_div_four_le_arcsin","def_path":"Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean","def_pos":[226,8],"def_end_pos":[226,29]}]}]}
{"url":"Mathlib/Data/Set/Lattice.lean","commit":"","full_name":"Set.biUnion_preimage_singleton","start":[1468,0],"end":[1470,47],"file_path":"Mathlib/Data/Set/Lattice.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\nf : α → β\ns : Set β\n⊢ ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s","state_after":"no goals","tactic":"rw [← preimage_iUnion₂, biUnion_of_singleton]","premises":[{"full_name":"Set.biUnion_of_singleton","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[792,8],"def_end_pos":[792,28]},{"full_name":"Set.preimage_iUnion₂","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[1440,8],"def_end_pos":[1440,24]}]}]}
{"url":"Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean","commit":"","full_name":"CategoryTheory.PresheafOfGroups.OneCocycle.equivalence_isCohomologous","start":[183,0],"end":[191,25],"file_path":"Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean","tactics":[{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\n⊢ ∀ {x y : OneCocycle G U}, x.IsCohomologous y → y.IsCohomologous x","state_after":"case intro\nC : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\nγ₁ γ₂ : OneCocycle G U\nα : ZeroCochain G U\nh : OneCohomologyRelation γ₁.toOneCochain γ₂.toOneCochain α\n⊢ γ₂.IsCohomologous γ₁","tactic":"rintro γ₁ γ₂ ⟨α, h⟩","premises":[]},{"state_before":"case intro\nC : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\nγ₁ γ₂ : OneCocycle G U\nα : ZeroCochain G U\nh : OneCohomologyRelation γ₁.toOneCochain γ₂.toOneCochain α\n⊢ γ₂.IsCohomologous γ₁","state_after":"no goals","tactic":"exact ⟨_, h.symm⟩","premises":[{"full_name":"CategoryTheory.PresheafOfGroups.OneCohomologyRelation.symm","def_path":"Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean","def_pos":[160,6],"def_end_pos":[160,10]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]},{"state_before":"C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\n⊢ ∀ {x y z : OneCocycle G U}, x.IsCohomologous y → y.IsCohomologous z → x.IsCohomologous z","state_after":"case intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\nγ₁ γ₂✝ γ₂ : OneCocycle G U\nα : ZeroCochain G U\nh : OneCohomologyRelation γ₁.toOneCochain γ₂✝.toOneCochain α\nβ : ZeroCochain G U\nh' : OneCohomologyRelation γ₂✝.toOneCochain γ₂.toOneCochain β\n⊢ γ₁.IsCohomologous γ₂","tactic":"rintro γ₁ γ₂ γ₂ ⟨α, h⟩ ⟨β, h'⟩","premises":[]},{"state_before":"case intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\nγ₁ γ₂✝ γ₂ : OneCocycle G U\nα : ZeroCochain G U\nh : OneCohomologyRelation γ₁.toOneCochain γ₂✝.toOneCochain α\nβ : ZeroCochain G U\nh' : OneCohomologyRelation γ₂✝.toOneCochain γ₂.toOneCochain β\n⊢ γ₁.IsCohomologous γ₂","state_after":"no goals","tactic":"exact ⟨_, h.trans h'⟩","premises":[{"full_name":"CategoryTheory.PresheafOfGroups.OneCohomologyRelation.trans","def_path":"Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean","def_pos":[166,6],"def_end_pos":[166,11]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]}]}]}
{"url":"Mathlib/Topology/Instances/Complex.lean","commit":"","full_name":"Complex.subfield_eq_of_closed","start":[24,0],"end":[44,5],"file_path":"Mathlib/Topology/Instances/Complex.lean","tactics":[{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ K = ofReal.fieldRange ∨ K = ⊤","state_after":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ↑K","tactic":"suffices range (ofReal' : ℝ → ℂ) ⊆ K by\n    rw [range_subset_iff, ← coe_algebraMap] at this\n    have :=\n      (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top\n        (Subfield.toIntermediateField K this).toSubalgebra\n    simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] at this ⊢\n    exact this","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.coe_algebraMap","def_path":"Mathlib/Data/Complex/Module.lean","def_pos":[104,8],"def_end_pos":[104,22]},{"full_name":"Complex.finrank_real_complex","def_path":"Mathlib/Data/Complex/FiniteDimensional.lean","def_pos":[26,8],"def_end_pos":[26,28]},{"full_name":"Complex.ofReal'","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[79,4],"def_end_pos":[79,11]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"IntermediateField.coe_toSubalgebra","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[90,8],"def_end_pos":[90,24]},{"full_name":"IsSimpleOrder.eq_bot_or_eq_top","def_path":"Mathlib/Order/Atoms.lean","def_pos":[569,2],"def_end_pos":[569,18]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"Set.range_subset_iff","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[623,8],"def_end_pos":[623,24]},{"full_name":"SetLike.coe_set_eq","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[149,8],"def_end_pos":[149,18]},{"full_name":"Subalgebra.isSimpleOrder_of_finrank","def_path":"Mathlib/LinearAlgebra/FiniteDimensional.lean","def_pos":[303,8],"def_end_pos":[303,43]},{"full_name":"Subfield.toIntermediateField","def_path":"Mathlib/FieldTheory/IntermediateField.lean","def_pos":[282,4],"def_end_pos":[282,32]}]},{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ↑K","state_after":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast))","tactic":"suffices range (ofReal' : ℝ → ℂ) ⊆ closure (Set.range ((ofReal' : ℝ → ℂ) ∘ ((↑) : ℚ → ℝ))) by\n    refine subset_trans this ?_\n    rw [← IsClosed.closure_eq hc]\n    apply closure_mono\n    rintro _ ⟨_, rfl⟩\n    simp only [Function.comp_apply, ofReal_ratCast, SetLike.mem_coe, SubfieldClass.ratCast_mem]","premises":[{"full_name":"Complex","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[27,10],"def_end_pos":[27,17]},{"full_name":"Complex.ofReal'","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[79,4],"def_end_pos":[79,11]},{"full_name":"Complex.ofReal_ratCast","def_path":"Mathlib/Data/Complex/Basic.lean","def_pos":[427,25],"def_end_pos":[427,39]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]},{"full_name":"HasSubset.Subset","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[384,2],"def_end_pos":[384,8]},{"full_name":"IsClosed.closure_eq","def_path":"Mathlib/Topology/Basic.lean","def_pos":[364,8],"def_end_pos":[364,27]},{"full_name":"Rat","def_path":".lake/packages/batteries/Batteries/Data/Rat/Basic.lean","def_pos":[17,10],"def_end_pos":[17,13]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Set.range","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[144,4],"def_end_pos":[144,9]},{"full_name":"SetLike.mem_coe","def_path":"Mathlib/Data/SetLike/Basic.lean","def_pos":[168,8],"def_end_pos":[168,15]},{"full_name":"SubfieldClass.ratCast_mem","def_path":"Mathlib/Algebra/Field/Subfield.lean","def_pos":[92,6],"def_end_pos":[92,17]},{"full_name":"closure","def_path":"Mathlib/Topology/Defs/Basic.lean","def_pos":[112,4],"def_end_pos":[112,11]},{"full_name":"closure_mono","def_path":"Mathlib/Topology/Basic.lean","def_pos":[378,8],"def_end_pos":[378,20]},{"full_name":"subset_trans","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[546,6],"def_end_pos":[546,18]}]},{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast))","state_after":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ closure (ofReal' '' range Rat.cast)","tactic":"nth_rw 1 [range_comp]","premises":[{"full_name":"Set.range_comp","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[621,8],"def_end_pos":[621,18]}]},{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ closure (ofReal' '' range Rat.cast)","state_after":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ofReal' '' closure (range Rat.cast)","tactic":"refine subset_trans ?_ (image_closure_subset_closure_image continuous_ofReal)","premises":[{"full_name":"Complex.continuous_ofReal","def_path":"Mathlib/Analysis/Complex/Basic.lean","def_pos":[343,8],"def_end_pos":[343,25]},{"full_name":"image_closure_subset_closure_image","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1494,8],"def_end_pos":[1494,42]},{"full_name":"subset_trans","def_path":"Mathlib/Order/RelClasses.lean","def_pos":[546,6],"def_end_pos":[546,18]}]},{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ofReal' '' closure (range Rat.cast)","state_after":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ofReal' '' univ","tactic":"rw [DenseRange.closure_range Rat.denseEmbedding_coe_real.dense]","premises":[{"full_name":"DenseInducing.dense","def_path":"Mathlib/Topology/DenseEmbedding.lean","def_pos":[38,12],"def_end_pos":[38,17]},{"full_name":"DenseRange.closure_range","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1545,8],"def_end_pos":[1545,32]},{"full_name":"Rat.denseEmbedding_coe_real","def_path":"Mathlib/Topology/Instances/Rat.lean","def_pos":[36,8],"def_end_pos":[36,31]}]},{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ofReal' '' univ","state_after":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ range ofReal'","tactic":"simp only [image_univ]","premises":[{"full_name":"Set.image_univ","def_path":"Mathlib/Data/Set/Image.lean","def_pos":[596,8],"def_end_pos":[596,18]}]},{"state_before":"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ range ofReal'","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Data/List/Sigma.lean","commit":"","full_name":"List.dlookup_kinsert_ne","start":[517,0],"end":[518,60],"file_path":"Mathlib/Data/List/Sigma.lean","tactics":[{"state_before":"α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : a ≠ a'\n⊢ dlookup a (kinsert a' b' l) = dlookup a l","state_after":"no goals","tactic":"simp [h]","premises":[]}]}
{"url":"Mathlib/Data/List/Basic.lean","commit":"","full_name":"List.intercalate_splitOn","start":[1774,0],"end":[1785,41],"file_path":"Mathlib/Data/List/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\n⊢ [x].intercalate (splitOn x xs) = xs","state_after":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) xs)).join = xs","tactic":"simp only [intercalate, splitOn]","premises":[{"full_name":"List.intercalate","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,15]},{"full_name":"List.splitOn","def_path":".lake/packages/batteries/Batteries/Data/List/Basic.lean","def_pos":[226,14],"def_end_pos":[226,21]}]},{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) xs)).join = xs","state_after":"case nil\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) [])).join = []\n\ncase cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl : List α\nih : (intersperse [x] (splitOnP (fun x_1 => x_1 == x) tl)).join = tl\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) (hd :: tl))).join = hd :: tl","tactic":"induction' xs with hd tl ih","premises":[]},{"state_before":"case cons.cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl : List α\nih : (intersperse [x] (splitOnP (fun x_1 => x_1 == x) tl)).join = tl\nhd' : List α\ntl' : List (List α)\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) (hd :: tl))).join = hd :: tl","state_after":"case cons.cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) (hd :: tl))).join = hd :: tl","tactic":"rw [h'] at ih","premises":[]},{"state_before":"case cons.cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) (hd :: tl))).join = hd :: tl","state_after":"case cons.cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\n⊢ (intersperse [x]\n        (if (hd == x) = true then [] :: splitOnP (fun x_1 => x_1 == x) tl\n        else modifyHead (cons hd) (splitOnP (fun x_1 => x_1 == x) tl))).join =\n    hd :: tl","tactic":"rw [splitOnP_cons]","premises":[{"full_name":"List.splitOnP_cons","def_path":"Mathlib/Data/List/Basic.lean","def_pos":[1731,8],"def_end_pos":[1731,21]}]},{"state_before":"case cons.cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\n⊢ (intersperse [x]\n        (if (hd == x) = true then [] :: splitOnP (fun x_1 => x_1 == x) tl\n        else modifyHead (cons hd) (splitOnP (fun x_1 => x_1 == x) tl))).join =\n    hd :: tl","state_after":"case pos\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\nh : (hd == x) = true\n⊢ (intersperse [x] ([] :: splitOnP (fun x_1 => x_1 == x) tl)).join = hd :: tl\n\ncase neg\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\nh : ¬(hd == x) = true\n⊢ (intersperse [x] (modifyHead (cons hd) (splitOnP (fun x_1 => x_1 == x) tl))).join = hd :: tl","tactic":"split_ifs with h","premises":[{"full_name":"dite","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[945,20],"def_end_pos":[945,24]}]},{"state_before":"case neg\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\nh : ¬(hd == x) = true\n⊢ (intersperse [x] (modifyHead (cons hd) (splitOnP (fun x_1 => x_1 == x) tl))).join = hd :: tl","state_after":"no goals","tactic":"cases tl' <;> simpa [join, h'] using ih","premises":[{"full_name":"List.join","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[532,4],"def_end_pos":[532,8]}]}]}
{"url":"Mathlib/Topology/Algebra/UniformGroup.lean","commit":"","full_name":"uniformity_eq_comap_nhds_zero","start":[226,0],"end":[239,44],"file_path":"Mathlib/Topology/Algebra/UniformGroup.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (fun x => x.2 / x.1) (𝓝 1)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α)","tactic":"rw [nhds_eq_comap_uniformity, Filter.comap_comap]","premises":[{"full_name":"Filter.comap_comap","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1934,8],"def_end_pos":[1934,19]},{"full_name":"nhds_eq_comap_uniformity","def_path":"Mathlib/Topology/UniformSpace/Basic.lean","def_pos":[401,8],"def_end_pos":[401,32]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α)","state_after":"case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ map (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α) ≤ 𝓤 α\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ comap (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α) ≤ 𝓤 α","tactic":"refine le_antisymm (Filter.map_le_iff_le_comap.1 ?_) ?_","premises":[{"full_name":"Filter.map_le_iff_le_comap","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2023,8],"def_end_pos":[2023,27]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]}]}
{"url":"Mathlib/Topology/Filter.lean","commit":"","full_name":"Filter.nhds_iInf","start":[121,0],"end":[124,39],"file_path":"Mathlib/Topology/Filter.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\nf : ι → Filter α\n⊢ 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i)","state_after":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\nf : ι → Filter α\n⊢ (⨅ i, f i).lift' (Iic ∘ 𝓟) = ⨅ i, (f i).lift' (Iic ∘ 𝓟)","tactic":"simp only [nhds_eq]","premises":[{"full_name":"Filter.nhds_eq","def_path":"Mathlib/Topology/Filter.lean","def_pos":[68,8],"def_end_pos":[68,15]}]},{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\nf : ι → Filter α\n⊢ (⨅ i, f i).lift' (Iic ∘ 𝓟) = ⨅ i, (f i).lift' (Iic ∘ 𝓟)","state_after":"no goals","tactic":"apply lift'_iInf_of_map_univ <;> simp","premises":[{"full_name":"Filter.lift'_iInf_of_map_univ","def_path":"Mathlib/Order/Filter/Lift.lean","def_pos":[328,8],"def_end_pos":[328,30]}]}]}
{"url":"Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean","commit":"","full_name":"aemeasurable_coe_nnreal_real_iff","start":[192,0],"end":[195,97],"file_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0\nμ : Measure α\nh : AEMeasurable (fun x => ↑(f x)) μ\n⊢ AEMeasurable f μ","state_after":"no goals","tactic":"simpa only [Real.toNNReal_coe] using h.real_toNNReal","premises":[{"full_name":"AEMeasurable.real_toNNReal","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean","def_pos":[145,8],"def_end_pos":[145,34]},{"full_name":"Real.toNNReal_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[345,8],"def_end_pos":[345,32]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","commit":"","full_name":"CategoryTheory.StructuredArrow.hom_eq_iff","start":[60,0],"end":[62,34],"file_path":"Mathlib/CategoryTheory/Comma/StructuredArrow.lean","tactics":[{"state_before":"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nS S' S'' : D\nY✝ Y' Y'' : C\nT T' : C ⥤ D\nX Y : StructuredArrow S T\nf g : X ⟶ Y\nh : f = g\n⊢ f.right = g.right","state_after":"no goals","tactic":"rw [h]","premises":[]}]}
{"url":"Mathlib/Data/List/ProdSigma.lean","commit":"","full_name":"List.sigma_nil","start":[67,0],"end":[70,47],"file_path":"Mathlib/Data/List/ProdSigma.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nσ : α → Type u_3\nhead✝ : α\nl : List α\n⊢ ((head✝ :: l).sigma fun a => []) = []","state_after":"no goals","tactic":"simp [sigma_cons, sigma_nil l]","premises":[{"full_name":"List.sigma_cons","def_path":"Mathlib/Data/List/ProdSigma.lean","def_pos":[63,8],"def_end_pos":[63,18]}]}]}
{"url":"Mathlib/Topology/MetricSpace/Isometry.lean","commit":"","full_name":"IsometryEquiv.image_closedBall","start":[528,0],"end":[531,63],"file_path":"Mathlib/Topology/MetricSpace/Isometry.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nh✝ h : α ≃ᵢ β\nx : α\nr : ℝ\n⊢ ⇑h '' Metric.closedBall x r = Metric.closedBall (h x) r","state_after":"no goals","tactic":"rw [← h.preimage_symm, h.symm.preimage_closedBall, symm_symm]","premises":[{"full_name":"IsometryEquiv.preimage_closedBall","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[515,8],"def_end_pos":[515,27]},{"full_name":"IsometryEquiv.preimage_symm","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[381,8],"def_end_pos":[381,21]},{"full_name":"IsometryEquiv.symm","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[336,14],"def_end_pos":[336,18]},{"full_name":"IsometryEquiv.symm_symm","def_path":"Mathlib/Topology/MetricSpace/Isometry.lean","def_pos":[351,8],"def_end_pos":[351,17]}]}]}
{"url":"Mathlib/Data/Seq/WSeq.lean","commit":"","full_name":"Stream'.WSeq.tail_nil","start":[612,0],"end":[613,62],"file_path":"Mathlib/Data/Seq/WSeq.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\n⊢ nil.tail = nil","state_after":"no goals","tactic":"simp [tail]","premises":[{"full_name":"Stream'.WSeq.tail","def_path":"Mathlib/Data/Seq/WSeq.lean","def_pos":[134,4],"def_end_pos":[134,8]}]}]}
{"url":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean","commit":"","full_name":"CstarRing.norm_mem_spectrum_of_nonneg","start":[113,0],"end":[115,85],"file_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean","tactics":[{"state_before":"A : Type u_1\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CstarRing A\ninst✝⁵ : CompleteSpace A\ninst✝⁴ : NormedAlgebra ℂ A\ninst✝³ : StarModule ℂ A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nontrivial A\na : A\nha : autoParam (0 ≤ a) _auto✝\n⊢ ‖a‖ ∈ spectrum ℝ a","state_after":"no goals","tactic":"simpa using spectrum.algebraMap_mem ℝ <| CstarRing.nnnorm_mem_spectrum_of_nonneg ha","premises":[{"full_name":"CstarRing.nnnorm_mem_spectrum_of_nonneg","def_path":"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean","def_pos":[106,6],"def_end_pos":[106,45]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]}]}]}
{"url":"Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean","commit":"","full_name":"Real.one_half_lt_eulerMascheroniSeq_six","start":[62,0],"end":[69,10],"file_path":"Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean","tactics":[{"state_before":"⊢ 1 / 2 < eulerMascheroniSeq 6","state_after":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 1 / 2 < eulerMascheroniSeq 6","tactic":"have : eulerMascheroniSeq 6 = 49 / 20 - log 7 := by\n    rw [eulerMascheroniSeq]\n    norm_num","premises":[{"full_name":"Real.eulerMascheroniSeq","def_path":"Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean","def_pos":[45,18],"def_end_pos":[45,36]},{"full_name":"Real.log","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[39,18],"def_end_pos":[39,21]}]},{"state_before":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 1 / 2 < eulerMascheroniSeq 6","state_after":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 < rexp (49 / 20 - 1 / 2)","tactic":"rw [this, lt_sub_iff_add_lt, ← lt_sub_iff_add_lt', log_lt_iff_lt_exp (by positivity)]","premises":[{"full_name":"Real.log_lt_iff_lt_exp","def_path":"Mathlib/Analysis/SpecialFunctions/Log/Basic.lean","def_pos":[135,8],"def_end_pos":[135,25]},{"full_name":"lt_sub_iff_add_lt","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[617,2],"def_end_pos":[617,13]},{"full_name":"lt_sub_iff_add_lt'","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Basic.lean","def_pos":[665,2],"def_end_pos":[665,13]}]},{"state_before":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 < rexp (49 / 20 - 1 / 2)","state_after":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 < ∑ i ∈ Finset.range 7, (49 / 20 - 1 / 2) ^ i / ↑i.factorial","tactic":"refine lt_of_lt_of_le ?_ (Real.sum_le_exp_of_nonneg (by norm_num) 7)","premises":[{"full_name":"Real.sum_le_exp_of_nonneg","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[947,8],"def_end_pos":[947,28]},{"full_name":"lt_of_lt_of_le","def_path":"Mathlib/Order/Defs.lean","def_pos":[93,8],"def_end_pos":[93,22]}]},{"state_before":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 < ∑ i ∈ Finset.range 7, (49 / 20 - 1 / 2) ^ i / ↑i.factorial","state_after":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 <\n    ∑ i ∈ Finset.range 0, (49 / 20 - 1 / 2) ^ i / ↑i.factorial + (49 / 20 - 1 / 2) ^ 0 / ↑(Nat.factorial 0) +\n                (49 / 20 - 1 / 2) ^ 1 / ↑((0 + 1) * Nat.factorial 0) +\n              (49 / 20 - 1 / 2) ^ 2 / ↑((1 + 1) * ((0 + 1) * Nat.factorial 0)) +\n            (49 / 20 - 1 / 2) ^ 3 / ↑((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0))) +\n          (49 / 20 - 1 / 2) ^ 4 / ↑((3 + 1) * ((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0)))) +\n        (49 / 20 - 1 / 2) ^ 5 / ↑((4 + 1) * ((3 + 1) * ((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0))))) +\n      (49 / 20 - 1 / 2) ^ 6 / ↑((5 + 1) * ((4 + 1) * ((3 + 1) * ((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0))))))","tactic":"simp_rw [Finset.sum_range_succ, Nat.factorial_succ]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Finset.sum_range_succ","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1296,2],"def_end_pos":[1296,13]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Nat.factorial_succ","def_path":"Mathlib/Data/Nat/Factorial/Basic.lean","def_pos":[43,8],"def_end_pos":[43,22]}]},{"state_before":"this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 <\n    ∑ i ∈ Finset.range 0, (49 / 20 - 1 / 2) ^ i / ↑i.factorial + (49 / 20 - 1 / 2) ^ 0 / ↑(Nat.factorial 0) +\n                (49 / 20 - 1 / 2) ^ 1 / ↑((0 + 1) * Nat.factorial 0) +\n              (49 / 20 - 1 / 2) ^ 2 / ↑((1 + 1) * ((0 + 1) * Nat.factorial 0)) +\n            (49 / 20 - 1 / 2) ^ 3 / ↑((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0))) +\n          (49 / 20 - 1 / 2) ^ 4 / ↑((3 + 1) * ((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0)))) +\n        (49 / 20 - 1 / 2) ^ 5 / ↑((4 + 1) * ((3 + 1) * ((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0))))) +\n      (49 / 20 - 1 / 2) ^ 6 / ↑((5 + 1) * ((4 + 1) * ((3 + 1) * ((2 + 1) * ((1 + 1) * ((0 + 1) * Nat.factorial 0))))))","state_after":"no goals","tactic":"norm_num","premises":[]}]}
{"url":"Mathlib/Data/List/Chain.lean","commit":"","full_name":"List.chain_iff_get","start":[116,0],"end":[138,53],"file_path":"Mathlib/Data/List/Chain.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\nl : List α\ni : ℕ\nh : i < l.length - 1\n⊢ i < l.length","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\nl : List α\ni : ℕ\nh : i < l.length - 1\n⊢ i + 1 < l.length","state_after":"no goals","tactic":"omega","premises":[]},{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\n⊢ Chain R a []","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\nh : 0 < [].length\n⊢ R a ([].get ⟨0, h⟩)","state_after":"no goals","tactic":"simp at h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\nx✝ : ℕ\nh : x✝ < [].length - 1\n⊢ R ([].get ⟨x✝, ⋯⟩) ([].get ⟨x✝ + 1, ⋯⟩)","state_after":"no goals","tactic":"simp at h","premises":[]},{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ Chain R a (b :: t) ↔\n    (∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)","state_after":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ (R a b ∧\n      (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧\n        ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)) ↔\n    (∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)","tactic":"rw [chain_cons, @chain_iff_get _ _ t]","premises":[{"full_name":"List.chain_cons","def_path":".lake/packages/batteries/Batteries/Data/List/Lemmas.lean","def_pos":[1267,8],"def_end_pos":[1267,18]}]},{"state_before":"α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ (R a b ∧\n      (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧\n        ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)) ↔\n    (∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)","state_after":"case mp\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ (R a b ∧\n      (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧\n        ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)) →\n    (∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n\ncase mpr\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ ((∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)) →\n    R a b ∧\n      (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","tactic":"constructor","premises":[]},{"state_before":"case mpr\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ ((∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)) →\n    R a b ∧\n      (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","state_after":"case mpr.intro\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ R a b ∧\n    (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","tactic":"rintro ⟨h0, h⟩","premises":[]},{"state_before":"case mpr.intro\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ R a b ∧\n    (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","state_after":"case mpr.intro.left\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ R a b\n\ncase mpr.intro.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","tactic":"constructor","premises":[]},{"state_before":"case mpr.intro.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","state_after":"case mpr.intro.right.left\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ ∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)\n\ncase mpr.intro.right.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","tactic":"constructor","premises":[]},{"state_before":"case mpr.intro.right.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\n⊢ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","state_after":"case mpr.intro.right.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\ni : ℕ\nw : i < t.length - 1\n⊢ R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","tactic":"intro i w","premises":[]},{"state_before":"case mpr.intro.right.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\ni : ℕ\nw : i < t.length - 1\n⊢ R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)","state_after":"no goals","tactic":"exact h (i+1) (by simp only [length_cons]; omega)","premises":[{"full_name":"List.length_cons","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean","def_pos":[70,16],"def_end_pos":[70,27]}]}]}
{"url":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","commit":"","full_name":"CategoryTheory.Limits.cokernelIsoOfEq_refl","start":[793,0],"end":[795,29],"file_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","tactics":[{"state_before":"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasCokernel f\nh : f = f\n⊢ cokernelIsoOfEq h = Iso.refl (cokernel f)","state_after":"case w.h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasCokernel f\nh : f = f\n⊢ coequalizer.π f 0 ≫ (cokernelIsoOfEq h).hom = coequalizer.π f 0 ≫ (Iso.refl (cokernel f)).hom","tactic":"ext","premises":[]},{"state_before":"case w.h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasCokernel f\nh : f = f\n⊢ coequalizer.π f 0 ≫ (cokernelIsoOfEq h).hom = coequalizer.π f 0 ≫ (Iso.refl (cokernel f)).hom","state_after":"no goals","tactic":"simp [cokernelIsoOfEq]","premises":[{"full_name":"CategoryTheory.Limits.cokernelIsoOfEq","def_path":"Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean","def_pos":[789,4],"def_end_pos":[789,19]}]}]}
{"url":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","commit":"","full_name":"CochainComplex.HomComplex.Cocycle.equivHom_apply","start":[693,0],"end":[700,26],"file_path":"Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean","tactics":[{"state_before":"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\n⊢ ∀ (x y : F ⟶ G),\n    { toFun := ofHom, invFun := homOf, left_inv := ⋯, right_inv := ⋯ }.toFun (x + y) =\n      { toFun := ofHom, invFun := homOf, left_inv := ⋯, right_inv := ⋯ }.toFun x +\n        { toFun := ofHom, invFun := homOf, left_inv := ⋯, right_inv := ⋯ }.toFun y","state_after":"no goals","tactic":"aesop_cat","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]},{"full_name":"Lean.Meta.TransparencyMode.default","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[27,4],"def_end_pos":[27,11]},{"full_name":"Option.some","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[2244,4],"def_end_pos":[2244,8]}]}]}
{"url":"Mathlib/CategoryTheory/Localization/Construction.lean","commit":"","full_name":"CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.unitIso_inv","start":[325,0],"end":[337,13],"file_path":"Mathlib/CategoryTheory/Localization/Construction.lean","tactics":[{"state_before":"C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\n⊢ 𝟭 (W.Localization ⥤ D) = functor W D ⋙ inverse W D","state_after":"case refine_1\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG✝ : C ⥤ D\nhG : W.IsInvertedBy G✝\nG : W.Localization ⥤ D\n⊢ (𝟭 (W.Localization ⥤ D)).obj G = (functor W D ⋙ inverse W D).obj G\n\ncase refine_2\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\nG₁ G₂ : W.Localization ⥤ D\nτ : G₁ ⟶ G₂\n⊢ (𝟭 (W.Localization ⥤ D)).map τ = eqToHom ⋯ ≫ (functor W D ⋙ inverse W D).map τ ≫ eqToHom ⋯","tactic":"refine Functor.ext (fun G => ?_) fun G₁ G₂ τ => ?_","premises":[{"full_name":"CategoryTheory.Functor.ext","def_path":"Mathlib/CategoryTheory/EqToHom.lean","def_pos":[179,8],"def_end_pos":[179,11]}]}]}
{"url":"Mathlib/GroupTheory/Perm/Fin.lean","commit":"","full_name":"Equiv.Perm.decomposeFin_symm_apply_zero","start":[35,0],"end":[37,82],"file_path":"Mathlib/GroupTheory/Perm/Fin.lean","tactics":[{"state_before":"n : ℕ\np : Fin (n + 1)\ne : Perm (Fin n)\n⊢ (decomposeFin.symm (p, e)) 0 = p","state_after":"no goals","tactic":"simp [Equiv.Perm.decomposeFin]","premises":[{"full_name":"Equiv.Perm.decomposeFin","def_path":"Mathlib/GroupTheory/Perm/Fin.lean","def_pos":[21,4],"def_end_pos":[21,27]}]}]}
{"url":"Mathlib/RingTheory/Multiplicity.lean","commit":"","full_name":"multiplicity.min_le_multiplicity_add","start":[358,0],"end":[366,78],"file_path":"Mathlib/RingTheory/Multiplicity.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≤ multiplicity p b\n⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≤ multiplicity p b\n⊢ ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ a + b","tactic":"rw [min_eq_left h, multiplicity_le_multiplicity_iff]","premises":[{"full_name":"min_eq_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[92,8],"def_end_pos":[92,19]},{"full_name":"multiplicity.multiplicity_le_multiplicity_iff","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[200,8],"def_end_pos":[200,40]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≤ multiplicity p b\n⊢ ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ a + b","state_after":"no goals","tactic":"exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"dvd_add","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[44,8],"def_end_pos":[44,15]},{"full_name":"multiplicity.multiplicity_le_multiplicity_iff","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[200,8],"def_end_pos":[200,40]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b ≤ multiplicity p a\n⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)","state_after":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b ≤ multiplicity p a\n⊢ ∀ (n : ℕ), p ^ n ∣ b → p ^ n ∣ a + b","tactic":"rw [min_eq_right h, multiplicity_le_multiplicity_iff]","premises":[{"full_name":"min_eq_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[95,8],"def_end_pos":[95,20]},{"full_name":"multiplicity.multiplicity_le_multiplicity_iff","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[200,8],"def_end_pos":[200,40]}]},{"state_before":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b ≤ multiplicity p a\n⊢ ∀ (n : ℕ), p ^ n ∣ b → p ^ n ∣ a + b","state_after":"no goals","tactic":"exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn","premises":[{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"dvd_add","def_path":"Mathlib/Algebra/Ring/Divisibility/Basic.lean","def_pos":[44,8],"def_end_pos":[44,15]},{"full_name":"multiplicity.multiplicity_le_multiplicity_iff","def_path":"Mathlib/RingTheory/Multiplicity.lean","def_pos":[200,8],"def_end_pos":[200,40]}]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.tendsto_div_const_atTop_of_neg","start":[994,0],"end":[998,59],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\nhr : r < 0\n⊢ Tendsto (fun x => f x / r) l atTop ↔ Tendsto f l atBot","state_after":"no goals","tactic":"simp [div_eq_mul_inv, tendsto_mul_const_atTop_of_neg, hr]","premises":[{"full_name":"Filter.tendsto_mul_const_atTop_of_neg","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[990,8],"def_end_pos":[990,38]},{"full_name":"div_eq_mul_inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[907,8],"def_end_pos":[907,22]}]}]}
{"url":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","commit":"","full_name":"CategoryTheory.ShortComplex.RightHomologyData.p_comp_opcyclesIso_inv","start":[799,0],"end":[802,44],"file_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","tactics":[{"state_before":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.p ≫ h.opcyclesIso.inv = S.pOpcycles","state_after":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.p ≫ opcyclesMap' (𝟙 S) h S.rightHomologyData = S.rightHomologyData.p","tactic":"dsimp [pOpcycles, RightHomologyData.opcyclesIso]","premises":[{"full_name":"CategoryTheory.ShortComplex.RightHomologyData.opcyclesIso","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[796,18],"def_end_pos":[796,29]},{"full_name":"CategoryTheory.ShortComplex.pOpcycles","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[495,18],"def_end_pos":[495,27]}]},{"state_before":"C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.p ≫ opcyclesMap' (𝟙 S) h S.rightHomologyData = S.rightHomologyData.p","state_after":"no goals","tactic":"simp only [p_opcyclesMap', id_τ₂, id_comp]","premises":[{"full_name":"CategoryTheory.Category.id_comp","def_path":"Mathlib/CategoryTheory/Category/Basic.lean","def_pos":[151,2],"def_end_pos":[151,9]},{"full_name":"CategoryTheory.ShortComplex.id_τ₂","def_path":"Mathlib/Algebra/Homology/ShortComplex/Basic.lean","def_pos":[100,14],"def_end_pos":[100,19]},{"full_name":"CategoryTheory.ShortComplex.p_opcyclesMap'","def_path":"Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean","def_pos":[589,6],"def_end_pos":[589,20]}]}]}
{"url":"Mathlib/Algebra/Module/Injective.lean","commit":"","full_name":"Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_eq","start":[294,0],"end":[299,82],"file_path":"Mathlib/Algebra/Module/Injective.lean","tactics":[{"state_before":"R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nr : R\nhr : r • y ∈ (extensionOfMax i f).domain\n⊢ (extendIdealTo i f h y) r = ↑(extensionOfMax i f).toLinearPMap ⟨r • y, hr⟩","state_after":"no goals","tactic":"simp only [ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h _ _ hr,\n    ExtensionOfMaxAdjoin.idealTo, LinearMap.coe_mk, Subtype.coe_mk, AddHom.coe_mk]","premises":[{"full_name":"AddHom.coe_mk","def_path":"Mathlib/Algebra/Group/Hom/Defs.lean","def_pos":[490,2],"def_end_pos":[490,13]},{"full_name":"LinearMap.coe_mk","def_path":"Mathlib/Algebra/Module/LinearMap/Defs.lean","def_pos":[253,8],"def_end_pos":[253,14]},{"full_name":"Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_is_extension","def_path":"Mathlib/Algebra/Module/Injective.lean","def_pos":[271,8],"def_end_pos":[271,55]},{"full_name":"Module.Baer.ExtensionOfMaxAdjoin.idealTo","def_path":"Mathlib/Algebra/Module/Injective.lean","def_pos":[253,4],"def_end_pos":[253,32]},{"full_name":"Subtype.coe_mk","def_path":"Mathlib/Data/Subtype.lean","def_pos":[86,8],"def_end_pos":[86,14]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Basic.lean","commit":"","full_name":"Finset.Ico_filter_le_of_left_le","start":[284,0],"end":[288,51],"file_path":"Mathlib/Order/Interval/Finset/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na✝ a₁ a₂ b✝ b₁ b₂ c✝ x a b c : α\ninst✝ : DecidablePred fun x => c ≤ x\nhac : a ≤ c\n⊢ filter (fun x => c ≤ x) (Ico a b) = Ico c b","state_after":"case a\nι : Type u_1\nα : Type u_2\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na✝ a₁ a₂ b✝ b₁ b₂ c✝ x✝ a b c : α\ninst✝ : DecidablePred fun x => c ≤ x\nhac : a ≤ c\nx : α\n⊢ x ∈ filter (fun x => c ≤ x) (Ico a b) ↔ x ∈ Ico c b","tactic":"ext x","premises":[]},{"state_before":"case a\nι : Type u_1\nα : Type u_2\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na✝ a₁ a₂ b✝ b₁ b₂ c✝ x✝ a b c : α\ninst✝ : DecidablePred fun x => c ≤ x\nhac : a ≤ c\nx : α\n⊢ x ∈ filter (fun x => c ≤ x) (Ico a b) ↔ x ∈ Ico c b","state_after":"case a\nι : Type u_1\nα : Type u_2\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na✝ a₁ a₂ b✝ b₁ b₂ c✝ x✝ a b c : α\ninst✝ : DecidablePred fun x => c ≤ x\nhac : a ≤ c\nx : α\n⊢ a ≤ x ∧ c ≤ x ∧ x < b ↔ c ≤ x ∧ x < b","tactic":"rw [mem_filter, mem_Ico, mem_Ico, and_comm, and_left_comm]","premises":[{"full_name":"Finset.mem_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[299,8],"def_end_pos":[299,15]},{"full_name":"Finset.mem_filter","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2158,8],"def_end_pos":[2158,18]},{"full_name":"and_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[819,8],"def_end_pos":[819,16]},{"full_name":"and_left_comm","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[56,8],"def_end_pos":[56,21]}]},{"state_before":"case a\nι : Type u_1\nα : Type u_2\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na✝ a₁ a₂ b✝ b₁ b₂ c✝ x✝ a b c : α\ninst✝ : DecidablePred fun x => c ≤ x\nhac : a ≤ c\nx : α\n⊢ a ≤ x ∧ c ≤ x ∧ x < b ↔ c ≤ x ∧ x < b","state_after":"no goals","tactic":"exact and_iff_right_of_imp fun h => hac.trans h.1","premises":[{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"and_iff_right_of_imp","def_path":".lake/packages/lean4/src/lean/Init/SimpLemmas.lean","def_pos":[162,8],"def_end_pos":[162,28]}]}]}
{"url":"Mathlib/GroupTheory/SpecificGroups/Dihedral.lean","commit":"","full_name":"DihedralGroup.card_conjClasses_odd","start":[238,0],"end":[241,98],"file_path":"Mathlib/GroupTheory/SpecificGroups/Dihedral.lean","tactics":[{"state_before":"n : ℕ\nhn : Odd n\n⊢ Nat.card (ConjClasses (DihedralGroup n)) = (n + 3) / 2","state_after":"no goals","tactic":"rw [← Nat.mul_div_mul_left _ 2 hn.pos, ← card_commute_odd hn, mul_comm,\n    card_comm_eq_card_conjClasses_mul_card, nat_card, Nat.mul_div_left _ (mul_pos two_pos hn.pos)]","premises":[{"full_name":"DihedralGroup.card_commute_odd","def_path":"Mathlib/GroupTheory/SpecificGroups/Dihedral.lean","def_pos":[232,6],"def_end_pos":[232,22]},{"full_name":"DihedralGroup.nat_card","def_path":"Mathlib/GroupTheory/SpecificGroups/Dihedral.lean","def_pos":[120,8],"def_end_pos":[120,16]},{"full_name":"Nat.mul_div_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[370,16],"def_end_pos":[370,28]},{"full_name":"Nat.mul_div_mul_left","def_path":".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean","def_pos":[383,18],"def_end_pos":[383,34]},{"full_name":"Odd.pos","def_path":"Mathlib/Algebra/Order/Ring/Canonical.lean","def_pos":[44,6],"def_end_pos":[44,13]},{"full_name":"card_comm_eq_card_conjClasses_mul_card","def_path":"Mathlib/GroupTheory/GroupAction/Quotient.lean","def_pos":[430,8],"def_end_pos":[430,46]},{"full_name":"mul_comm","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[294,8],"def_end_pos":[294,16]}]}]}
{"url":"Mathlib/Logic/Equiv/Defs.lean","commit":"","full_name":"Equiv.symm_symm","start":[295,0],"end":[295,74],"file_path":"Mathlib/Logic/Equiv/Defs.lean","tactics":[{"state_before":"α : Sort u\nβ : Sort v\nγ : Sort w\ne : α ≃ β\n⊢ e.symm.symm = e","state_after":"case mk\nα : Sort u\nβ : Sort v\nγ : Sort w\ntoFun✝ : α → β\ninvFun✝ : β → α\nleft_inv✝ : LeftInverse invFun✝ toFun✝\nright_inv✝ : Function.RightInverse invFun✝ toFun✝\n⊢ { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }.symm.symm =\n    { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }","tactic":"cases e","premises":[]},{"state_before":"case mk\nα : Sort u\nβ : Sort v\nγ : Sort w\ntoFun✝ : α → β\ninvFun✝ : β → α\nleft_inv✝ : LeftInverse invFun✝ toFun✝\nright_inv✝ : Function.RightInverse invFun✝ toFun✝\n⊢ { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }.symm.symm =\n    { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Combinatorics/Enumerative/Composition.lean","commit":"","full_name":"Composition.sizeUpTo_ofLength_le","start":[187,0],"end":[190,24],"file_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","tactics":[{"state_before":"n : ℕ\nc : Composition n\ni : ℕ\nh : c.length ≤ i\n⊢ c.sizeUpTo i = n","state_after":"n : ℕ\nc : Composition n\ni : ℕ\nh : c.length ≤ i\n⊢ (take i c.blocks).sum = n","tactic":"dsimp [sizeUpTo]","premises":[{"full_name":"Composition.sizeUpTo","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[181,4],"def_end_pos":[181,12]}]},{"state_before":"n : ℕ\nc : Composition n\ni : ℕ\nh : c.length ≤ i\n⊢ (take i c.blocks).sum = n","state_after":"case h.e'_2.h.e'_4\nn : ℕ\nc : Composition n\ni : ℕ\nh : c.length ≤ i\n⊢ take i c.blocks = c.blocks","tactic":"convert c.blocks_sum","premises":[{"full_name":"Composition.blocks_sum","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[102,2],"def_end_pos":[102,12]}]},{"state_before":"case h.e'_2.h.e'_4\nn : ℕ\nc : Composition n\ni : ℕ\nh : c.length ≤ i\n⊢ take i c.blocks = c.blocks","state_after":"no goals","tactic":"exact take_all_of_le h","premises":[{"full_name":"List.take_all_of_le","def_path":".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean","def_pos":[1573,8],"def_end_pos":[1573,22]}]}]}
{"url":"Mathlib/Algebra/Group/Basic.lean","commit":"","full_name":"comp_mul_right","start":[105,0],"end":[112,18],"file_path":"Mathlib/Algebra/Group/Basic.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Semigroup α\nx y : α\n⊢ ((fun x_1 => x_1 * x) ∘ fun x => x * y) = fun x_1 => x_1 * (y * x)","state_after":"case h\nα : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Semigroup α\nx y z : α\n⊢ ((fun x_1 => x_1 * x) ∘ fun x => x * y) z = z * (y * x)","tactic":"ext z","premises":[]},{"state_before":"case h\nα : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Semigroup α\nx y z : α\n⊢ ((fun x_1 => x_1 * x) ∘ fun x => x * y) z = z * (y * x)","state_after":"no goals","tactic":"simp [mul_assoc]","premises":[{"full_name":"mul_assoc","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[259,8],"def_end_pos":[259,17]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","commit":"","full_name":"MeasureTheory.exists_measurable_superset","start":[187,0],"end":[190,92],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s","state_after":"no goals","tactic":"simpa only [← measure_eq_trim] using μ.toOuterMeasure.exists_measurable_superset_eq_trim s","premises":[{"full_name":"MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim","def_path":"Mathlib/MeasureTheory/OuterMeasure/Induced.lean","def_pos":[356,8],"def_end_pos":[356,42]},{"full_name":"MeasureTheory.measure_eq_trim","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[149,8],"def_end_pos":[149,23]}]}]}
{"url":"Mathlib/Order/Disjoint.lean","commit":"","full_name":"Codisjoint.ne","start":[233,0],"end":[234,56],"file_path":"Mathlib/Order/Disjoint.lean","tactics":[{"state_before":"α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : OrderTop α\na b c d : α\nha : a ≠ ⊤\nhab : Codisjoint a b\nh : a = b\n⊢ Codisjoint a a","state_after":"no goals","tactic":"rwa [← h] at hab","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","commit":"","full_name":"IntervalIntegrable.trans_iterate_Ico","start":[156,0],"end":[163,93],"file_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","tactics":[{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na✝ b c d : ℝ\nμ ν : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\nhint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ IntervalIntegrable f μ (a m) (a n)","state_after":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na✝ b c d : ℝ\nμ ν : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a n)","tactic":"revert hint","premises":[]},{"state_before":"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na✝ b c d : ℝ\nμ ν : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a n)","state_after":"case refine_1\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na✝ b c d : ℝ\nμ ν : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ k ∈ Ico m m, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a m)\n\ncase refine_2\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na✝ b c d : ℝ\nμ ν : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ ∀ (n : ℕ),\n    m ≤ n →\n      ((∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a n)) →\n        (∀ k ∈ Ico m (n + 1), IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a (n + 1))","tactic":"refine Nat.le_induction ?_ ?_ n hmn","premises":[{"full_name":"Nat.le_induction","def_path":"Mathlib/Data/Nat/Defs.lean","def_pos":[829,6],"def_end_pos":[829,18]}]}]}
{"url":"Mathlib/Data/Num/Lemmas.lean","commit":"","full_name":"ZNum.dvd_to_int","start":[1370,0],"end":[1372,85],"file_path":"Mathlib/Data/Num/Lemmas.lean","tactics":[{"state_before":"α : Type u_1\nm n : ZNum\nx✝ : ↑m ∣ ↑n\nk : ℤ\ne : ↑n = ↑m * k\n⊢ n = m * ↑k","state_after":"α : Type u_1\nm n : ZNum\nx✝ : ↑m ∣ ↑n\nk : ℤ\ne : ↑n = ↑m * k\n⊢ ↑(↑m * k) = m * ↑k","tactic":"rw [← of_to_int n, e]","premises":[{"full_name":"ZNum.of_to_int","def_path":"Mathlib/Data/Num/Lemmas.lean","def_pos":[1348,8],"def_end_pos":[1348,17]}]},{"state_before":"α : Type u_1\nm n : ZNum\nx✝ : ↑m ∣ ↑n\nk : ℤ\ne : ↑n = ↑m * k\n⊢ ↑(↑m * k) = m * ↑k","state_after":"no goals","tactic":"simp","premises":[]},{"state_before":"α : Type u_1\nm n : ZNum\nx✝ : m ∣ n\nk : ZNum\ne : n = m * k\n⊢ ↑n = ↑m * ↑k","state_after":"no goals","tactic":"simp [e]","premises":[]}]}
{"url":"Mathlib/Order/Filter/AtTopBot.lean","commit":"","full_name":"Filter.map_atTop_eq_of_gc","start":[1402,0],"end":[1414,86],"file_path":"Mathlib/Order/Filter/AtTopBot.lean","tactics":[{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\n⊢ map f atTop = atTop","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\n⊢ atTop ≤ map f atTop","tactic":"refine\n    le_antisymm\n      (hf.tendsto_atTop_atTop fun b => ⟨g (b ⊔ b'), le_sup_left.trans <| hgi _ le_sup_right⟩) ?_","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Sup.sup","def_path":"Mathlib/Order/Notation.lean","def_pos":[47,2],"def_end_pos":[47,5]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_sup_left","def_path":"Mathlib/Order/Lattice.lean","def_pos":[106,8],"def_end_pos":[106,19]},{"full_name":"le_sup_right","def_path":"Mathlib/Order/Lattice.lean","def_pos":[112,8],"def_end_pos":[112,20]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\n⊢ atTop ≤ map f atTop","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\n⊢ atTop ≤ ⨅ a, 𝓟 (f '' {a' | a ≤ a'})","tactic":"rw [@map_atTop_eq _ _ ⟨g b'⟩]","premises":[{"full_name":"Filter.map_atTop_eq","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[334,8],"def_end_pos":[334,20]},{"full_name":"Nonempty.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[711,4],"def_end_pos":[711,9]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\n⊢ atTop ≤ ⨅ a, 𝓟 (f '' {a' | a ≤ a'})","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\na : α\nb : β\nhb : b ∈ Ici (f a ⊔ b')\n⊢ b ∈ f '' {a' | a ≤ a'}","tactic":"refine le_iInf fun a => iInf_le_of_le (f a ⊔ b') <| principal_mono.2 fun b hb => ?_","premises":[{"full_name":"Filter.principal_mono","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[595,8],"def_end_pos":[595,22]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"Sup.sup","def_path":"Mathlib/Order/Notation.lean","def_pos":[47,2],"def_end_pos":[47,5]},{"full_name":"iInf_le_of_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[644,8],"def_end_pos":[644,21]},{"full_name":"le_iInf","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[664,8],"def_end_pos":[664,15]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\na : α\nb : β\nhb : b ∈ Ici (f a ⊔ b')\n⊢ b ∈ f '' {a' | a ≤ a'}","state_after":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\na : α\nb : β\nhb : f a ≤ b ∧ b' ≤ b\n⊢ b ∈ f '' {a' | a ≤ a'}","tactic":"rw [mem_Ici, sup_le_iff] at hb","premises":[{"full_name":"Set.mem_Ici","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[122,8],"def_end_pos":[122,15]},{"full_name":"sup_le_iff","def_path":"Mathlib/Order/Lattice.lean","def_pos":[133,8],"def_end_pos":[133,18]}]},{"state_before":"ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\ng : β → α\nb' : β\nhf : Monotone f\ngc : ∀ (a : α), ∀ b ≥ b', f a ≤ b ↔ a ≤ g b\nhgi : ∀ b ≥ b', b ≤ f (g b)\na : α\nb : β\nhb : f a ≤ b ∧ b' ≤ b\n⊢ b ∈ f '' {a' | a ≤ a'}","state_after":"no goals","tactic":"exact ⟨g b, (gc _ _ hb.2).1 hb.1, le_antisymm ((gc _ _ hb.2).2 le_rfl) (hgi _ hb.2)⟩","premises":[{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"And.left","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[521,2],"def_end_pos":[521,6]},{"full_name":"And.right","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[524,2],"def_end_pos":[524,7]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]},{"full_name":"le_rfl","def_path":"Mathlib/Order/Defs.lean","def_pos":[43,8],"def_end_pos":[43,14]}]}]}
{"url":"Mathlib/Data/Int/Cast/Lemmas.lean","commit":"","full_name":"Int.cast_inj","start":[66,0],"end":[67,100],"file_path":"Mathlib/Data/Int/Cast/Lemmas.lean","tactics":[{"state_before":"F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : AddGroupWithOne α\ninst✝ : CharZero α\nm n : ℤ\n⊢ ↑m = ↑n ↔ m = n","state_after":"no goals","tactic":"rw [← sub_eq_zero, ← cast_sub, cast_eq_zero, sub_eq_zero]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"Int.cast_eq_zero","def_path":"Mathlib/Data/Int/Cast/Lemmas.lean","def_pos":[57,14],"def_end_pos":[57,26]},{"full_name":"Int.cast_sub","def_path":"Mathlib/Data/Int/Cast/Basic.lean","def_pos":[107,8],"def_end_pos":[107,16]},{"full_name":"sub_eq_zero","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[738,2],"def_end_pos":[738,13]}]}]}
{"url":"Mathlib/RingTheory/Trace/Basic.lean","commit":"","full_name":"Algebra.traceMatrix_reindex","start":[293,0],"end":[294,85],"file_path":"Mathlib/RingTheory/Trace/Basic.lean","tactics":[{"state_before":"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : CommRing T\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra K L\nι κ : Type w\ninst✝¹⁰ : Fintype ι\nF : Type u_6\ninst✝⁹ : Field F\ninst✝⁸ : Algebra R L\ninst✝⁷ : Algebra L F\ninst✝⁶ : Algebra R F\ninst✝⁵ : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : CommRing C\ninst✝ : Algebra A C\nκ' : Type u_7\nb : Basis κ A B\nf : κ ≃ κ'\n⊢ traceMatrix A ⇑(b.reindex f) = (reindex f f) (traceMatrix A ⇑b)","state_after":"case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : CommRing T\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra K L\nι κ : Type w\ninst✝¹⁰ : Fintype ι\nF : Type u_6\ninst✝⁹ : Field F\ninst✝⁸ : Algebra R L\ninst✝⁷ : Algebra L F\ninst✝⁶ : Algebra R F\ninst✝⁵ : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : CommRing C\ninst✝ : Algebra A C\nκ' : Type u_7\nb : Basis κ A B\nf : κ ≃ κ'\nx y : κ'\n⊢ traceMatrix A (⇑(b.reindex f)) x y = (reindex f f) (traceMatrix A ⇑b) x y","tactic":"ext (x y)","premises":[]},{"state_before":"case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : CommRing T\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : Algebra R T\nK : Type u_4\nL : Type u_5\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra K L\nι κ : Type w\ninst✝¹⁰ : Fintype ι\nF : Type u_6\ninst✝⁹ : Field F\ninst✝⁸ : Algebra R L\ninst✝⁷ : Algebra L F\ninst✝⁶ : Algebra R F\ninst✝⁵ : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : CommRing C\ninst✝ : Algebra A C\nκ' : Type u_7\nb : Basis κ A B\nf : κ ≃ κ'\nx y : κ'\n⊢ traceMatrix A (⇑(b.reindex f)) x y = (reindex f f) (traceMatrix A ⇑b) x y","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/CategoryTheory/Shift/Induced.lean","commit":"","full_name":"CategoryTheory.shiftFunctorAdd_inv_app_obj_of_induced","start":[194,0],"end":[203,78],"file_path":"Mathlib/CategoryTheory/Shift/Induced.lean","tactics":[{"state_before":"C : Type u_4\nD : Type u_2\ninst✝⁵ : Category.{u_5, u_4} C\ninst✝⁴ : Category.{u_1, u_2} D\nF : C ⥤ D\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D D).obj F).Faithful\na b : A\nX : C\n⊢ (shiftFunctorAdd D a b).inv.app (F.obj X) =\n    (s b).map ((i a).hom.app X) ≫\n      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X","state_after":"C : Type u_4\nD : Type u_2\ninst✝⁵ : Category.{u_5, u_4} C\ninst✝⁴ : Category.{u_1, u_2} D\nF : C ⥤ D\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D D).obj F).Faithful\na b : A\nX : C\nthis : HasShift D A := HasShift.induced F A s i\n⊢ (shiftFunctorAdd D a b).inv.app (F.obj X) =\n    (s b).map ((i a).hom.app X) ≫\n      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X","tactic":"letI := HasShift.induced F A s i","premises":[{"full_name":"CategoryTheory.HasShift.induced","def_path":"Mathlib/CategoryTheory/Shift/Induced.lean","def_pos":[90,18],"def_end_pos":[90,25]}]},{"state_before":"C : Type u_4\nD : Type u_2\ninst✝⁵ : Category.{u_5, u_4} C\ninst✝⁴ : Category.{u_1, u_2} D\nF : C ⥤ D\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D D).obj F).Faithful\na b : A\nX : C\nthis : HasShift D A := HasShift.induced F A s i\n⊢ (shiftFunctorAdd D a b).inv.app (F.obj X) =\n    (s b).map ((i a).hom.app X) ≫\n      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X","state_after":"no goals","tactic":"simp only [ShiftMkCore.shiftFunctorAdd_eq, HasShift.Induced.add_inv_app_obj]","premises":[{"full_name":"CategoryTheory.HasShift.Induced.add_inv_app_obj","def_path":"Mathlib/CategoryTheory/Shift/Induced.lean","def_pos":[76,6],"def_end_pos":[76,21]},{"full_name":"CategoryTheory.ShiftMkCore.shiftFunctorAdd_eq","def_path":"Mathlib/CategoryTheory/Shift/Basic.lean","def_pos":[204,6],"def_end_pos":[204,36]}]}]}
{"url":"Mathlib/Analysis/Analytic/Composition.lean","commit":"","full_name":"HasFPowerSeriesAt.comp","start":[683,0],"end":[804,9],"file_path":"Mathlib/Analysis/Analytic/Composition.lean","tactics":[{"state_before":"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhg : HasFPowerSeriesAt g q (f x)\nhf : HasFPowerSeriesAt f p x\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f p x\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","tactic":"rcases hg with ⟨rg, Hg⟩","premises":[]},{"state_before":"case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f p x\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","tactic":"rcases hf with ⟨rf, Hf⟩","premises":[]},{"state_before":"case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","tactic":"rcases q.comp_summable_nnreal p Hg.radius_pos Hf.radius_pos with ⟨r, r_pos : 0 < r, hr⟩","premises":[{"full_name":"FormalMultilinearSeries.comp_summable_nnreal","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[430,8],"def_end_pos":[430,28]},{"full_name":"HasFPowerSeriesOnBall.radius_pos","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[436,8],"def_end_pos":[436,40]}]},{"state_before":"case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","tactic":"obtain ⟨δ, δpos, hδ⟩ :\n    ∃ δ : ℝ≥0∞, 0 < δ ∧ ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg := by\n    have : EMetric.ball (f x) rg ∈ 𝓝 (f x) := EMetric.ball_mem_nhds _ Hg.r_pos\n    rcases EMetric.mem_nhds_iff.1 (Hf.analyticAt.continuousAt this) with ⟨δ, δpos, Hδ⟩\n    exact ⟨δ, δpos, fun hz => Hδ hz⟩","premises":[{"full_name":"AnalyticAt.continuousAt","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[839,18],"def_end_pos":[839,41]},{"full_name":"And","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[516,10],"def_end_pos":[516,13]},{"full_name":"And.intro","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[518,2],"def_end_pos":[518,7]},{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"EMetric.ball_mem_nhds","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[623,8],"def_end_pos":[623,21]},{"full_name":"EMetric.mem_nhds_iff","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[583,8],"def_end_pos":[583,20]},{"full_name":"ENNReal","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[96,4],"def_end_pos":[96,11]},{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"HasFPowerSeriesOnBall.analyticAt","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[409,8],"def_end_pos":[409,40]},{"full_name":"HasFPowerSeriesOnBall.r_pos","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[353,2],"def_end_pos":[353,7]},{"full_name":"Iff.mp","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[118,2],"def_end_pos":[118,4]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","tactic":"let rf' := min rf δ","premises":[{"full_name":"Min.min","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1142,2],"def_end_pos":[1142,5]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","tactic":"have min_pos : 0 < min rf' r := by\n    simp only [rf', r_pos, Hf.r_pos, δpos, lt_min_iff, ENNReal.coe_pos, and_self_iff]","premises":[{"full_name":"ENNReal.coe_pos","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[338,27],"def_end_pos":[338,34]},{"full_name":"HasFPowerSeriesOnBall.r_pos","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[353,2],"def_end_pos":[353,7]},{"full_name":"Min.min","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1142,2],"def_end_pos":[1142,5]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"lt_min_iff","def_path":"Mathlib/Order/MinMax.lean","def_pos":[47,8],"def_end_pos":[47,18]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\n⊢ HasFPowerSeriesAt (g ∘ f) (q.comp p) x","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\n⊢ HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x (min rf' ↑r)","tactic":"refine ⟨min rf' r, ?_⟩","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Min.min","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1142,2],"def_end_pos":[1142,5]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\n⊢ HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x (min rf' ↑r)","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"refine\n    ⟨le_trans (min_le_right rf' r) (FormalMultilinearSeries.le_comp_radius_of_summable q p r hr),\n      min_pos, @fun y hy => ?_⟩","premises":[{"full_name":"FormalMultilinearSeries.le_comp_radius_of_summable","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[491,8],"def_end_pos":[491,34]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"min_le_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[42,8],"def_end_pos":[42,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have y_mem : y ∈ EMetric.ball (0 : E) rf :=\n    (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_left _ _))) hy","premises":[{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"EMetric.ball_subset_ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[520,8],"def_end_pos":[520,24]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"min_le_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[36,8],"def_end_pos":[36,19]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have fy_mem : f (x + y) ∈ EMetric.ball (f x) rg := by\n    apply hδ\n    have : y ∈ EMetric.ball (0 : E) δ :=\n      (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_right _ _))) hy\n    simpa [edist_eq_coe_nnnorm_sub, edist_eq_coe_nnnorm]","premises":[{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"EMetric.ball_subset_ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[520,8],"def_end_pos":[520,24]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"edist_eq_coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[694,14],"def_end_pos":[694,33]},{"full_name":"edist_eq_coe_nnnorm_sub","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[690,2],"def_end_pos":[690,13]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"min_le_left","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[36,8],"def_end_pos":[36,19]},{"full_name":"min_le_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[42,8],"def_end_pos":[42,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have A : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, p a fun _b => y)\n      atTop (𝓝 (f (x + y) - f x)) := by\n    have L :\n      ∀ᶠ n in atTop, (∑ a ∈ Finset.range n, p a fun _b => y) - f x\n        = ∑ a ∈ Finset.Ico 1 n, p a fun _b => y := by\n      rw [eventually_atTop]\n      refine ⟨1, fun n hn => ?_⟩\n      symm\n      rw [eq_sub_iff_add_eq', Finset.range_eq_Ico, ← Hf.coeff_zero fun _i => y,\n        Finset.sum_eq_sum_Ico_succ_bot hn]\n    have :\n      Tendsto (fun n => (∑ a ∈ Finset.range n, p a fun _b => y) - f x) atTop\n        (𝓝 (f (x + y) - f x)) :=\n      (Hf.hasSum y_mem).tendsto_sum_nat.sub tendsto_const_nhds\n    exact Tendsto.congr' L this","premises":[{"full_name":"Exists.intro","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[233,4],"def_end_pos":[233,9]},{"full_name":"Filter.Eventually","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[948,14],"def_end_pos":[948,24]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.Tendsto.congr'","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2634,8],"def_end_pos":[2634,22]},{"full_name":"Filter.Tendsto.sub","def_path":"Mathlib/Topology/Algebra/Group/Basic.lean","def_pos":[927,14],"def_end_pos":[927,17]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Filter.eventually_atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[155,8],"def_end_pos":[155,24]},{"full_name":"Finset.Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[281,4],"def_end_pos":[281,7]},{"full_name":"Finset.range","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[2442,4],"def_end_pos":[2442,9]},{"full_name":"Finset.range_eq_Ico","def_path":"Mathlib/Order/Interval/Finset/Nat.lean","def_pos":[66,8],"def_end_pos":[66,34]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.sum_eq_sum_Ico_succ_bot","def_path":"Mathlib/Algebra/BigOperators/Intervals.lean","def_pos":[107,2],"def_end_pos":[107,13]},{"full_name":"HasFPowerSeriesOnBall.coeff_zero","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[584,8],"def_end_pos":[584,40]},{"full_name":"HasFPowerSeriesOnBall.hasSum","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[354,2],"def_end_pos":[354,8]},{"full_name":"HasSum.tendsto_sum_nat","def_path":"Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean","def_pos":[42,2],"def_end_pos":[42,13]},{"full_name":"eq_sub_iff_add_eq'","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[917,14],"def_end_pos":[917,32]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]},{"full_name":"tendsto_const_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[844,8],"def_end_pos":[844,26]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have B :\n    Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, p a fun _b => y)) atTop\n      (𝓝 (g (f (x + y)))) := by\n    -- we use the fact that the partial sums of `q` converge locally uniformly to `g`, and that\n    -- composition passes to the limit under locally uniform convergence.\n    have B₁ : ContinuousAt (fun z : F => g (f x + z)) (f (x + y) - f x) := by\n      refine ContinuousAt.comp ?_ (continuous_const.add continuous_id).continuousAt\n      simp only [add_sub_cancel, _root_.id]\n      exact Hg.continuousOn.continuousAt (IsOpen.mem_nhds EMetric.isOpen_ball fy_mem)\n    have B₂ : f (x + y) - f x ∈ EMetric.ball (0 : F) rg := by\n      simpa [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub] using fy_mem\n    rw [← EMetric.isOpen_ball.nhdsWithin_eq B₂] at A\n    convert Hg.tendstoLocallyUniformlyOn.tendsto_comp B₁.continuousWithinAt B₂ A\n    simp only [add_sub_cancel]","premises":[{"full_name":"Continuous.add","def_path":"Mathlib/Topology/Algebra/Monoid.lean","def_pos":[93,2],"def_end_pos":[93,13]},{"full_name":"Continuous.continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1424,8],"def_end_pos":[1424,31]},{"full_name":"ContinuousAt","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[152,4],"def_end_pos":[152,16]},{"full_name":"ContinuousAt.comp","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1401,15],"def_end_pos":[1401,32]},{"full_name":"ContinuousAt.continuousWithinAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[752,8],"def_end_pos":[752,39]},{"full_name":"ContinuousOn.continuousAt","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[768,8],"def_end_pos":[768,33]},{"full_name":"EMetric.ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[484,4],"def_end_pos":[484,8]},{"full_name":"EMetric.isOpen_ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[615,8],"def_end_pos":[615,19]},{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Finset.Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[281,4],"def_end_pos":[281,7]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"FormalMultilinearSeries.partialSum","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[102,4],"def_end_pos":[102,14]},{"full_name":"HasFPowerSeriesOnBall.continuousOn","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[828,18],"def_end_pos":[828,52]},{"full_name":"HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn","def_path":"Mathlib/Analysis/Analytic/Basic.lean","def_pos":[793,8],"def_end_pos":[793,55]},{"full_name":"IsOpen.mem_nhds","def_path":"Mathlib/Topology/Basic.lean","def_pos":[744,8],"def_end_pos":[744,23]},{"full_name":"IsOpen.nhdsWithin_eq","def_path":"Mathlib/Topology/ContinuousOn.lean","def_pos":[177,8],"def_end_pos":[177,28]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"TendstoLocallyUniformlyOn.tendsto_comp","def_path":"Mathlib/Topology/UniformSpace/UniformConvergence.lean","def_pos":[850,8],"def_end_pos":[850,46]},{"full_name":"add_sub_cancel","def_path":"Mathlib/Algebra/Group/Basic.lean","def_pos":[926,2],"def_end_pos":[926,13]},{"full_name":"continuous_const","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1436,8],"def_end_pos":[1436,24]},{"full_name":"continuous_id","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1382,8],"def_end_pos":[1382,21]},{"full_name":"edist_eq_coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[694,14],"def_end_pos":[694,33]},{"full_name":"edist_eq_coe_nnnorm_sub","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[690,2],"def_end_pos":[690,13]},{"full_name":"id","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[33,14],"def_end_pos":[33,16]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have C :\n    Tendsto\n      (fun n => ∑ i ∈ compPartialSumTarget 0 n n, q.compAlongComposition p i.2 fun _j => y)\n      atTop (𝓝 (g (f (x + y)))) := by\n    simpa [comp_partialSum] using B","premises":[{"full_name":"Filter.Tendsto","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[2567,4],"def_end_pos":[2567,11]},{"full_name":"Filter.atTop","def_path":"Mathlib/Order/Filter/AtTopBot.lean","def_pos":[40,4],"def_end_pos":[40,9]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"FormalMultilinearSeries.compAlongComposition","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[209,4],"def_end_pos":[209,24]},{"full_name":"FormalMultilinearSeries.compPartialSumTarget","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[582,4],"def_end_pos":[582,24]},{"full_name":"FormalMultilinearSeries.comp_partialSum","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[657,8],"def_end_pos":[657,23]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]},{"full_name":"nhds","def_path":"Mathlib/Topology/Defs/Filter.lean","def_pos":[113,16],"def_end_pos":[113,20]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\nD : HasSum (fun i => (q.compAlongComposition p i.snd) fun _j => y) (g (f (x + y)))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have D :\n    HasSum (fun i : Σ n, Composition n => q.compAlongComposition p i.2 fun _j => y)\n      (g (f (x + y))) :=\n    haveI cau :\n      CauchySeq fun s : Finset (Σ n, Composition n) =>\n        ∑ i ∈ s, q.compAlongComposition p i.2 fun _j => y := by\n      apply cauchySeq_finset_of_norm_bounded _ (NNReal.summable_coe.2 hr) _\n      simp only [coe_nnnorm, NNReal.coe_mul, NNReal.coe_pow]\n      rintro ⟨n, c⟩\n      calc\n        ‖(compAlongComposition q p c) fun _j : Fin n => y‖ ≤\n            ‖compAlongComposition q p c‖ * ∏ _j : Fin n, ‖y‖ := by\n          apply ContinuousMultilinearMap.le_opNorm\n        _ ≤ ‖compAlongComposition q p c‖ * (r : ℝ) ^ n := by\n          apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)\n          rw [Finset.prod_const, Finset.card_fin]\n          apply pow_le_pow_left (norm_nonneg _)\n          rw [EMetric.mem_ball, edist_eq_coe_nnnorm] at hy\n          have := le_trans (le_of_lt hy) (min_le_right _ _)\n          rwa [ENNReal.coe_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this\n    tendsto_nhds_of_cauchySeq_of_subseq cau compPartialSumTarget_tendsto_atTop C","premises":[{"full_name":"CauchySeq","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[176,4],"def_end_pos":[176,13]},{"full_name":"Composition","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[96,10],"def_end_pos":[96,21]},{"full_name":"ContinuousMultilinearMap.le_opNorm","def_path":"Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean","def_pos":[348,8],"def_end_pos":[348,17]},{"full_name":"EMetric.mem_ball","def_path":"Mathlib/Topology/EMetricSpace/Basic.lean","def_pos":[487,16],"def_end_pos":[487,24]},{"full_name":"ENNReal.coe_le_coe","def_path":"Mathlib/Data/ENNReal/Basic.lean","def_pos":[314,27],"def_end_pos":[314,37]},{"full_name":"Fin","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1831,10],"def_end_pos":[1831,13]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.card_fin","def_path":"Mathlib/Data/Fintype/Card.lean","def_pos":[291,8],"def_end_pos":[291,23]},{"full_name":"Finset.prod","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[58,14],"def_end_pos":[58,18]},{"full_name":"Finset.prod_const","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[1470,8],"def_end_pos":[1470,18]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Finset.univ","def_path":"Mathlib/Data/Fintype/Basic.lean","def_pos":[67,4],"def_end_pos":[67,8]},{"full_name":"FormalMultilinearSeries.compAlongComposition","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[209,4],"def_end_pos":[209,24]},{"full_name":"FormalMultilinearSeries.compPartialSumTarget_tendsto_atTop","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[638,8],"def_end_pos":[638,42]},{"full_name":"HasSum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[72,2],"def_end_pos":[72,13]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"NNReal.coe_le_coe","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[325,25],"def_end_pos":[325,35]},{"full_name":"NNReal.coe_mul","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[164,18],"def_end_pos":[164,25]},{"full_name":"NNReal.coe_pow","def_path":"Mathlib/Data/NNReal/Basic.lean","def_pos":[258,8],"def_end_pos":[258,15]},{"full_name":"NNReal.summable_coe","def_path":"Mathlib/Topology/Instances/NNReal.lean","def_pos":[179,8],"def_end_pos":[179,20]},{"full_name":"Norm.norm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[59,2],"def_end_pos":[59,6]},{"full_name":"Real","def_path":"Mathlib/Data/Real/Basic.lean","def_pos":[32,10],"def_end_pos":[32,14]},{"full_name":"Sigma","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[169,10],"def_end_pos":[169,15]},{"full_name":"Sigma.snd","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[177,2],"def_end_pos":[177,5]},{"full_name":"cauchySeq_finset_of_norm_bounded","def_path":"Mathlib/Analysis/Normed/Group/InfiniteSum.lean","def_pos":[66,8],"def_end_pos":[66,40]},{"full_name":"coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[615,40],"def_end_pos":[615,50]},{"full_name":"edist_eq_coe_nnnorm","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[694,14],"def_end_pos":[694,33]},{"full_name":"le_of_lt","def_path":"Mathlib/Order/Defs.lean","def_pos":[89,8],"def_end_pos":[89,16]},{"full_name":"le_trans","def_path":"Mathlib/Order/Defs.lean","def_pos":[48,8],"def_end_pos":[48,16]},{"full_name":"min_le_right","def_path":"Mathlib/Init/Order/LinearOrder.lean","def_pos":[42,8],"def_end_pos":[42,20]},{"full_name":"mul_le_mul_of_nonneg_left","def_path":"Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean","def_pos":[190,8],"def_end_pos":[190,33]},{"full_name":"norm_nonneg","def_path":"Mathlib/Analysis/Normed/Group/Basic.lean","def_pos":[401,29],"def_end_pos":[401,40]},{"full_name":"pow_le_pow_left","def_path":"Mathlib/Algebra/Order/Ring/Basic.lean","def_pos":[90,8],"def_end_pos":[90,23]},{"full_name":"tendsto_nhds_of_cauchySeq_of_subseq","def_path":"Mathlib/Topology/UniformSpace/Cauchy.lean","def_pos":[268,8],"def_end_pos":[268,43]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\nD : HasSum (fun i => (q.compAlongComposition p i.snd) fun _j => y) (g (f (x + y)))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE✝ : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E✝ → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E✝ F\nx : E✝\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E✝}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E✝\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\nD : HasSum (fun i => (q.compAlongComposition p i.snd) fun _j => y) (g (f (x + y)))\nE : HasSum (fun n => (q.comp p n) fun _j => y) (g (f (x + y)))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","tactic":"have E : HasSum (fun n => (q.comp p) n fun _j => y) (g (f (x + y))) := by\n    apply D.sigma\n    intro n\n    dsimp [FormalMultilinearSeries.comp]\n    convert hasSum_fintype (α := G) (β := Composition n) _\n    simp only [ContinuousMultilinearMap.sum_apply]\n    rfl","premises":[{"full_name":"Composition","def_path":"Mathlib/Combinatorics/Enumerative/Composition.lean","def_pos":[96,10],"def_end_pos":[96,21]},{"full_name":"ContinuousMultilinearMap.sum_apply","def_path":"Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean","def_pos":[199,8],"def_end_pos":[199,17]},{"full_name":"FormalMultilinearSeries.comp","def_path":"Mathlib/Analysis/Analytic/Composition.lean","def_pos":[229,14],"def_end_pos":[229,18]},{"full_name":"HasSum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[72,2],"def_end_pos":[72,13]},{"full_name":"HasSum.sigma","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean","def_pos":[79,2],"def_end_pos":[79,13]},{"full_name":"hasSum_fintype","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[131,2],"def_end_pos":[131,13]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE✝ : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E✝ → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E✝ F\nx : E✝\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E✝}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E✝\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\nD : HasSum (fun i => (q.compAlongComposition p i.snd) fun _j => y) (g (f (x + y)))\nE : HasSum (fun n => (q.comp p n) fun _j => y) (g (f (x + y)))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))","state_after":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE✝ : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E✝ → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E✝ F\nx : E✝\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E✝}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E✝\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\nD : HasSum (fun i => (q.compAlongComposition p i.snd) fun _j => y) (g (f (x + y)))\nE : HasSum (fun n => (q.comp p n) fun _j => y) (g (f (x + y)))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) (g (f (x + y)))","tactic":"rw [Function.comp_apply]","premises":[{"full_name":"Function.comp_apply","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[35,16],"def_end_pos":[35,35]}]},{"state_before":"case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE✝ : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E✝\ninst✝⁶ : NormedSpace 𝕜 E✝\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E✝ → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E✝ F\nx : E✝\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E✝}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E✝\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\nC :\n  Tendsto (fun n => ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j => y) atTop\n    (𝓝 (g (f (x + y))))\nD : HasSum (fun i => (q.compAlongComposition p i.snd) fun _j => y) (g (f (x + y)))\nE : HasSum (fun n => (q.comp p n) fun _j => y) (g (f (x + y)))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) (g (f (x + y)))","state_after":"no goals","tactic":"exact E","premises":[]}]}
{"url":"Mathlib/Data/PFunctor/Multivariate/M.lean","commit":"","full_name":"MvPFunctor.M.dest_corec","start":[190,0],"end":[197,5],"file_path":"Mathlib/Data/PFunctor/Multivariate/M.lean","tactics":[{"state_before":"n : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\n⊢ dest P (corec P g x) = (TypeVec.id ::: corec P g) <$$> g x","state_after":"n : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\n⊢ dest P (corec P g x) = ?m.10810\n\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\n⊢ ?m.10810 = (TypeVec.id ::: corec P g) <$$> g x\n\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\n⊢ ↑P (α ::: P.M α)","tactic":"trans","premises":[]},{"state_before":"n : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\n⊢ ⟨(g x).fst,\n      splitFun (dropFun (g x).snd)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun (g x).snd)⟩ =\n    (TypeVec.id ::: corec P g) <$$> g x","state_after":"case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨⟨a, f⟩.fst,\n      splitFun (dropFun ⟨a, f⟩.snd)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘\n          lastFun ⟨a, f⟩.snd)⟩ =\n    (TypeVec.id ::: corec P g) <$$> ⟨a, f⟩","tactic":"cases' g x with a f","premises":[]},{"state_before":"case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨⟨a, f⟩.fst,\n      splitFun (dropFun ⟨a, f⟩.snd)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘\n          lastFun ⟨a, f⟩.snd)⟩ =\n    (TypeVec.id ::: corec P g) <$$> ⟨a, f⟩","state_after":"case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨a,\n      splitFun (dropFun f)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f)⟩ =\n    (TypeVec.id ::: corec P g) <$$> ⟨a, f⟩","tactic":"dsimp","premises":[]},{"state_before":"case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨a,\n      splitFun (dropFun f)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f)⟩ =\n    (TypeVec.id ::: corec P g) <$$> ⟨a, f⟩","state_after":"case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨a,\n      splitFun (dropFun f)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f)⟩ =\n    ⟨a, (TypeVec.id ::: corec P g) ⊚ f⟩","tactic":"rw [MvPFunctor.map_eq]","premises":[{"full_name":"MvPFunctor.map_eq","def_path":"Mathlib/Data/PFunctor/Multivariate/Basic.lean","def_pos":[59,8],"def_end_pos":[59,14]}]},{"state_before":"case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨a,\n      splitFun (dropFun f)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f)⟩ =\n    ⟨a, (TypeVec.id ::: corec P g) ⊚ f⟩","state_after":"case mk.e_snd\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ splitFun (dropFun f)\n      ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f) =\n    (TypeVec.id ::: corec P g) ⊚ f","tactic":"congr","premises":[]},{"state_before":"case mk.e_snd\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ splitFun (dropFun f)\n      ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f) =\n    (TypeVec.id ::: corec P g) ⊚ f","state_after":"case mk.e_snd\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ splitFun (dropFun f)\n      ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f) =\n    splitFun (TypeVec.id ⊚ dropFun f) (corec P g ∘ lastFun f)","tactic":"conv_rhs => rw [← split_dropFun_lastFun f, appendFun_comp_splitFun]","premises":[{"full_name":"TypeVec.appendFun_comp_splitFun","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[208,8],"def_end_pos":[208,31]},{"full_name":"TypeVec.split_dropFun_lastFun","def_path":"Mathlib/Data/TypeVec.lean","def_pos":[190,8],"def_end_pos":[190,29]}]},{"state_before":"case mk.e_snd\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ splitFun (dropFun f)\n      ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f) =\n    splitFun (TypeVec.id ⊚ dropFun f) (corec P g ∘ lastFun f)","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Analysis/MellinTransform.lean","commit":"","full_name":"mellinConvergent_of_isBigO_rpow","start":[263,0],"end":[269,99],"file_path":"Mathlib/Analysis/MellinTransform.lean","tactics":[{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s.re\n⊢ MellinConvergent f s","state_after":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s.re\n⊢ IntegrableOn (fun t => t ^ (s.re - 1) * ‖f t‖) (Ioi 0) volume","tactic":"rw [MellinConvergent,\n    mellin_convergent_iff_norm Subset.rfl measurableSet_Ioi hfc.aestronglyMeasurable]","premises":[{"full_name":"MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable","def_path":"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean","def_pos":[123,8],"def_end_pos":[123,48]},{"full_name":"MellinConvergent","def_path":"Mathlib/Analysis/MellinTransform.lean","def_pos":[41,4],"def_end_pos":[41,20]},{"full_name":"Set.Subset.rfl","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[292,8],"def_end_pos":[292,18]},{"full_name":"measurableSet_Ioi","def_path":"Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean","def_pos":[170,8],"def_end_pos":[170,25]},{"full_name":"mellin_convergent_iff_norm","def_path":"Mathlib/Analysis/MellinTransform.lean","def_pos":[177,8],"def_end_pos":[177,34]}]},{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s.re\n⊢ IntegrableOn (fun t => t ^ (s.re - 1) * ‖f t‖) (Ioi 0) volume","state_after":"no goals","tactic":"exact mellin_convergent_of_isBigO_scalar hfc.norm hf_top.norm_left hs_top hf_bot.norm_left hs_bot","premises":[{"full_name":"MeasureTheory.LocallyIntegrableOn.norm","def_path":"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean","def_pos":[47,8],"def_end_pos":[47,32]},{"full_name":"mellin_convergent_of_isBigO_scalar","def_path":"Mathlib/Analysis/MellinTransform.lean","def_pos":[246,8],"def_end_pos":[246,42]}]}]}
{"url":"Mathlib/Algebra/Category/Grp/Basic.lean","commit":"","full_name":"Grp.uliftFunctor_map","start":[143,0],"end":[151,32],"file_path":"Mathlib/Algebra/Category/Grp/Basic.lean","tactics":[{"state_before":"X : Grp\n⊢ { obj := fun X => of (ULift.{v, u} ↑X),\n          map := fun {X Y} f =>\n            ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map\n      (𝟙 X) =\n    𝟙\n      ({ obj := fun X => of (ULift.{v, u} ↑X),\n            map := fun {X Y} f =>\n              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.obj\n        X)","state_after":"no goals","tactic":"rfl","premises":[]},{"state_before":"X Y Z : Grp\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ { obj := fun X => of (ULift.{v, u} ↑X),\n          map := fun {X Y} f =>\n            ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map\n      (f ≫ g) =\n    { obj := fun X => of (ULift.{v, u} ↑X),\n            map := fun {X Y} f =>\n              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map\n        f ≫\n      { obj := fun X => of (ULift.{v, u} ↑X),\n            map := fun {X Y} f =>\n              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map\n        g","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/Order/Filter/CountableInter.lean","commit":"","full_name":"Filter.mem_countableGenerate_iff","start":[240,0],"end":[258,42],"file_path":"Mathlib/Order/Filter/CountableInter.lean","tactics":[{"state_before":"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\n⊢ s ∈ countableGenerate g ↔ ∃ S ⊆ g, S.Countable ∧ ⋂₀ S ⊆ s","state_after":"case mp\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nh : s ∈ countableGenerate g\n⊢ ∃ S ⊆ g, S.Countable ∧ ⋂₀ S ⊆ s\n\ncase mpr\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nh : ∃ S ⊆ g, S.Countable ∧ ⋂₀ S ⊆ s\n⊢ s ∈ countableGenerate g","tactic":"constructor <;> intro h","premises":[]},{"state_before":"case mpr\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nh : ∃ S ⊆ g, S.Countable ∧ ⋂₀ S ⊆ s\n⊢ s ∈ countableGenerate g","state_after":"case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s\n⊢ s ∈ countableGenerate g","tactic":"rcases h with ⟨S, Sg, Sct, hS⟩","premises":[]},{"state_before":"case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s\n⊢ s ∈ countableGenerate g","state_after":"case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s\n⊢ ∀ s ∈ S, s ∈ countableGenerate g","tactic":"refine mem_of_superset ((countable_sInter_mem Sct).mpr ?_) hS","premises":[{"full_name":"Filter.mem_of_superset","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[139,8],"def_end_pos":[139,23]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"countable_sInter_mem","def_path":"Mathlib/Order/Filter/CountableInter.lean","def_pos":[46,8],"def_end_pos":[46,28]}]},{"state_before":"case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s\n⊢ ∀ s ∈ S, s ∈ countableGenerate g","state_after":"case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns✝ : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s✝\ns : Set α\nH : s ∈ S\n⊢ s ∈ countableGenerate g","tactic":"intro s H","premises":[]},{"state_before":"case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns✝ : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s✝\ns : Set α\nH : s ∈ S\n⊢ s ∈ countableGenerate g","state_after":"no goals","tactic":"exact CountableGenerateSets.basic (Sg H)","premises":[{"full_name":"Filter.CountableGenerateSets.basic","def_path":"Mathlib/Order/Filter/CountableInter.lean","def_pos":[222,4],"def_end_pos":[222,9]}]}]}
{"url":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","commit":"","full_name":"LinearMap.isUnit_iff_range_eq_top","start":[642,0],"end":[644,57],"file_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","tactics":[{"state_before":"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ IsUnit f ↔ range f = ⊤","state_after":"no goals","tactic":"rw [isUnit_iff_ker_eq_bot, ker_eq_bot_iff_range_eq_top]","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"LinearMap.isUnit_iff_ker_eq_bot","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[631,8],"def_end_pos":[631,29]},{"full_name":"LinearMap.ker_eq_bot_iff_range_eq_top","def_path":"Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean","def_pos":[537,8],"def_end_pos":[537,35]}]}]}
{"url":"Mathlib/Computability/Halting.lean","commit":"","full_name":"ComputablePred.computable_iff_re_compl_re'","start":[249,0],"end":[251,44],"file_path":"Mathlib/Computability/Halting.lean","tactics":[{"state_before":"α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\np : α → Prop\n⊢ ComputablePred p ↔ RePred p ∧ RePred fun a => ¬p a","state_after":"no goals","tactic":"classical exact computable_iff_re_compl_re","premises":[{"full_name":"ComputablePred.computable_iff_re_compl_re","def_path":"Mathlib/Computability/Halting.lean","def_pos":[232,8],"def_end_pos":[232,34]}]}]}
{"url":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","commit":"","full_name":"Matrix.det_smul_inv_vecMul_eq_cramer_transpose","start":[632,0],"end":[637,66],"file_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","tactics":[{"state_before":"l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nb : n → α\nh : IsUnit A.det\n⊢ A.det • b ᵥ* A⁻¹ = Aᵀ.cramer b","state_after":"no goals","tactic":"rw [← A⁻¹.transpose_transpose, vecMul_transpose, transpose_nonsing_inv, ← det_transpose,\n    Aᵀ.det_smul_inv_mulVec_eq_cramer _ (isUnit_det_transpose A h)]","premises":[{"full_name":"Inv.inv","def_path":"Mathlib/Algebra/Group/Defs.lean","def_pos":[163,2],"def_end_pos":[163,5]},{"full_name":"Matrix.det_smul_inv_mulVec_eq_cramer","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[627,8],"def_end_pos":[627,37]},{"full_name":"Matrix.det_transpose","def_path":"Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean","def_pos":[198,8],"def_end_pos":[198,21]},{"full_name":"Matrix.isUnit_det_transpose","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[183,8],"def_end_pos":[183,28]},{"full_name":"Matrix.transpose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[140,4],"def_end_pos":[140,13]},{"full_name":"Matrix.transpose_nonsing_inv","def_path":"Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean","def_pos":[224,8],"def_end_pos":[224,29]},{"full_name":"Matrix.transpose_transpose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1797,8],"def_end_pos":[1797,27]},{"full_name":"Matrix.vecMul_transpose","def_path":"Mathlib/Data/Matrix/Basic.lean","def_pos":[1753,8],"def_end_pos":[1753,24]}]}]}
{"url":"Mathlib/Algebra/Group/Subgroup/Finite.lean","commit":"","full_name":"AddSubgroup.card_eq_iff_eq_top","start":[117,0],"end":[119,76],"file_path":"Mathlib/Algebra/Group/Subgroup/Finite.lean","tactics":[{"state_before":"G : Type u_1\ninst✝² : Group G\nA : Type u_2\ninst✝¹ : AddGroup A\nH K : Subgroup G\ninst✝ : Finite ↥H\nh : H = ⊤\n⊢ Nat.card ↥H = Nat.card G","state_after":"no goals","tactic":"simpa only [h] using card_top","premises":[{"full_name":"Subgroup.card_top","def_path":"Mathlib/Algebra/Group/Subgroup/Finite.lean","def_pos":[101,8],"def_end_pos":[101,16]}]}]}
{"url":"Mathlib/ModelTheory/Satisfiability.lean","commit":"","full_name":"FirstOrder.Language.Theory.models_of_models_theory","start":[327,0],"end":[334,13],"file_path":"Mathlib/ModelTheory/Satisfiability.lean","tactics":[{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nh : ∀ φ ∈ T', T ⊨ᵇ φ\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\n⊢ T ⊨ᵇ φ","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\n⊢ T ⊨ᵇ φ","tactic":"simp only [models_sentence_iff] at h","premises":[{"full_name":"FirstOrder.Language.Theory.models_sentence_iff","def_path":"Mathlib/ModelTheory/Satisfiability.lean","def_pos":[293,8],"def_end_pos":[293,27]}]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\n⊢ T ⊨ᵇ φ","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs","tactic":"intro M","premises":[]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\nhM : ↑M ⊨ T'\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs","tactic":"have hM : M ⊨ T' := T'.model_iff.2 (fun ψ hψ => h ψ hψ M)","premises":[{"full_name":"FirstOrder.Language.Theory.Model","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[658,6],"def_end_pos":[658,18]},{"full_name":"FirstOrder.Language.Theory.model_iff","def_path":"Mathlib/ModelTheory/Semantics.lean","def_pos":[668,8],"def_end_pos":[668,24]},{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]}]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\nhM : ↑M ⊨ T'\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs","state_after":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\nhM : ↑M ⊨ T'\nM' : T'.ModelType := ModelType.mk ↑M\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs","tactic":"let M' : ModelType T' := ⟨M⟩","premises":[{"full_name":"FirstOrder.Language.Theory.ModelType","def_path":"Mathlib/ModelTheory/Bundled.lean","def_pos":[67,10],"def_end_pos":[67,19]}]},{"state_before":"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\nhM : ↑M ⊨ T'\nM' : T'.ModelType := ModelType.mk ↑M\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs","state_after":"no goals","tactic":"exact hφ M'","premises":[]}]}
{"url":"Mathlib/Data/Int/Bitwise.lean","commit":"","full_name":"Int.bodd_negOfNat","start":[129,0],"end":[132,5],"file_path":"Mathlib/Data/Int/Bitwise.lean","tactics":[{"state_before":"n : ℕ\n⊢ (negOfNat n).bodd = n.bodd","state_after":"case succ\nn✝ : ℕ\n⊢ (negOfNat (n✝ + 1)).bodd = !n✝.bodd","tactic":"cases n <;> simp (config := {decide := true})","premises":[{"full_name":"Bool.true","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[571,4],"def_end_pos":[571,8]}]},{"state_before":"case succ\nn✝ : ℕ\n⊢ (negOfNat (n✝ + 1)).bodd = !n✝.bodd","state_after":"no goals","tactic":"rfl","premises":[]}]}
{"url":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","commit":"","full_name":"CircleIntegrable.out","start":[223,0],"end":[231,20],"file_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","tactics":[{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : CircleIntegrable f c R\n⊢ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)","state_after":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume","tactic":"simp only [CircleIntegrable, deriv_circleMap, intervalIntegrable_iff] at *","premises":[{"full_name":"CircleIntegrable","def_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","def_pos":[205,4],"def_end_pos":[205,20]},{"full_name":"deriv_circleMap","def_path":"Mathlib/MeasureTheory/Integral/CircleIntegral.lean","def_pos":[172,8],"def_end_pos":[172,23]},{"full_name":"intervalIntegrable_iff","def_path":"Mathlib/MeasureTheory/Integral/IntervalIntegral.lean","def_pos":[80,8],"def_end_pos":[80,30]}]},{"state_before":"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume","state_after":"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ AEStronglyMeasurable (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (volume.restrict (Ι 0 (2 * π)))\n\ncase refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ι 0 (2 * π)), ‖(circleMap 0 R a * I) • f (circleMap c R a)‖ ≤ |R| * ‖f (circleMap c R a)‖","tactic":"refine (hf.norm.const_mul |R|).mono' ?_ ?_","premises":[{"full_name":"MeasureTheory.Integrable.const_mul","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[1080,8],"def_end_pos":[1080,28]},{"full_name":"MeasureTheory.Integrable.mono'","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[410,8],"def_end_pos":[410,24]},{"full_name":"MeasureTheory.Integrable.norm","def_path":"Mathlib/MeasureTheory/Function/L1Space.lean","def_pos":[659,8],"def_end_pos":[659,23]},{"full_name":"abs","def_path":"Mathlib/Algebra/Order/Group/Unbundled/Abs.lean","def_pos":[33,2],"def_end_pos":[33,13]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/DiracProba.lean","commit":"","full_name":"MeasureTheory.continuous_diracProbaEquivSymm","start":[164,0],"end":[170,79],"file_path":"Mathlib/MeasureTheory/Measure/DiracProba.lean","tactics":[{"state_before":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\n⊢ Continuous ⇑diracProbaEquiv.symm","state_after":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\n⊢ ∀ (x : ↑(range diracProba)), ContinuousAt (⇑diracProbaEquiv.symm) x","tactic":"apply continuous_iff_continuousAt.mpr","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"continuous_iff_continuousAt","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1427,8],"def_end_pos":[1427,35]}]},{"state_before":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\n⊢ ∀ (x : ↑(range diracProba)), ContinuousAt (⇑diracProbaEquiv.symm) x","state_after":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\nμ : ↑(range diracProba)\n⊢ ContinuousAt (⇑diracProbaEquiv.symm) μ","tactic":"intro μ","premises":[]},{"state_before":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\nμ : ↑(range diracProba)\n⊢ ContinuousAt (⇑diracProbaEquiv.symm) μ","state_after":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\nμ : ↑(range diracProba)\n⊢ Tendsto (⇑diracProbaEquiv.symm) (𝓝 μ) (𝓝 (diracProbaInverse μ))","tactic":"apply continuousAt_of_tendsto_nhds (y := diracProbaInverse μ)","premises":[{"full_name":"MeasureTheory.diracProbaInverse","def_path":"Mathlib/MeasureTheory/Measure/DiracProba.lean","def_pos":[122,18],"def_end_pos":[122,35]},{"full_name":"continuousAt_of_tendsto_nhds","def_path":"Mathlib/Topology/Separation.lean","def_pos":[849,8],"def_end_pos":[849,36]}]},{"state_before":"X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : T0Space X\ninst✝ : CompletelyRegularSpace X\nμ : ↑(range diracProba)\n⊢ Tendsto (⇑diracProbaEquiv.symm) (𝓝 μ) (𝓝 (diracProbaInverse μ))","state_after":"no goals","tactic":"exact (tendsto_diracProbaEquivSymm_iff_tendsto _).mpr fun _ mem_nhd ↦ mem_nhd","premises":[{"full_name":"Iff.mpr","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[120,2],"def_end_pos":[120,5]},{"full_name":"MeasureTheory.tendsto_diracProbaEquivSymm_iff_tendsto","def_path":"Mathlib/MeasureTheory/Measure/DiracProba.lean","def_pos":[148,6],"def_end_pos":[148,45]}]}]}
{"url":"Mathlib/Data/Set/Function.lean","commit":"","full_name":"Set.piecewise_insert_of_ne","start":[1400,0],"end":[1402,78],"file_path":"Mathlib/Data/Set/Function.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\nδ : α → Sort u_6\ns : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ni j : α\nh : i ≠ j\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ (insert j s).piecewise f g i = s.piecewise f g i","state_after":"no goals","tactic":"simp [piecewise, h]","premises":[{"full_name":"Set.piecewise","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[884,4],"def_end_pos":[884,17]}]}]}
{"url":"Mathlib/CategoryTheory/Comma/Over.lean","commit":"","full_name":"CategoryTheory.Under.under_left","start":[380,0],"end":[380,64],"file_path":"Mathlib/CategoryTheory/Comma/Over.lean","tactics":[{"state_before":"T : Type u₁\ninst✝ : Category.{v₁, u₁} T\nX : T\nU : Under X\n⊢ U.left = { as := PUnit.unit }","state_after":"no goals","tactic":"simp only","premises":[]}]}
{"url":"Mathlib/Data/Fin/Basic.lean","commit":"","full_name":"Fin.castSucc_pos'","start":[688,0],"end":[693,35],"file_path":"Mathlib/Data/Fin/Basic.lean","tactics":[{"state_before":"n m : ℕ\ninst✝ : NeZero n\ni : Fin n\nh : 0 < i\n⊢ 0 < i.castSucc","state_after":"no goals","tactic":"simpa [lt_iff_val_lt_val] using h","premises":[{"full_name":"Fin.lt_iff_val_lt_val","def_path":".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean","def_pos":[88,8],"def_end_pos":[88,25]}]}]}
{"url":"Mathlib/Order/Interval/Finset/Basic.lean","commit":"","full_name":"Finset.Icc_subset_Ico_right","start":[190,0],"end":[192,34],"file_path":"Mathlib/Order/Interval/Finset/Basic.lean","tactics":[{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Icc a b₁ ⊆ Ico a b₂","state_after":"ι : Type u_1\nα : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Set.Icc a b₁ ⊆ Set.Ico a b₂","tactic":"rw [← coe_subset, coe_Icc, coe_Ico]","premises":[{"full_name":"Finset.coe_Icc","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[311,8],"def_end_pos":[311,15]},{"full_name":"Finset.coe_Ico","def_path":"Mathlib/Order/Interval/Finset/Defs.lean","def_pos":[315,8],"def_end_pos":[315,15]},{"full_name":"Finset.coe_subset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[336,8],"def_end_pos":[336,18]}]},{"state_before":"ι : Type u_1\nα : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Set.Icc a b₁ ⊆ Set.Ico a b₂","state_after":"no goals","tactic":"exact Set.Icc_subset_Ico_right h","premises":[{"full_name":"Set.Icc_subset_Ico_right","def_path":"Mathlib/Order/Interval/Set/Basic.lean","def_pos":[415,8],"def_end_pos":[415,28]}]}]}
{"url":"Mathlib/Probability/Moments.lean","commit":"","full_name":"ProbabilityTheory.IndepFun.integrable_exp_mul_add","start":[238,0],"end":[243,60],"file_path":"Mathlib/Probability/Moments.lean","tactics":[{"state_before":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_indep : IndepFun X Y μ\nh_int_X : Integrable (fun ω => rexp (t * X ω)) μ\nh_int_Y : Integrable (fun ω => rexp (t * Y ω)) μ\n⊢ Integrable (fun ω => rexp (t * (X + Y) ω)) μ","state_after":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_indep : IndepFun X Y μ\nh_int_X : Integrable (fun ω => rexp (t * X ω)) μ\nh_int_Y : Integrable (fun ω => rexp (t * Y ω)) μ\n⊢ Integrable (fun ω => rexp (t * X ω) * rexp (t * Y ω)) μ","tactic":"simp_rw [Pi.add_apply, mul_add, exp_add]","premises":[{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"Pi.add_apply","def_path":"Mathlib/Algebra/Group/Pi/Basic.lean","def_pos":[81,2],"def_end_pos":[81,13]},{"full_name":"Real.exp_add","def_path":"Mathlib/Data/Complex/Exponential.lean","def_pos":[687,15],"def_end_pos":[687,22]}]},{"state_before":"Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_indep : IndepFun X Y μ\nh_int_X : Integrable (fun ω => rexp (t * X ω)) μ\nh_int_Y : Integrable (fun ω => rexp (t * Y ω)) μ\n⊢ Integrable (fun ω => rexp (t * X ω) * rexp (t * Y ω)) μ","state_after":"no goals","tactic":"exact (h_indep.exp_mul t t).integrable_mul h_int_X h_int_Y","premises":[{"full_name":"ProbabilityTheory.IndepFun.exp_mul","def_path":"Mathlib/Probability/Moments.lean","def_pos":[187,8],"def_end_pos":[187,24]},{"full_name":"ProbabilityTheory.IndepFun.integrable_mul","def_path":"Mathlib/Probability/Integration.lean","def_pos":[131,8],"def_end_pos":[131,31]}]}]}
{"url":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","commit":"","full_name":"MeasureTheory.measure_iUnion_eq_iSup","start":[420,0],"end":[454,45],"file_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\n⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nval✝ : Encodable ι\n⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)","tactic":"cases nonempty_encodable ι","premises":[{"full_name":"nonempty_encodable","def_path":"Mathlib/Logic/Encodable/Basic.lean","def_pos":[399,8],"def_end_pos":[399,26]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nval✝ : Encodable ι\n⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nval✝ : Encodable ι\nt : ℕ → Set α\nht : Function.extend Encodable.encode s ⊥ = t\n⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)","tactic":"generalize ht : Function.extend Encodable.encode s ⊥ = t","premises":[{"full_name":"Bot.bot","def_path":"Mathlib/Order/Notation.lean","def_pos":[100,2],"def_end_pos":[100,5]},{"full_name":"Encodable.encode","def_path":"Mathlib/Logic/Encodable/Basic.lean","def_pos":[45,2],"def_end_pos":[45,8]},{"full_name":"Function.extend","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[602,4],"def_end_pos":[602,10]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nval✝ : Encodable ι\nt : ℕ → Set α\nht : Function.extend Encodable.encode s ⊥ = t\nhd : Directed (fun x x_1 => x ⊆ x_1) t\n⊢ μ (⋃ n, t n) = ⨆ n, μ (t n)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\n⊢ μ (⋃ n, t n) = ⨆ n, μ (t n)","tactic":"clear! ι","premises":[]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\n⊢ μ (⋃ n, t n) = ⨆ n, μ (t n)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","tactic":"refine le_antisymm ?_ (iSup_le fun i => measure_mono <| subset_iUnion _ _)","premises":[{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"Set.subset_iUnion","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[246,8],"def_end_pos":[246,21]},{"full_name":"iSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[661,8],"def_end_pos":[661,15]},{"full_name":"le_antisymm","def_path":"Mathlib/Order/Defs.lean","def_pos":[156,8],"def_end_pos":[156,19]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","tactic":"set T : ℕ → Set α := fun n => toMeasurable μ (t n)","premises":[{"full_name":"MeasureTheory.toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[294,16],"def_end_pos":[294,28]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","tactic":"set Td : ℕ → Set α := disjointed T","premises":[{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Set","def_path":"Mathlib/Init/Set.lean","def_pos":[53,4],"def_end_pos":[53,7]},{"full_name":"disjointed","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[47,4],"def_end_pos":[47,14]},{"full_name":"rfl","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[294,21],"def_end_pos":[294,24]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","state_after":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\nhm : ∀ (n : ℕ), MeasurableSet (Td n)\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","tactic":"have hm : ∀ n, MeasurableSet (Td n) :=\n    MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _","premises":[{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"MeasurableSet.disjointed","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[210,18],"def_end_pos":[210,42]},{"full_name":"MeasureTheory.measurableSet_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[310,8],"def_end_pos":[310,34]}]},{"state_before":"case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\nhm : ∀ (n : ℕ), MeasurableSet (Td n)\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)","state_after":"no goals","tactic":"calc\n    μ (⋃ n, t n) ≤ μ (⋃ n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _)\n    _ = μ (⋃ n, Td n) := by rw [iUnion_disjointed]\n    _ ≤ ∑' n, μ (Td n) := measure_iUnion_le _\n    _ = ⨆ I : Finset ℕ, ∑ n ∈ I, μ (Td n) := ENNReal.tsum_eq_iSup_sum\n    _ ≤ ⨆ n, μ (t n) := iSup_le fun I => by\n      rcases hd.finset_le I with ⟨N, hN⟩\n      calc\n        (∑ n ∈ I, μ (Td n)) = μ (⋃ n ∈ I, Td n) :=\n          (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n        _ ≤ μ (⋃ n ∈ I, T n) := measure_mono (iUnion₂_mono fun n _hn => disjointed_subset _ _)\n        _ = μ (⋃ n ∈ I, t n) := measure_biUnion_toMeasurable I.countable_toSet _\n        _ ≤ μ (t N) := measure_mono (iUnion₂_subset hN)\n        _ ≤ ⨆ n, μ (t n) := le_iSup (μ ∘ t) N","premises":[{"full_name":"Directed.finset_le","def_path":"Mathlib/Data/Finset/Order.lean","def_pos":[17,8],"def_end_pos":[17,26]},{"full_name":"ENNReal.tsum_eq_iSup_sum","def_path":"Mathlib/Topology/Instances/ENNReal.lean","def_pos":[715,18],"def_end_pos":[715,34]},{"full_name":"Eq.symm","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[323,8],"def_end_pos":[323,15]},{"full_name":"Finset","def_path":"Mathlib/Data/Finset/Basic.lean","def_pos":[133,10],"def_end_pos":[133,16]},{"full_name":"Finset.countable_toSet","def_path":"Mathlib/Data/Set/Countable.lean","def_pos":[300,8],"def_end_pos":[300,30]},{"full_name":"Finset.sum","def_path":"Mathlib/Algebra/BigOperators/Group/Finset.lean","def_pos":[53,2],"def_end_pos":[53,13]},{"full_name":"Function.comp","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[52,14],"def_end_pos":[52,27]},{"full_name":"MeasureTheory.measure_biUnion_finset","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[166,8],"def_end_pos":[166,30]},{"full_name":"MeasureTheory.measure_biUnion_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[350,8],"def_end_pos":[350,36]},{"full_name":"MeasureTheory.measure_iUnion_le","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[58,8],"def_end_pos":[58,25]},{"full_name":"MeasureTheory.measure_mono","def_path":"Mathlib/MeasureTheory/OuterMeasure/Basic.lean","def_pos":[49,8],"def_end_pos":[49,20]},{"full_name":"MeasureTheory.subset_toMeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[300,8],"def_end_pos":[300,27]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Pairwise.set_pairwise","def_path":"Mathlib/Logic/Pairwise.lean","def_pos":[88,8],"def_end_pos":[88,29]},{"full_name":"Set.iUnion","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[178,4],"def_end_pos":[178,10]},{"full_name":"Set.iUnion_mono","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[285,8],"def_end_pos":[285,19]},{"full_name":"Set.iUnion₂_mono","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[294,8],"def_end_pos":[294,20]},{"full_name":"Set.iUnion₂_subset","def_path":"Mathlib/Data/Set/Lattice.lean","def_pos":[217,8],"def_end_pos":[217,22]},{"full_name":"disjoint_disjointed","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[67,8],"def_end_pos":[67,27]},{"full_name":"disjointed_subset","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[150,8],"def_end_pos":[150,25]},{"full_name":"iSup","def_path":"Mathlib/Order/SetNotation.lean","def_pos":[56,4],"def_end_pos":[56,8]},{"full_name":"iSup_le","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[661,8],"def_end_pos":[661,15]},{"full_name":"iUnion_disjointed","def_path":"Mathlib/Order/Disjointed.lean","def_pos":[153,8],"def_end_pos":[153,25]},{"full_name":"le_iSup","def_path":"Mathlib/Order/CompleteLattice.lean","def_pos":[617,8],"def_end_pos":[617,15]},{"full_name":"tsum","def_path":"Mathlib/Topology/Algebra/InfiniteSum/Defs.lean","def_pos":[94,2],"def_end_pos":[94,13]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/AssocList.lean","commit":"","full_name":"Batteries.AssocList.toList_mapKey","start":[97,0],"end":[99,26],"file_path":".lake/packages/batteries/Batteries/Data/AssocList.lean","tactics":[{"state_before":"α : Type u_1\nδ : Type u_2\nβ : Type u_3\nf : α → δ\nl : AssocList α β\n⊢ (mapKey f l).toList =\n    List.map\n      (fun x =>\n        match x with\n        | (a, b) => (f a, b))\n      l.toList","state_after":"no goals","tactic":"induction l <;> simp [*]","premises":[]}]}
{"url":"Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean","commit":"","full_name":"ProbabilityTheory.toRatCDF_unit_prod","start":[246,0],"end":[249,40],"file_path":"Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nf : α → ℚ → ℝ\na : α\n⊢ toRatCDF (fun p => f p.2) ((), a) = toRatCDF f a","state_after":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nf : α → ℚ → ℝ\na : α\n⊢ (if IsRatStieltjesPoint (fun p => f p.2) ((), a) then (fun p => f p.2) ((), a) else defaultRatCDF) =\n    if IsRatStieltjesPoint f a then f a else defaultRatCDF","tactic":"unfold toRatCDF","premises":[{"full_name":"ProbabilityTheory.toRatCDF","def_path":"Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean","def_pos":[228,4],"def_end_pos":[228,12]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nf : α → ℚ → ℝ\na : α\n⊢ (if IsRatStieltjesPoint (fun p => f p.2) ((), a) then (fun p => f p.2) ((), a) else defaultRatCDF) =\n    if IsRatStieltjesPoint f a then f a else defaultRatCDF","state_after":"no goals","tactic":"rw [isRatStieltjesPoint_unit_prod_iff]","premises":[{"full_name":"ProbabilityTheory.isRatStieltjesPoint_unit_prod_iff","def_path":"Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean","def_pos":[65,6],"def_end_pos":[65,39]}]}]}
{"url":"Mathlib/Algebra/Lie/Submodule.lean","commit":"","full_name":"LieSubmodule.map_incl_le","start":[1332,0],"end":[1336,36],"file_path":"Mathlib/Algebra/Lie/Submodule.lean","tactics":[{"state_before":"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nN' : LieSubmodule R L ↥↑N\n⊢ map N.incl N' ≤ N","state_after":"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nN' : LieSubmodule R L ↥↑N\n⊢ map N.incl N' ≤ map N.incl ⊤","tactic":"conv_rhs => rw [← N.map_incl_top]","premises":[{"full_name":"LieSubmodule.map_incl_top","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[1315,8],"def_end_pos":[1315,20]}]},{"state_before":"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nN' : LieSubmodule R L ↥↑N\n⊢ map N.incl N' ≤ map N.incl ⊤","state_after":"no goals","tactic":"exact LieSubmodule.map_mono le_top","premises":[{"full_name":"LieSubmodule.map_mono","def_path":"Mathlib/Algebra/Lie/Submodule.lean","def_pos":[833,8],"def_end_pos":[833,16]},{"full_name":"le_top","def_path":"Mathlib/Order/BoundedOrder.lean","def_pos":[62,8],"def_end_pos":[62,14]}]}]}
{"url":"Mathlib/Analysis/RCLike/Basic.lean","commit":"","full_name":"RCLike.conj_eq_iff_real","start":[335,0],"end":[336,60],"file_path":"Mathlib/Analysis/RCLike/Basic.lean","tactics":[{"state_before":"K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nz : K\n⊢ (∃ r, ↑r = z) ↔ ∃ r, z = ↑r","state_after":"no goals","tactic":"simp only [eq_comm]","premises":[{"full_name":"eq_comm","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[811,8],"def_end_pos":[811,15]}]}]}
{"url":"Mathlib/Topology/Order/OrderClosed.lean","commit":"","full_name":"bddBelow_closure","start":[360,0],"end":[361,41],"file_path":"Mathlib/Topology/Order/OrderClosed.lean","tactics":[{"state_before":"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : ClosedIciTopology α\nf : β → α\na b : α\ns : Set α\n⊢ BddBelow (closure s) ↔ BddBelow s","state_after":"no goals","tactic":"simp_rw [BddBelow, lowerBounds_closure]","premises":[{"full_name":"BddBelow","def_path":"Mathlib/Order/Bounds/Basic.lean","def_pos":[56,4],"def_end_pos":[56,12]},{"full_name":"Bool.false","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[569,4],"def_end_pos":[569,9]},{"full_name":"Lean.Meta.Simp.Config","def_path":".lake/packages/lean4/src/lean/Init/MetaTypes.lean","def_pos":[121,10],"def_end_pos":[121,16]},{"full_name":"lowerBounds_closure","def_path":"Mathlib/Topology/Order/OrderClosed.lean","def_pos":[357,14],"def_end_pos":[357,33]}]}]}
{"url":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","commit":"","full_name":"Sum.getRight_eq_iff","start":[62,0],"end":[63,25],"file_path":".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean","tactics":[{"state_before":"β✝ : Type u_1\nb : β✝\nα✝ : Type u_2\nx : α✝ ⊕ β✝\nh : x.isRight = true\n⊢ x.getRight h = b ↔ x = inr b","state_after":"no goals","tactic":"cases x <;> simp at h ⊢","premises":[]}]}
{"url":"Mathlib/Computability/TuringMachine.lean","commit":"","full_name":"Turing.Tape.map_move","start":[634,0],"end":[638,57],"file_path":"Mathlib/Computability/TuringMachine.lean","tactics":[{"state_before":"Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nT : Tape Γ\nd : Dir\n⊢ map f (move d T) = move d (map f T)","state_after":"case mk\nΓ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nd : Dir\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) =\n    move d (map f { head := head✝, left := left✝, right := right✝ })","tactic":"cases T","premises":[]},{"state_before":"case mk\nΓ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nd : Dir\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) =\n    move d (map f { head := head✝, left := left✝, right := right✝ })","state_after":"no goals","tactic":"cases d <;> simp only [Tape.move, Tape.map, ListBlank.head_map, eq_self_iff_true,\n    ListBlank.map_cons, and_self_iff, ListBlank.tail_map]","premises":[{"full_name":"Turing.ListBlank.head_map","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[361,8],"def_end_pos":[361,26]},{"full_name":"Turing.ListBlank.map_cons","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[373,8],"def_end_pos":[373,26]},{"full_name":"Turing.ListBlank.tail_map","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[367,8],"def_end_pos":[367,26]},{"full_name":"Turing.Tape.map","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[609,4],"def_end_pos":[609,12]},{"full_name":"Turing.Tape.move","def_path":"Mathlib/Computability/TuringMachine.lean","def_pos":[479,4],"def_end_pos":[479,13]},{"full_name":"and_self_iff","def_path":".lake/packages/lean4/src/lean/Init/PropLemmas.lean","def_pos":[35,8],"def_end_pos":[35,20]},{"full_name":"eq_self_iff_true","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[1380,8],"def_end_pos":[1380,24]}]}]}
{"url":"Mathlib/MeasureTheory/Constructions/Cylinders.lean","commit":"","full_name":"MeasureTheory.union_cylinder","start":[197,0],"end":[203,64],"file_path":"Mathlib/MeasureTheory/Constructions/Cylinders.lean","tactics":[{"state_before":"ι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\n⊢ cylinder s₁ S₁ ∪ cylinder s₂ S₂ = cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∪ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)","state_after":"case h\nι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\n⊢ f ∈ cylinder s₁ S₁ ∪ cylinder s₂ S₂ ↔\n    f ∈ cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∪ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)","tactic":"ext1 f","premises":[]},{"state_before":"case h\nι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\n⊢ f ∈ cylinder s₁ S₁ ∪ cylinder s₂ S₂ ↔\n    f ∈ cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∪ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)","state_after":"case h\nι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\n⊢ (fun i => f ↑i) ∈ S₁ ∨ (fun i => f ↑i) ∈ S₂ ↔\n    (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∨ (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂","tactic":"simp only [mem_union, mem_cylinder, mem_setOf_eq]","premises":[{"full_name":"MeasureTheory.mem_cylinder","def_path":"Mathlib/MeasureTheory/Constructions/Cylinders.lean","def_pos":[156,8],"def_end_pos":[156,20]},{"full_name":"Set.mem_setOf_eq","def_path":"Mathlib/Data/Set/Defs.lean","def_pos":[78,28],"def_end_pos":[78,40]},{"full_name":"Set.mem_union","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[609,8],"def_end_pos":[609,17]}]},{"state_before":"case h\nι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\n⊢ (fun i => f ↑i) ∈ S₁ ∨ (fun i => f ↑i) ∈ S₂ ↔\n    (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∨ (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂","state_after":"no goals","tactic":"rfl","premises":[{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]}]}]}
{"url":"Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean","commit":"","full_name":"MeasureTheory.Measure.rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular","start":[204,0],"end":[218,100],"file_path":"Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν","tactic":"let t := hνν'.nullSet","premises":[{"full_name":"MeasureTheory.Measure.MutuallySingular.nullSet","def_path":"Mathlib/MeasureTheory/Measure/MutuallySingular.lean","def_pos":[52,4],"def_end_pos":[52,11]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν","tactic":"have ht : MeasurableSet t := hνν'.measurableSet_nullSet","premises":[{"full_name":"MeasurableSet","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[61,4],"def_end_pos":[61,17]},{"full_name":"MeasureTheory.Measure.MutuallySingular.measurableSet_nullSet","def_path":"Mathlib/MeasureTheory/Measure/MutuallySingular.lean","def_pos":[54,6],"def_end_pos":[54,27]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ ∀ᵐ (x : α) ∂ν.restrict tᶜ, μ.rnDeriv (ν + ν') x = μ.rnDeriv ν x","tactic":"refine ae_of_ae_restrict_of_ae_restrict_compl t (by simp [t]) ?_","premises":[{"full_name":"MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[571,8],"def_end_pos":[571,46]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ ∀ᵐ (x : α) ∂ν.restrict tᶜ, μ.rnDeriv (ν + ν') x = μ.rnDeriv ν x","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae (ν.restrict tᶜ)] μ.rnDeriv ν","tactic":"change μ.rnDeriv (ν + ν') =ᵐ[ν.restrict tᶜ] μ.rnDeriv ν","premises":[{"full_name":"Filter.EventuallyEq","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[1260,4],"def_end_pos":[1260,16]},{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.Measure.rnDeriv","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[75,30],"def_end_pos":[75,37]},{"full_name":"MeasureTheory.ae","def_path":"Mathlib/MeasureTheory/OuterMeasure/AE.lean","def_pos":[43,4],"def_end_pos":[43,6]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae (ν.restrict tᶜ)] μ.rnDeriv ν","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)","tactic":"rw [← withDensity_eq_iff_of_sigmaFinite (μ := ν.restrict tᶜ)\n    (Measure.measurable_rnDeriv _ _).aemeasurable (Measure.measurable_rnDeriv _ _).aemeasurable]","premises":[{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"Measurable.aemeasurable","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean","def_pos":[380,8],"def_end_pos":[380,31]},{"full_name":"MeasureTheory.Measure.measurable_rnDeriv","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[95,8],"def_end_pos":[95,26]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.withDensity_eq_iff_of_sigmaFinite","def_path":"Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean","def_pos":[680,8],"def_end_pos":[680,41]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)","state_after":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\nthis : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))\n⊢ (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)","tactic":"have : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))\n      = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) := by simp [t]","premises":[{"full_name":"HasCompl.compl","def_path":"Mathlib/Order/Notation.lean","def_pos":[34,2],"def_end_pos":[34,7]},{"full_name":"MeasureTheory.Measure.restrict","def_path":"Mathlib/MeasureTheory/Measure/Restrict.lean","def_pos":[43,18],"def_end_pos":[43,26]},{"full_name":"MeasureTheory.Measure.rnDeriv","def_path":"Mathlib/MeasureTheory/Decomposition/Lebesgue.lean","def_pos":[75,30],"def_end_pos":[75,37]},{"full_name":"MeasureTheory.Measure.withDensity","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[33,4],"def_end_pos":[33,23]}]},{"state_before":"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\nthis : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))\n⊢ (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)","state_after":"no goals","tactic":"rw [this, ← restrict_withDensity ht.compl, ← restrict_withDensity ht.compl,\n      Measure.withDensity_rnDeriv_eq _ _ (hμν.add_right ν'), Measure.withDensity_rnDeriv_eq _ _ hμν]","premises":[{"full_name":"MeasurableSet.compl","def_path":"Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","def_pos":[84,18],"def_end_pos":[84,37]},{"full_name":"MeasureTheory.Measure.AbsolutelyContinuous.add_right","def_path":"Mathlib/MeasureTheory/Measure/MeasureSpace.lean","def_pos":[1515,6],"def_end_pos":[1515,15]},{"full_name":"MeasureTheory.Measure.withDensity_rnDeriv_eq","def_path":"Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean","def_pos":[54,8],"def_end_pos":[54,30]},{"full_name":"MeasureTheory.restrict_withDensity","def_path":"Mathlib/MeasureTheory/Measure/WithDensity.lean","def_pos":[192,8],"def_end_pos":[192,28]}]}]}
{"url":"Mathlib/Data/Multiset/Powerset.lean","commit":"","full_name":"Multiset.mem_powersetCard","start":[223,0],"end":[225,61],"file_path":"Mathlib/Data/Multiset/Powerset.lean","tactics":[{"state_before":"α : Type u_1\nn : ℕ\ns t : Multiset α\nl : List α\n⊢ s ∈ powersetCard n ⟦l⟧ ↔ s ≤ ⟦l⟧ ∧ card s = n","state_after":"no goals","tactic":"simp [powersetCard_coe']","premises":[{"full_name":"Multiset.powersetCard_coe'","def_path":"Mathlib/Data/Multiset/Powerset.lean","def_pos":[200,8],"def_end_pos":[200,25]}]}]}
{"url":"Mathlib/Algebra/Group/Submonoid/Pointwise.lean","commit":"","full_name":"AddSubmonoid.nsmul_vadd_mem_closure_vadd","start":[85,0],"end":[95,23],"file_path":"Mathlib/Algebra/Group/Submonoid/Pointwise.lean","tactics":[{"state_before":"α : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid A\ns✝ t u : Set M\nN : Type u_6\ninst✝² : CommMonoid N\ninst✝¹ : MulAction M N\ninst✝ : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x ∈ closure s\n⊢ ∃ n, r ^ n • x ∈ closure (r • s)","state_after":"case refine_1\nα : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid A\ns✝ t u : Set M\nN : Type u_6\ninst✝² : CommMonoid N\ninst✝¹ : MulAction M N\ninst✝ : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x ∈ closure s\n⊢ ∀ x ∈ s, (fun x => ∃ n, r ^ n • x ∈ closure (r • s)) x\n\ncase refine_2\nα : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid A\ns✝ t u : Set M\nN : Type u_6\ninst✝² : CommMonoid N\ninst✝¹ : MulAction M N\ninst✝ : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x ∈ closure s\n⊢ (fun x => ∃ n, r ^ n • x ∈ closure (r • s)) 1\n\ncase refine_3\nα : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid A\ns✝ t u : Set M\nN : Type u_6\ninst✝² : CommMonoid N\ninst✝¹ : MulAction M N\ninst✝ : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x ∈ closure s\n⊢ ∀ (x y : N),\n    (fun x => ∃ n, r ^ n • x ∈ closure (r • s)) x →\n      (fun x => ∃ n, r ^ n • x ∈ closure (r • s)) y → (fun x => ∃ n, r ^ n • x ∈ closure (r • s)) (x * y)","tactic":"refine @closure_induction N _ s (fun x : N => ∃ n : ℕ, r ^ n • x ∈ closure (r • s)) _ hx ?_ ?_ ?_","premises":[{"full_name":"Exists","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[230,10],"def_end_pos":[230,16]},{"full_name":"Membership.mem","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1518,2],"def_end_pos":[1518,5]},{"full_name":"Nat","def_path":".lake/packages/lean4/src/lean/Init/Prelude.lean","def_pos":[1073,10],"def_end_pos":[1073,13]},{"full_name":"Submonoid.closure","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[329,4],"def_end_pos":[329,11]},{"full_name":"Submonoid.closure_induction","def_path":"Mathlib/Algebra/Group/Submonoid/Basic.lean","def_pos":[375,8],"def_end_pos":[375,25]}]}]}
{"url":"Mathlib/Topology/Basic.lean","commit":"","full_name":"not_mem_closure_iff_nhdsWithin_eq_bot","start":[1094,0],"end":[1095,50],"file_path":"Mathlib/Topology/Basic.lean","tactics":[{"state_before":"X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\n⊢ x ∉ closure s ↔ 𝓝[s] x = ⊥","state_after":"no goals","tactic":"rw [mem_closure_iff_nhdsWithin_neBot, not_neBot]","premises":[{"full_name":"Filter.not_neBot","def_path":"Mathlib/Order/Filter/Basic.lean","def_pos":[467,16],"def_end_pos":[467,25]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"mem_closure_iff_nhdsWithin_neBot","def_path":"Mathlib/Topology/Basic.lean","def_pos":[1082,8],"def_end_pos":[1082,40]}]}]}
{"url":"Mathlib/Data/Set/Card.lean","commit":"","full_name":"Set.encard_eq_zero","start":[89,0],"end":[92,32],"file_path":"Mathlib/Data/Set/Card.lean","tactics":[{"state_before":"α : Type u_1\nβ : Type u_2\ns t : Set α\n⊢ s.encard = 0 ↔ s = ∅","state_after":"no goals","tactic":"rw [encard, ← PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply,\n    PartENat.withTopEquiv_symm_zero, PartENat.card_eq_zero_iff_empty, isEmpty_subtype,\n    eq_empty_iff_forall_not_mem]","premises":[{"full_name":"Equiv.injective","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[175,18],"def_end_pos":[175,27]},{"full_name":"Equiv.symm","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[146,14],"def_end_pos":[146,18]},{"full_name":"Equiv.symm_apply_apply","def_path":"Mathlib/Logic/Equiv/Defs.lean","def_pos":[241,16],"def_end_pos":[241,32]},{"full_name":"Function.Injective.eq_iff","def_path":"Mathlib/Logic/Function/Basic.lean","def_pos":[69,8],"def_end_pos":[69,24]},{"full_name":"Iff.rfl","def_path":".lake/packages/lean4/src/lean/Init/Core.lean","def_pos":[796,18],"def_end_pos":[796,25]},{"full_name":"PartENat.card_eq_zero_iff_empty","def_path":"Mathlib/SetTheory/Cardinal/Finite.lean","def_pos":[295,8],"def_end_pos":[295,30]},{"full_name":"PartENat.withTopEquiv","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[663,18],"def_end_pos":[663,30]},{"full_name":"PartENat.withTopEquiv_symm_zero","def_path":"Mathlib/Data/Nat/PartENat.lean","def_pos":[697,8],"def_end_pos":[697,30]},{"full_name":"Set.encard","def_path":"Mathlib/Data/Set/Card.lean","def_pos":[62,18],"def_end_pos":[62,24]},{"full_name":"Set.eq_empty_iff_forall_not_mem","def_path":"Mathlib/Data/Set/Basic.lean","def_pos":[463,8],"def_end_pos":[463,35]},{"full_name":"isEmpty_subtype","def_path":"Mathlib/Logic/IsEmpty.lean","def_pos":[163,8],"def_end_pos":[163,23]}]}]}
{"url":"Mathlib/Algebra/Order/Ring/Rat.lean","commit":"","full_name":"Rat.mkRat_nonneg","start":[44,0],"end":[45,53],"file_path":"Mathlib/Algebra/Order/Ring/Rat.lean","tactics":[{"state_before":"a✝ b✝ c p q : ℚ\na : ℤ\nha : 0 ≤ a\nb : ℕ\n⊢ 0 ≤ mkRat a b","state_after":"no goals","tactic":"simpa using divInt_nonneg ha (Int.natCast_nonneg _)","premises":[{"full_name":"Int.natCast_nonneg","def_path":"Mathlib/Data/Int/Defs.lean","def_pos":[119,6],"def_end_pos":[119,20]},{"full_name":"Rat.divInt_nonneg","def_path":"Mathlib/Algebra/Order/Ring/Rat.lean","def_pos":[38,14],"def_end_pos":[38,27]}]}]}
{"url":"Mathlib/Logic/Equiv/Set.lean","commit":"","full_name":"Equiv.Set.union_symm_apply_left","start":[218,0],"end":[221,5],"file_path":"Mathlib/Logic/Equiv/Set.lean","tactics":[{"state_before":"α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type ?u.15873\ns t : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nH : s ∩ t ⊆ ∅\na : ↑s\n⊢ ↑a ∈ s ∪ t","state_after":"no goals","tactic":"simp","premises":[]}]}
{"url":"Mathlib/SetTheory/Lists.lean","commit":"","full_name":"Lists.Equiv.symm","start":[267,0],"end":[269,65],"file_path":"Mathlib/SetTheory/Lists.lean","tactics":[{"state_before":"α : Type u_1\nl₁ l₂ : Lists α\nh : l₁ ~ l₂\n⊢ l₂ ~ l₁","state_after":"no goals","tactic":"cases' h with _ _ _ h₁ h₂ <;> [rfl; exact Equiv.antisymm h₂ h₁]","premises":[]}]}